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Inverted state sums & inverted Habiro series

This Mathematica file illustrates the inverted state sum technique developed in arXiv:2106.03942. 
Evaluate the initialization cells before use.

Initialization (q-binomial coefficients and R-matrices)

$Assumptions=q>0&&x>0&&y>0&&z>0&&t>0

QBin[n_,k_,q_]:=Which[k≥0,Product[((1-(q^(n-l+1)))/(1-(q^l))),{l,1,k}],n-k≥0,Product[((1-(q^(n-l+1)))/(1-(q^l))),{l,1,n-k}],True,0]

R[i_,j_,i2_,j2_,q_,x_]:=If[i+j==i2+j2,If[i2≥j,((q^((j+j2+1)/2))/(x^((j+j2+1)/2)))(q^(j j2))QBin[i,j2,q]Product[1-(q^(j+l))(x^(-1)),{l,1,i2-j}],((q^((j+j2+1)/2))/(x^((j+j2+1)/2)))(q^(j j2))((QBin[i,j2,q])/(Product[1-(q^(j-l))(x^(-1)),{l,0,j-i2-1}]))],0]

Rinv[i_,j_,i2_,j2_,q_,x_]:=If[i+j==i2+j2,If[j2≥i,((x^((i+i2+1)/2))/(q^((i+i2+1)/2)))(q^(-i i2))QBin[j,i2,(q^(-1))]Product[1-(q^(-i-l))x,{l,1,j2-i}],((x^((i+i2+1)/2))/(q^((i+i2+1)/2)))(q^(-i i2))((QBin[j,i2,(q^(-1))])/(Product[1-(q^(-i+l))x,{l,0,i-j2-1}]))],0]

For the R-matrix for two differently colored strands, we use the one such that the leftover term is (q^(nm/4)) instead of (q^((nm-1)/4)).

Rxy[i_,j_,i2_,j2_,q_,x_,y_]:=If[i+j==i2+j2,If[i2≥j,((q^((j+j2+(1/2))/2))/((x^((i2+j+1)/4))(y^((3j2-i+1)/4))))(q^(j j2))QBin[i,j2,q]Product[1-(q^(j+l))(y^(-1)),{l,1,i2-j}],((q^((j+j2+(1/2))/2))/((x^((i2+j+1)/4))(y^((3j2-i+1)/4))))(q^(j j2))((QBin[i,j2,q])/(Product[1-(q^(j-l))(y^(-1)),{l,0,j-i2-1}]))],0]

Rxyinv[i_,j_,i2_,j2_,q_,x_,y_]:=If[i+j==i2+j2,If[j2≥i,(((x^((3i2-j+1)/4))(y^((j2+i+1)/4)))/(q^((i+i2+(1/2))/2)))(q^(-i i2))QBin[j,i2,(q^(-1))]Product[1-(q^(-i-l))x,{l,1,j2-i}],(((x^((3i2-j+1)/4))(y^((j2+i+1)/4)))/(q^((i+i2+(1/2))/2)))(q^(-i i2))((QBin[j,i2,(q^(-1))])/(Product[1-(q^(-i+l))x,{l,0,i-j2-1}]))],0]

R-matrices for the multi-variable Alexander polynomial (from J. Murakami’s state model):

RAlex[i_,j_,i2_,j2_,t1_,t2_]:=If[0≤Min[i,j,i2,j2]&&Max[i,j,i2,j2]≤1&&i+j==i2+j2,Which[i==0&&j==0&&i2==0&&j2==0,t1,i==0&&j==1&&i2==0&&j2==1,(((t1^2)-1)/t2),i==0&&j==1&&i2==1&&j2==0,1,i==1&&j==0&&i2==0&&j2==1,(t1/t2),i==1&&j==1&&i2==1&&j2==1,((-1)/t2),True,0],0]

RAlexinv[i_,j_,i2_,j2_,t1_,t2_]:=If[0≤Min[i,j,i2,j2]&&Max[i,j,i2,j2]≤1&&i+j==i2+j2,Which[i==0&&j==0&&i2==0&&j2==0,(1/t2),i==0&&j==1&&i2==1&&j2==0,(t1/t2),i==1&&j==0&&i2==0&&j2==1,1,i==1&&j==0&&i2==1&&j2==0,(t2^(-1))-t2,i==1&&j==1&&i2==1&&j2==1,-t1,True,0],0]

Note, since Murakami’s convention is basically mirrored from ours, R corresponds to RAlexinv and Rinv corresponds to RAlex.

Crucial observation

Negative stabilization makes sense after inversion

Let’s start with the most basic case, when we put weight 0 on the open strand.

Do[Rinv[0,w,0,w,q,x](x^(1/2))(q^(-1/2))(q^(-w))//Simplify//Print,{w,0,3}]

The w-th term is (x/(q^(w+1)))((∏_(k=1))^w)(1-(x/(q^k))), and we can easily extend this to negative w. 
As a result of inversion, we get 1, as a power series in (x^(-1)):

Series[-(x^(1/2))(q^(-1/2))Sum[(q^(-w))Rinv[0,w,0,w,q,x],{w,-1,-11,-1}],{x,∞,10}]//Simplify

Now let’s consider the case when we put weight 1 on the open strand.

Do[Rinv[1,w,1,w,q,x](x^(1/2))(q^(-1/2))(q^(-w))//Simplify//Print,{w,0,4}]

The w-th term is ((1-(q^w))/(1-q)) ((x^2)/(q^(2w+2))) ((∏_(k=2))^w)(1-(x/(q^k))). 
As a result of inversion, we get 1 again as a power series in (x^(-1)) (as expected!)

Series[-(x^(1/2))(q^(-1/2))Sum[(q^(-w))Rinv[1,w,1,w,q,x],{w,-1,-11,-1}],{x,∞,10}]//Simplify

If the weight is a

With[{a=2},Series[-(x^(1/2))(q^(-1/2))Sum[(q^(-w))Rinv[3,w,3,w,q,x],{w,-1,-7,-1}],{x,∞,6}]//Simplify]

With[{a=3},Series[-(x^(1/2))(q^(-1/2))Sum[(q^(-w))Rinv[3,w,3,w,q,x],{w,-1,-7,-1}],{x,∞,6}]//Simplify]

When a<0, the summation is finite, so we don’t even need to invert :)

(x^(1/2))(q^(-1/2))Sum[(q^(-w))Rinv[-1,w,-1,w,q,x],{w,-1,-1,-1}]//Simplify

(x^(1/2))(q^(-1/2))Sum[(q^(-w))Rinv[-2,w,-2,w,q,x],{w,-1,-2,-1}]//Simplify

(x^(1/2))(q^(-1/2))Sum[(q^(-w))Rinv[-3,w,-3,w,q,x],{w,-1,-3,-1}]//Simplify

(x^(1/2))(q^(-1/2))Sum[(q^(-w))Rinv[-4,w,-4,w,q,x],{w,-1,-4,-1}]//Simplify

Positive stabilization revisited

The “standard” case (a≥0)

With[{a=0},(x^(1/2)) (q^((-1)/2))Sum[(q^(-w))R[a,w,a,w,q,x],{w,0,a}]//Simplify]

With[{a=1},(x^(1/2)) (q^((-1)/2))Sum[(q^(-w))R[a,w,a,w,q,x],{w,0,a}]//Simplify]

With[{a=2},(x^(1/2)) (q^((-1)/2))Sum[(q^(-w))R[a,w,a,w,q,x],{w,0,a}]//Simplify]

What if a<0? -- Sum over non-negative w works just fine. (It makes sense as a power series in (x^(-1)).)

With[{a=-1},Series[(x^(1/2)) (q^((-1)/2))Sum[(q^(-w))R[a,w,a,w,q,x],{w,0,10}],{x,∞,5}]//Simplify]

With[{a=-2},Series[(x^(1/2)) (q^((-1)/2))Sum[(q^(-w))R[a,w,a,w,q,x],{w,0,10}],{x,∞,5}]//Simplify]

With[{a=-3},Series[(x^(1/2)) (q^((-1)/2))Sum[(q^(-w))R[a,w,a,w,q,x],{w,0,10}],{x,∞,5}]//Simplify]

Revisiting old examples

(Subscript[4, 1]): Figure-eight knot

Take the braid (Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 2])^(-1)). 
Typically, the state sum formula would be R[0,m,m,0,q,x]R[m,0,0,m,q,x]Rinv[0,k,0,k,q,x]Rinv[m,k,m,k,q,x]x (q^(-1-m-k)), but we’re going to invert k.

Series[((x^(1/2))-(x^(-1/2)))Sum[-R[0,m,m,0,q,x]R[m,0,0,m,q,x]Rinv[0,k,0,k,q,x]Rinv[m,k,m,k,q,x]x (q^(-1-m-k)),{m,0,10},{k,-10,-1}],{x,∞,10}]//Simplify
Do[-Coefficient[%,x,((-1)/2)-j]//Print,{j,0,6}]

It works! This is the first state-sum computation of (Subscript[F, (Subscript[4, 1])])(x,q).

Below, we double check by computing the inverted Habiro coefficients.

Series[((x^(1/2))-(x^(-1/2)))(Sum[-R[0,m,m,0,q,x]R[m,0,0,m,q,x]Rinv[0,k,0,k,q,x]Rinv[m,k,m,k,q,x]x (q^(-1-m-k)),{m,0,10},{k,-10,-1}]/.{x->(x^(-1))}),{x,0,10}]//Simplify
f41=Table[Coefficient[%,x,(1/2)+j],{j,0,9}];
a41=Table[Sum[(((-1))^(i+j))Product[(((q^((2i-l+1)/2))-(q^(-((2i-l+1)/2))))/((q^(l/2))-(q^(-(l/2))))),{l,1,i-j}](((q^((2j+1)/2))-(q^(-((2j+1)/2))))/((q^((i+j+1)/2))-(q^(-((i+j+1)/2)))))%[[j+1]],{j,0,i}]//Simplify,{i,0,9}]

((Subscript[3, 1])^l): Left-handed trefoil

Take the braid ((Subscript[σ, 1])^(-3))

What if we take all the weights to be negative? That is, we are not inverting anything, but we just change the range of summation to negative numbers.

With[{w=-1,jmax=10},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[w,j,w+j-j1,j1,q,x]Rinv[w+j-j1,j1,w+j-j2,j2,q,x]Rinv[w+j-j2,j2,w,j,q,x](x^(1/2))(q^(((-1)/2)-j)),{j,-jmax,-1},{j1,w,-1},{j2,w,-1}],{x,∞,10}]]//Simplify

It works!!!

With[{w=-2,jmax=10},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[w,j,w+j-j1,j1,q,x]Rinv[w+j-j1,j1,w+j-j2,j2,q,x]Rinv[w+j-j2,j2,w,j,q,x](x^(1/2))(q^(((-1)/2)-j)),{j,-jmax,-1},{j1,w,-1},{j2,w,-1}],{x,∞,10}]]//Simplify

With[{w=-3,jmax=10},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[w,j,w+j-j1,j1,q,x]Rinv[w+j-j1,j1,w+j-j2,j2,q,x]Rinv[w+j-j2,j2,w,j,q,x](x^(1/2))(q^(((-1)/2)-j)),{j,-jmax,-1},{j1,w,-1},{j2,w,-1}],{x,∞,10}]]//Simplify

It works for w≥0 as well!

With[{w=0,jmax=7},Series[-((x^(1/2))-(x^(-1/2)))Sum[Rinv[w,j,w+j-j1,j1,q,x]Rinv[w+j-j1,j1,w+j-j2,j2,q,x]Rinv[w+j-j2,j2,w,j,q,x](x^(1/2))(q^(((-1)/2)-j)),{j,-jmax,-1},{j1,-jmax,-1},{j2,-jmax,-1}],{x,∞,7}]]//Simplify

The inverted Habiro coefficients:

With[{w=0,jmax=8},Series[-((x^(1/2))-(x^(-1/2)))(Sum[Rinv[w,j,w+j-j1,j1,q,x]Rinv[w+j-j1,j1,w+j-j2,j2,q,x]Rinv[w+j-j2,j2,w,j,q,x](x^(1/2))(q^(((-1)/2)-j)),{j,-jmax,-1},{j1,-jmax,-1},{j2,-jmax,-1}]/.{x->(x^(-1))}),{x,0,8}]]//Simplify
f31l=Table[Coefficient[%,x,(1/2)+j],{j,0,7}];
a31l=Table[Sum[(((-1))^(i+j))Product[(((q^((2i-l+1)/2))-(q^(-((2i-l+1)/2))))/((q^(l/2))-(q^(-(l/2))))),{l,1,i-j}](((q^((2j+1)/2))-(q^(-((2j+1)/2))))/((q^((i+j+1)/2))-(q^(-((i+j+1)/2)))))%[[j+1]],{j,0,i}]//Simplify,{i,0,7}]

m((Subscript[5, 2]))

Take the braid ((Subscript[σ, 1])^(-1))(Subscript[σ, 2])((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^(-3))

The homogeneity is mildly broken by (Subscript[σ, 2]), but in this case doesn’t affect the final answer.

With[{cutoff=6},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,m1,m1,-1,q,x]Rinv[m1,-1,-1,m1,q,x]R[-1,k,-1,k,q,x]Rinv[m1,k,m1+k-m2,m2,q,x]Rinv[m1+k-m2,m2,m1+k-m3,m3,q,x]Rinv[m1+k-m3,m3,m1,k,q,x]x (q^(-1-m1-k)),{m1,-cutoff,-1},{k,-cutoff,-1},{m2,-cutoff,-1},{m3,-cutoff,-1}],{x,∞,4}]//Simplify]

Inverting k locally doesn’t affect the result:

With[{cutoff=6},Series[-((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,m1,m1,-1,q,x]Rinv[m1,-1,-1,m1,q,x]R[-1,k,-1,k,q,x]Rinv[m1,k,m1+k-m2,m2,q,x]Rinv[m1+k-m2,m2,m1+k-m3,m3,q,x]Rinv[m1+k-m3,m3,m1,k,q,x]x (q^(-1-m1-k)),{m1,-cutoff,-1},{k,0,cutoff},{m2,-cutoff,-1},{m3,-cutoff,-1}],{x,∞,4}]//Simplify]

New examples (homogeneous braids)

(Subscript[6, 2]) knot

Take the braid ((Subscript[σ, 1])^3)((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 2])^(-1)). 
Typically, the state sum would be R[0,m1,m1,0,q,x]R[m1,0,m2,m1-m2,q,x]R[m2,m1-m2,m3,m1-m3,q,x]R[m3,0,0,m3,q,x]Rinv[m1-m3,k,0,m1-m3+k,q,x]Rinv[m3,m1-m3+k,m1,k,q,x]x (q^(-1-m1-k)), but we’re going to invert k.

Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,m1,m1,0,q,x]R[m1,0,m2,m1-m2,q,x]R[m2,m1-m2,m3,m1-m3,q,x]R[m3,0,0,m3,q,x]Rinv[m1-m3,k,0,m1-m3+k,q,x]Rinv[m3,m1-m3+k,m1,k,q,x]x (q^(-1-m1-k)),{m1,0,6},{m2,0,m1},{m3,m1-m2,m1},{k,-6,m3-m1-1}],{x,∞,8}]//Simplify
Do[-Coefficient[%,x,((-1)/2)-j]//Expand//Print,{j,0,7}]
Table[-Coefficient[%%,x,((-1)/2)-j]/.{q->1},{j,0,7}]

It works!
Each term seems to be a Laurent polynomial in q, just like (Subscript[4, 1])!
It’s classical limit agrees with the inverse Alexander:

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^2)+3x-3+3(x^(-1))-(x^(-2)))),{x,∞,8}]

Below, we compute the inverted Habiro coefficients.

Series[-((x^(1/2))-(x^(-1/2)))(Sum[R[0,m1,m1,0,q,x]R[m1,0,m2,m1-m2,q,x]R[m2,m1-m2,m3,m1-m3,q,x]R[m3,0,0,m3,q,x]Rinv[m1-m3,k,0,m1-m3+k,q,x]Rinv[m3,m1-m3+k,m1,k,q,x]x (q^(-1-m1-k)),{m1,0,6},{m2,0,m1},{m3,m1-m2,m1},{k,-6,m3-m1-1}]/.{x->(x^(-1))}),{x,0,8}]//Simplify
f62=Table[Coefficient[%,x,(1/2)+j],{j,0,7}];
a62=Table[Sum[(((-1))^(i+j))Product[(((q^((2i-l+1)/2))-(q^(-((2i-l+1)/2))))/((q^(l/2))-(q^(-(l/2))))),{l,1,i-j}](((q^((2j+1)/2))-(q^(-((2j+1)/2))))/((q^((i+j+1)/2))-(q^(-((i+j+1)/2)))))%[[j+1]],{j,0,i}]//Simplify//Expand,{i,0,7}]

Let’s just compute a few more terms; They can be useful in determining the quantum C-polynomial.

Series[-((x^(1/2))-(x^(-1/2)))(Sum[R[0,m1,m1,0,q,x]R[m1,0,m2,m1-m2,q,x]R[m2,m1-m2,m3,m1-m3,q,x]R[m3,0,0,m3,q,x]Rinv[m1-m3,k,0,m1-m3+k,q,x]Rinv[m3,m1-m3+k,m1,k,q,x]x (q^(-1-m1-k)),{m1,0,8},{m2,0,m1},{m3,m1-m2,m1},{k,-8,m3-m1-1}]/.{x->(x^(-1))}),{x,0,10}]//Simplify
f62=Table[Coefficient[%,x,(1/2)+j],{j,0,9}];
a62=Table[Sum[(((-1))^(i+j))Product[(((q^((2i-l+1)/2))-(q^(-((2i-l+1)/2))))/((q^(l/2))-(q^(-(l/2))))),{l,1,i-j}](((q^((2j+1)/2))-(q^(-((2j+1)/2))))/((q^((i+j+1)/2))-(q^(-((i+j+1)/2)))))%[[j+1]],{j,0,i}]//Simplify//Expand,{i,0,9}]

(Subscript[6, 3]) knot

Take the braid ((Subscript[σ, 1])^2)((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 2])^(-2)).

With[{cutoff=8},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,m1,m1,0,q,x]R[m1,0,m1-m2,m2,q,x]R[m1-m2,0,0,m1-m2,q,x]Rinv[m2,k1,0,m2+k1,q,x]Rinv[m1-m2,m2+k1,m1+k1-k2,k2,q,x]Rinv[m1+k1-k2,k2,m1,k1,q,x]x (q^(-1-m1-k1)),{m1,0,cutoff},{m2,0,m1},{k1,-cutoff,-1},{k2,-cutoff,-1}],{x,∞,7}]//Simplify]
Table[Coefficient[%,x,((-1)/2)-j]//Expand,{j,0,6}]
%/.{q->1}

This matches perfectly with the expansion of (1/((Subscript[Δ, K])(x))) :)

Series[(((x^(1/2))-(x^(-1/2)))/((x^2)-3x+5-3(x^(-1))+(x^(-2)))),{x,∞,7}]

Inverted Habiro coefficients:

With[{cutoff=8},Series[-((x^(1/2))-(x^(-1/2)))(Sum[R[0,m1,m1,0,q,x]R[m1,0,m1-m2,m2,q,x]R[m1-m2,0,0,m1-m2,q,x]Rinv[m2,k1,0,m2+k1,q,x]Rinv[m1-m2,m2+k1,m1+k1-k2,k2,q,x]Rinv[m1+k1-k2,k2,m1,k1,q,x]x (q^(-1-m1-k1)),{m1,0,cutoff},{m2,0,m1},{k1,-cutoff,-1},{k2,-cutoff,-1}])/.{x->(x^(-1))},{x,0,7}]//Simplify]
f63=Table[Coefficient[%,x,(1/2)+j],{j,0,6}];
a63=Table[Sum[(((-1))^(i+j))Product[(((q^((2i-l+1)/2))-(q^(-((2i-l+1)/2))))/((q^(l/2))-(q^(-(l/2))))),{l,1,i-j}](((q^((2j+1)/2))-(q^(-((2j+1)/2))))/((q^((i+j+1)/2))-(q^(-((i+j+1)/2)))))%[[j+1]],{j,0,i}]//Simplify,{i,0,6}]

(Subscript[7, 6]) knot

Take the braid ((Subscript[σ, 1])^2)((Subscript[σ, 2])^(-1))(Subscript[σ, 1])(Subscript[σ, 3])((Subscript[σ, 2])^(-1))(Subscript[σ, 3]).

With[{cutoff=7},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,m1,m1,0,q,x]R[m1,0,m1-m2,m2,q,x]R[m1-m2,0,0,m1-m2,q,x]Rinv[m2,k1,0,m2+k1,q,x]Rinv[m1-m2,k2,m1,k2-m2,q,x]R[m2+k1,n1,k2,m2+k1+n1-k2,q,x]R[k2-m2,m2+k1+n1-k2,k1,n1,q,x](x^(3/2)) (q^(((-3)/2)-m1-k1-n1)),{m1,0,cutoff},{m2,0,m1},{k1,-cutoff,-m2-1},{k2,-cutoff,-1},{n1,0,cutoff}],{x,∞,6}]//Simplify]
Table[Coefficient[%,x,((-1)/2)-j]//Expand,{j,0,5}]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^2)+5x-7+5(x^(-1))-(x^(-2)))),{x,∞,6}]

(Subscript[7, 7]) knot

Take the braid (Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 3])((Subscript[σ, 2])^(-1))(Subscript[σ, 3]).

With[{cutoff=7},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,m1,m1,0,q,x]R[m1,0,0,m1,q,x]Rinv[0,k1,0,k1,q,x]Rinv[m1,k1,m1+k1-k2,k2,q,x]Rinv[m1+k1-k2,k3,m1,k1-k2+k3,q,x]R[k2,n1,k3,k2+n1-k3,q,x]R[k1-k2+k3,k2+n1-k3,k1,n1,q,x](x^(3/2)) (q^(((-3)/2)-m1-k1-n1)),{m1,0,cutoff},{k1,-cutoff,-1},{k2,-cutoff,-1},{k3,-cutoff,Min[-1,-1+k2-k1]},{n1,Max[0,k3-k2],cutoff}],{x,∞,6}]//Simplify]
Table[Coefficient[%,x,((-1)/2)-j]//Expand,{j,0,5}]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^2)-5x+9-5(x^(-1))+(x^(-2)))),{x,∞,6}]

Fibered but possibly non-homogeneous braids

(Subscript[8, 20]) knot

From the braid ((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^(-3))((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^3),

With[{cutoff=6},Series[-((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[-1,b,b,-1,q,x]Rinv[b,-1,b,-1,q,x]Rinv[b,-1,-1,b,q,x]Rinv[a,-1,-1,a,q,x]R[a,b,a+b-p,p,q,x]R[a+b-p,p,a+b-r,r,q,x]R[a+b-r,r,a,b,q,x](x^(2/2)) (q^(((-2)/2)-a-b)),{a,-cutoff,-1},{b,0,cutoff},{p,Max[0,a+b],cutoff},{r,Max[0,a+b],cutoff}],{x,∞,8}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^2)-2x+3-2(x^(-1))+(x^(-2)))),{x,∞,8}]

(Subscript[8, 21]) knot

From the braid ((Subscript[σ, 1])^(-2))((Subscript[σ, 2])^2)((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^(-3)),

With[{cutoff=5},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[a,-1,a-1-m,m,q,x]R[m,b,b+m-n,n,q,x]R[b+m-n,n,-1,b+m+1,q,x]Rinv[a-1-m,-1,-1,a-1-m,q,x]Rinv[a-1-m,b+m+1,a+b-p,p,q,x]Rinv[a+b-p,p,a+b-r,r,q,x]Rinv[a+b-r,r,a,b,q,x](x^(2/2)) (q^(((-2)/2)-a-b)),{a,-cutoff,-1},{b,0,cutoff},{m,-cutoff,-1},{n,0,cutoff},{p,-cutoff,-1},{r,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^2)+4x-5+4(x^(-1))-(x^(-2)))),{x,∞,6}]

(Subscript[9, 42]) knot

From the braid (Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 3])^(-2))((Subscript[σ, 2])^(-1))((Subscript[σ, 3])^3),

With[{cutoff=5},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,a,a,0,q,x]Rinv[0,b,0,b,q,x]R[a,0,0,a,q,x]Rinv[b,c,b+c-m,m,q,x]Rinv[b+c-m,m,b+c-n,n,q,x]Rinv[a,b+c-n,a,b+c-n,q,x]R[b+c-n,n,b+c-p,p,q,x]R[b+c-p,p,b+c-r,r,q,x]R[b+c-r,r,b,c,q,x](x^(3/2)) (q^(((-3)/2)-a-b-c)),{a,0,cutoff},{b,-cutoff,-1},{c,0,cutoff},{m,-cutoff,-1},{n,0,cutoff},{p,0,cutoff},{r,0,cutoff}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^2)+2x-1+2(x^(-1))-(x^(-2)))),{x,∞,6}]

(Subscript[9, 44]) knot

From the braid (Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 3])^2)((Subscript[σ, 2])^(-1))((Subscript[σ, 3])^(-3)),

With[{cutoff=5},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,a,a,0,q,x]Rinv[0,b,0,b,q,x]R[a,0,0,a,q,x]R[b,c,b+c-m,m,q,x]R[b+c-m,m,b+c-n,n,q,x]Rinv[a,b+c-n,a,b+c-n,q,x]Rinv[b+c-n,n,b+c-p,p,q,x]Rinv[b+c-p,p,b+c-r,r,q,x]Rinv[b+c-r,r,b,c,q,x](x^(3/2)) (q^(((-3)/2)-a-b-c)),{a,0,cutoff},{b,-cutoff,-1},{c,0,cutoff},{m,0,cutoff},{n,0,cutoff},{p,-cutoff,-1},{r,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^2)-4x+7-4(x^(-1))+(x^(-2)))),{x,∞,6}]

(Subscript[9, 45]) knot

From the braid ((Subscript[σ, 1])^(-1))(Subscript[σ, 2])((Subscript[σ, 1])^(-1))((Subscript[σ, 3])^(-1))((Subscript[σ, 2])^(-1))(Subscript[σ, 3])((Subscript[σ, 2])^(-1))((Subscript[σ, 3])^(-2)),

With[{cutoff=5},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]R[-1,b,-1,b,q,x]Rinv[a,-1,-1,a,q,x]Rinv[b,c,b+c-m,m,q,x]Rinv[a,b+c-m,a+b+c-m-n,n,q,x]R[n,m,n+m-p,p,q,x]Rinv[a+b+c-m-n,n+m-p,a,b+c-p,q,x]Rinv[b+c-p,p,b+c-r,r,q,x]Rinv[b+c-r,r,b,c,q,x](x^(3/2)) (q^(((-3)/2)-a-b-c)),{a,-cutoff,-1},{b,0,cutoff},{c,-cutoff,-1},{m,0,cutoff},{n,-cutoff,-1},{p,0,cutoff},{r,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^2)+6x-9+6(x^(-1))-(x^(-2)))),{x,∞,6}]

(Subscript[9, 48]) knot

From the braid ((Subscript[σ, 1])^(-1))(Subscript[σ, 2])((Subscript[σ, 3])^(-1))(Subscript[σ, 2])((Subscript[σ, 1])^(-1))(Subscript[σ, 3])(Subscript[σ, 2])((Subscript[σ, 3])^(-1))(Subscript[σ, 2])((Subscript[σ, 3])^2),

With[{cutoff=5},Series[-((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]R[-1,b,b-1-m,m,q,x]Rinv[m,c,c+m-n,n,q,x]R[b-1-m,c+m-n,-1,b+c-n,q,x]Rinv[a,-1,-1,a,q,x]R[b+c-n,n,b+c-p,p,q,x]R[a,b+c-p,a+b+c-p-r,r,q,x]Rinv[r,p,p+r-s,s,q,x]R[a+b+c-p-r,p+r-s,a,b+c-s,q,x]R[b+c-s,s,b+c-t,t,q,x]R[b+c-t,t,b,c,q,x](x^(3/2)) (q^(((-3)/2)-a-b-c)),{a,-cutoff,-1},{b,0,cutoff},{c,0,cutoff},{m,0,cutoff},{n,0,cutoff},{p,0,cutoff},{r,0,cutoff},{s,0,cutoff},{t,0,cutoff}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^2)+7x-11+7(x^(-1))-(x^(-2)))),{x,∞,6}]

Now we have computed (Subscript[F, K]) for all fibered knots up to 9 crossings.

(Subscript[10, 60]) knot

From the braid ((Subscript[σ, 1])^(-1))(Subscript[σ, 2])((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^2)((Subscript[σ, 3])^(-1))(Subscript[σ, 2])((Subscript[σ, 3])^(-1))((Subscript[σ, 2])^(-1))((Subscript[σ, 4])^(-1))(Subscript[σ, 3])((Subscript[σ, 4])^(-1)),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]R[-1,b,-1,b,q,x]Rinv[a,-1,-1,a,q,x]R[a,b,a+b-m,m,q,x]R[a+b-m,m,a+b-n,n,q,x]Rinv[n,c,n+c-p,p,q,x]R[a+b-n,n+c-p,a+b+c-p-r,r,q,x]Rinv[r,p,p+r-s,s,q,x]Rinv[a+b+c-p-r,p+r-s,a,b+c-s,q,x]Rinv[s,d,s+d-t,t,q,x]R[b+c-s,s+d-t,b,c+d-t,q,x]Rinv[c+d-t,t,c,d,q,x](x^(4/2)) (q^(((-4)/2)-a-b-c-d)),{a,-cutoff,-1},{b,0,cutoff-1},{c,-cutoff,-1},{d,-cutoff,-1},{m,0,cutoff-1},{n,0,cutoff-1},{p,-cutoff,-1},{r,0,cutoff-1},{s,0,cutoff-1},{t,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^3)+7(x^2)-20x+29-20(x^(-1))+7(x^(-2))-(x^(-3)))),{x,∞,6}]

(Subscript[10, 69]) knot

From the braid (Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 3])((Subscript[σ, 2])^(-1))(Subscript[σ, 4])(Subscript[σ, 1])(Subscript[σ, 3])((Subscript[σ, 2])^(-1))((Subscript[σ, 4])^(-1))(Subscript[σ, 3])((Subscript[σ, 4])^2),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,a,a,0,q,x]Rinv[0,b,b-m,m,q,x]R[m,c,m+c-n,n,q,x]Rinv[b-m,m+c-n,0,b+c-n,q,x]R[n,d,n+d-p,p,q,x]R[a,0,0,a,q,x]R[b+c-n,n+d-p,b+c+d-p-r,r,q,x]Rinv[a,b+c+d-p-r,a,b+c+d-p-r,q,x]Rinv[r,p,p+r-s,s,q,x]R[b+c+d-p-r,p+r-s,b,c+d-s,q,x]R[c+d-s,s,c+d-t,t,q,x]R[c+d-t,t,c,d,q,x](x^(4/2)) (q^(((-4)/2)-a-b-c-d)),{a,0,cutoff-1},{b,-cutoff,-1},{c,0,cutoff-1},{d,0,cutoff-1},{m,-cutoff,-1},{n,0,cutoff-1},{p,0,cutoff-1},{r,0,cutoff-1},{s,0,cutoff-1},{t,0,cutoff-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^3)-7(x^2)+21x-29+21(x^(-1))-7(x^(-2))+(x^(-3)))),{x,∞,6}]

(Subscript[10, 73]) knot

From the braid ((Subscript[σ, 1])^(-1))(Subscript[σ, 2])((Subscript[σ, 1])^(-1))(Subscript[σ, 2])((Subscript[σ, 3])^(-1))(Subscript[σ, 2])((Subscript[σ, 4])^(-1))((Subscript[σ, 3])^(-1))(Subscript[σ, 4])((Subscript[σ, 3])^(-1))((Subscript[σ, 4])^(-2)),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]R[-1,b,-1,b,q,x]Rinv[a,-1,-1,a,q,x]R[a,b,a+b-m,m,q,x]Rinv[m,c,m+c-n,n,q,x]R[a+b-m,m+c-n,a,b+c-n,q,x]Rinv[n,d,n+d-p,p,q,x]Rinv[b+c-n,n+d-p,b+c+d-p-r,r,q,x]R[r,p,p+r-s,s,q,x]Rinv[b+c+d-p-r,p+r-s,b,c+d-s,q,x]Rinv[c+d-s,s,c+d-u,u,q,x]Rinv[c+d-u,u,c,d,q,x](x^(4/2)) (q^(((-4)/2)-a-b-c-d)),{a,-cutoff,-1},{b,0,cutoff-1},{c,-cutoff,-1},{d,-cutoff,-1},{m,0,cutoff-1},{n,-cutoff,-1},{p,-cutoff,-1},{r,-cutoff,-1},{s,-cutoff,-1},{u,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^3)-7(x^2)+20x-27+20(x^(-1))-7(x^(-2))+(x^(-3)))),{x,∞,6}]

(Subscript[10, 75]) knot

From the braid (Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 3])((Subscript[σ, 2])^(-2))(Subscript[σ, 4])((Subscript[σ, 3])^(-1))(Subscript[σ, 2])(Subscript[σ, 4])(Subscript[σ, 3]),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,a,a,0,q,x]Rinv[0,b,0,b,q,x]R[a,0,0,a,q,x]Rinv[a,b,a+b-m,m,q,x]R[m,c,m+c-n,n,q,x]Rinv[a+b-m,m+c-n,a+b+c-n-p,p,q,x]R[n,d,n+d-r,r,q,x]Rinv[a+b+c-n-p,p,a+b+c-n-s,s,q,x]Rinv[s,n+d-r,n+d+s-r-u,u,q,x]R[a+b+c-n-s,n+d+s-r-u,a,b+c+d-r-u,q,x]R[u,r,u+r-d,d,q,x]R[b+c+d-r-u,u+r-d,b,c,q,x](x^(4/2)) (q^(((-4)/2)-a-b-c-d)),{a,0,cutoff-1},{b,-cutoff,-1},{c,0,cutoff-1},{d,0,cutoff-1},{m,-cutoff,-1},{n,0,cutoff-1},{p,-cutoff,-1},{r,0,cutoff-1},{s,-cutoff,-1},{u,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^3)+7(x^2)-19x+27-19(x^(-1))+7(x^(-2))-(x^(-3)))),{x,∞,6}]

(Subscript[10, 78]) knot

From the braid ((Subscript[σ, 1])^(-2))(Subscript[σ, 2])((Subscript[σ, 1])^(-1))((Subscript[σ, 3])^(-1))(Subscript[σ, 2])((Subscript[σ, 4])^(-1))((Subscript[σ, 3])^(-1))(Subscript[σ, 4])((Subscript[σ, 3])^(-1))((Subscript[σ, 4])^(-2)),

With[{cutoff=4},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[a,-1,a-1-m,m,q,x]R[m,b,-1,m+b+1,q,x]Rinv[a-1-m,-1,-1,a-1-m,q,x]Rinv[m+b+1,c,m+b+c+1-n,n,q,x]R[a-1-m,m+b+c+1-n,a,b+c-n,q,x]Rinv[n,d,n+d-p,p,q,x]Rinv[b+c-n,n+d-p,b+c+d-p-r,r,q,x]R[r,p,p+r-s,s,q,x]Rinv[b+c+d-p-r,p+r-s,b,c+d-s,q,x]Rinv[c+d-s,s,c+d-u,u,q,x]Rinv[c+d-u,u,c,d,q,x](x^(4/2)) (q^(((-4)/2)-a-b-c-d)),{a,-cutoff,-1},{b,0,cutoff-1},{c,-cutoff,-1},{d,-cutoff,-1},{m,-cutoff,-1},{n,-cutoff,-1},{p,-cutoff,-1},{r,-cutoff,-1},{s,-cutoff,-1},{u,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^3)+7(x^2)-16x+21-16(x^(-1))+7(x^(-2))-(x^(-3)))),{x,∞,6}]

(Subscript[10, 81]) knot

From the braid ((Subscript[σ, 1])^2)((Subscript[σ, 2])^(-1))(Subscript[σ, 1])(Subscript[σ, 3])((Subscript[σ, 2])^2)((Subscript[σ, 4])^(-1))((Subscript[σ, 3])^(-3))((Subscript[σ, 4])^(-1)),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,a,a,0,q,x]R[a,0,a-m,m,q,x]Rinv[m,b,0,m+b,q,x]R[a-m,0,0,a-m,q,x]R[m+b,c,m+b+c-n,n,q,x]R[a-m,m+b+c-n,a+b+c-n-p,p,q,x]Rinv[n,d,n+d-r,r,q,x]R[a+b+c-n-p,p,a,b+c-n,q,x]Rinv[b+c-n,n+d-r,b+c+d-r-s,s,q,x]Rinv[b+c+d-r-s,s,b+c+d-r-u,u,q,x]Rinv[b+c+d-r-u,u,b,c+d-r,q,x]Rinv[c+d-r,r,c,d,q,x](x^(4/2)) (q^(((-4)/2)-a-b-c-d)),{a,0,cutoff-1},{b,-cutoff,-1},{c,0,cutoff-1},{d,-cutoff,-1},{m,0,cutoff-1},{n,-cutoff,-1},{p,0,cutoff-1},{r,-cutoff,-1},{s,-cutoff,-1},{u,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^3)+8(x^2)-20x+27-20(x^(-1))+8(x^(-2))-(x^(-3)))),{x,∞,6}]

(Subscript[10, 89]) knot

From the braid ((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^(-1))(Subscript[σ, 3])((Subscript[σ, 2])^(-1))((Subscript[σ, 4])^(-1))((Subscript[σ, 1])^(-1))((Subscript[σ, 3])^(-1))(Subscript[σ, 2])(Subscript[σ, 3])((Subscript[σ, 4])^(-1))(Subscript[σ, 3])((Subscript[σ, 4])^(-1)),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[-1,b,b-1-m,m,q,x]R[m,c,m+c-n,n,q,x]Rinv[b-1-m,m+c-n,-1,b+c-n,q,x]Rinv[n,d,n+d-p,p,q,x]Rinv[a,-1,-1,a,q,x]Rinv[b+c-n,n+d-p,b+c+d-p-r,r,q,x]R[a,b+c+d-p-r,a,b+c+d-p-r,q,x]R[b+c+d-p-r,r,b+c+d-p-s,s,q,x]Rinv[s,p,p+s-u,u,q,x]R[b+c+d-p-s,p+s-u,b,c+d-u,q,x]Rinv[c+d-u,u,c,d,q,x](x^(4/2)) (q^(((-4)/2)-a-b-c-d)),{a,-cutoff,-1},{b,-cutoff,-1},{c,0,cutoff-1},{d,-cutoff,-1},{m,-cutoff,-1},{n,0,cutoff-1},{p,-cutoff,-1},{r,-cutoff,-1},{s,0,cutoff-1},{u,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^3)-8(x^2)+24x-33+24(x^(-1))-8(x^(-2))+(x^(-3)))),{x,∞,6}]

(Subscript[10, 96]) knot

From the braid (Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 3])((Subscript[σ, 2])^(-1))(Subscript[σ, 4])(Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 3])((Subscript[σ, 2])^(-1))(Subscript[σ, 4])(Subscript[σ, 3])((Subscript[σ, 4])^(-1)),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,a,a,0,q,x]Rinv[0,b,b-m,m,q,x]R[m,c,m+c-n,n,q,x]Rinv[b-m,m+c-n,0,b+c-n,q,x]R[n,d,n+d-p,p,q,x]R[a,0,0,a,q,x]Rinv[a,b+c-n,a+b+c-n-r,r,q,x]R[r,n+d-p,d+n+r-p-s,s,q,x]Rinv[a+b+c-n-r,d+n+r-p-s,a,b+c+d-p-s,q,x]R[s,p,p+s-u,u,q,x]R[b+c+d-p-s,p+s-u,b,c+d-u,q,x]Rinv[c+d-u,u,c,d,q,x](x^(4/2)) (q^(((-4)/2)-a-b-c-d)),{a,0,cutoff-1},{b,-cutoff,-1},{c,0,cutoff-1},{d,0,cutoff-1},{m,-cutoff,-1},{n,0,cutoff-1},{p,0,cutoff-1},{r,-cutoff,-1},{s,0,cutoff-1},{u,0,cutoff-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^3)+7(x^2)-22x+33-22(x^(-1))+7(x^(-2))-(x^(-3)))),{x,∞,6}]

(Subscript[10, 105]) knot

From the braid ((Subscript[σ, 1])^2)((Subscript[σ, 2])^(-1))(Subscript[σ, 1])(Subscript[σ, 3])((Subscript[σ, 2])^2)((Subscript[σ, 4])^(-1 ))((Subscript[σ, 3])^(-1))(Subscript[σ, 2])((Subscript[σ, 3])^(-1))((Subscript[σ, 4])^(-1)),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,a,a,0,q,x]R[a,0,a-m,m,q,x]Rinv[m,b,0,b+m,q,x]R[a-m,0,0,a-m,q,x]R[b+m,c,b+c+m-n,n,q,x]R[a-m,b+c+m-n,a+b+c-n-p,p,q,x]R[a+b+c-n-p,p,a+b+c-n-r,r,q,x]Rinv[n,d,d+n-s,s,q,x]Rinv[r,d+n-s,d+n+r-s-u,u,q,x]R[a+b+c-n-r,d+n+r-s-u,a,b+c+d-s-u,q,x]Rinv[b+c+d-s-u,u,b,c+d-s,q,x]Rinv[c+d-s,s,c,d,q,x](x^(4/2)) (q^(((-4)/2)-a-b-c-d)),{a,0,cutoff-1},{b,-cutoff,-1},{c,0,cutoff-1},{d,-cutoff,-1},{m,0,Min[a,-b-1]},{n,-cutoff,b+m},{p,0,Min[a-m,a+b+c-n]},{r,0,a+b+c-n-p},{s,n,-1},{u,-cutoff,Min[-1,d+n+r-s]}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^3)-8(x^2)+22x-29+22(x^(-1))-8(x^(-2))+(x^(-3)))),{x,∞,6}]

(Subscript[10, 107])  knot

From the braid ((Subscript[σ, 1])^(-2)) (Subscript[σ, 2])((Subscript[σ, 1])^(-1))(Subscript[σ, 3])((Subscript[σ, 2])^2)((Subscript[σ, 4])^(-1))(Subscript[σ, 3])((Subscript[σ, 2])^(-1))(Subscript[σ, 3])((Subscript[σ, 4])^(-1)),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[a,-1,a-1-m,m,q,x]R[m,b,-1,b+m+1,q,x]Rinv[a-1-m,-1,-1,a-1-m,q,x]R[b+m+1,c,b+c+m-n+1,n,q,x]R[a-1-m,b+c+m-n+1,a+b+c-n-p,p,q,x]R[a+b+c-n-p,p,a+b+c-n-r,r,q,x]Rinv[n,d,d+n-s,s,q,x]R[r,d+n-s,d+n+r-s-u,u,q,x]Rinv[a+b+c-n-r,d+n+r-s-u,a,b+c+d-s-u,q,x]R[b+c+d-s-u,u,b,c+d-s,q,x]Rinv[c+d-s,s,c,d,q,x](x^(4/2)) (q^(((-4)/2)-a-b-c-d)),{a,-cutoff,-1},{b,0,cutoff-1},{c,0,cutoff-1},{d,-cutoff,-1},{m,a,-1},{n,0,Min[cutoff-1,b+m+1]},{p,Max[0,a+b+c-n+1],cutoff-1},{r,-cutoff,Min[-1,a+b+c-n-p]},{s,-cutoff,-1},{u,0,cutoff}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^3)+8(x^2)-22x+31-22(x^(-1))+8(x^(-2))-(x^(-3)))),{x,∞,6}]

(Subscript[10, 110]) knot

From the braid ((Subscript[σ, 1])^(-1))(Subscript[σ, 2])((Subscript[σ, 1])^(-1))((Subscript[σ, 3])^(-1))((Subscript[σ, 2])^(-3))(Subscript[σ, 4])(Subscript[σ, 3])((Subscript[σ, 2])^(-1))(Subscript[σ, 3])(Subscript[σ, 4]),

With[{cutoff=4},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]R[-1,b,-1,b,q,x]Rinv[a,-1,-1,a,q,x]Rinv[b,c,b+c-m,m,q,x]Rinv[a,b+c-m,a+b+c-m-n,n,q,x]Rinv[a+b+c-m-n,n,a+b+c-m-p,p,q,x]Rinv[a+b+c-m-p,p,a+b+c-m-r,r,q,x]R[m,d,d+m-s,s,q,x]R[r,d+m-s,d+m+r-s-u,u,q,x]Rinv[a+b+c-m-r,d+m+r-s-u,a,b+c+d-s-u,q,x]R[b+c+d-s-u,u,b,c+d-s,q,x]R[c+d-s,s,c,d,q,x](x^(4/2)) (q^(((-4)/2)-a-b-c-d)),{a,-cutoff,-1},{b,0,cutoff-1},{c,-cutoff,-1},{d,0,cutoff-1},{m,b,cutoff-1},{n,a,-1},{p,Max[-cutoff,a+b+c-m-n],-1},{r,Max[-cutoff,a+b+c-m-p],-1},{s,0,m},{u,0,cutoff}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^3)-8(x^2)+20x-25+20(x^(-1))-8(x^(-2))+(x^(-3)))),{x,∞,6}]

(Subscript[10, 115]) knot

From the braid (Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 1])(Subscript[σ, 3])((Subscript[σ, 2])^2)((Subscript[σ, 4])^(-1))((Subscript[σ, 3])^(-1))(Subscript[σ, 2])((Subscript[σ, 3])^(-2))((Subscript[σ, 4])^(-1)),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,a,a,0,q,x]Rinv[0,b,0,b,q,x]R[a,0,0,a,q,x]R[b,c,b+c-m,m,q,x]R[a,b+c-m,a+b+c-m-n,n,q,x]R[a+b+c-m-n,n,a+b+c-m-p,p,q,x]Rinv[m,d,d+m-r,r,q,x]Rinv[p,d+m-r,d+m+p-r-s,s,q,x]R[a+b+c-m-p,d+m+p-r-s,a,b+c+d-r-s,q,x]Rinv[b+c+d-r-s,s,b+c+d-r-u,u,q,x]Rinv[b+c+d-r-u,u,b,c+d-r,q,x]Rinv[c+d-r,r,c,d,q,x](x^(4/2)) (q^(((-4)/2)-a-b-c-d)),{a,0,cutoff-1},{b,-cutoff,-1},{c,0,cutoff-1},{d,-cutoff,-1},{m,-cutoff,b},{n,0,a},{p,0,Min[cutoff-1,a+b+c-m-n]},{r,m,-1},{s,-cutoff,-1},{u,b,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^3)+9(x^2)-26x+37-26(x^(-1))+9(x^(-2))-(x^(-3)))),{x,∞,6}]

(Subscript[10, 125]) knot

From the braid ((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^(-3))((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^5),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[-1,b,b-1-m,m,q,x]Rinv[b-1-m,m,b-1-n,n,q,x]Rinv[b-1-n,n,-1,b,q,x]Rinv[a,-1,-1,a,q,x]R[a,b,a+b-p,p,q,x]R[a+b-p,p,a+b-r,r,q,x]R[a+b-r,r,a+b-s,s,q,x]R[a+b-s,s,a+b-u,u,q,x]R[a+b-u,u,a,b,q,x](x^(2/2)) (q^(((-2)/2)-a-b)),{a,-cutoff,-1},{b,0,cutoff-1},{m,-1,-1},{n,-cutoff,-1},{p,0,cutoff-1},{r,0,cutoff-1},{s,0,cutoff-1},{u,0,cutoff-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^3)-2(x^2)+2x-1+2(x^(-1))-2(x^(-2))+(x^(-3)))),{x,∞,6}]

(Subscript[10, 126]) knot

From the braid ((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^3)((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^(-5)),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]R[-1,b,b-1-m,m,q,x]R[b-1-m,m,b-1-n,n,q,x]R[b-1-n,n,-1,b,q,x]Rinv[a,-1,-1,a,q,x]Rinv[a,b,a+b-p,p,q,x]Rinv[a+b-p,p,a+b-r,r,q,x]Rinv[a+b-r,r,a+b-s,s,q,x]Rinv[a+b-s,s,a+b-u,u,q,x]Rinv[a+b-u,u,a,b,q,x](x^(2/2)) (q^(((-2)/2)-a-b)),{a,-cutoff,-1},{b,0,cutoff-1},{m,0,cutoff-1},{n,0,cutoff-1},{p,a,-1},{r,-cutoff,-1},{s,-cutoff,-1},{u,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^3)-2(x^2)+4x-5+4(x^(-1))-2(x^(-2))+(x^(-3)))),{x,∞,6}]

(Subscript[10, 127]) knot

From the braid ((Subscript[σ, 1])^(-2))((Subscript[σ, 2])^2)((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^(-5)),

With[{cutoff=4},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[a,-1,a-1-m,m,q,x]R[m,b,b+m-n,n,q,x]R[b+m-n,n,-1,b+m+1,q,x]Rinv[a-1-m,-1,-1,a-1-m,q,x]Rinv[a-1-m,b+m+1,a+b-p,p,q,x]Rinv[a+b-p,p,a+b-r,r,q,x]Rinv[a+b-r,r,a+b-s,s,q,x]Rinv[a+b-s,s,a+b-u,u,q,x]Rinv[a+b-u,u,a,b,q,x](x^(2/2)) (q^(((-2)/2)-a-b)),{a,-cutoff,-1},{b,0,cutoff-1},{m,a,-1},{n,0,cutoff-1},{p,a-1-m,-1},{r,-cutoff,-1},{s,-cutoff,-1},{u,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^3)+4(x^2)-6x+7-6(x^(-1))+4(x^(-2))-(x^(-3)))),{x,∞,6}]

(Subscript[10, 132]) knot

From the braid ((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^(-1))((Subscript[σ, 3])^(-1))(Subscript[σ, 2])((Subscript[σ, 1])^3)((Subscript[σ, 3])^(-2))((Subscript[σ, 2])^(-1))((Subscript[σ, 1])^(-1)),

br10132=BR[4,{-1,-2,-3,2,1,1,1,-3,-3,-2,-1}];
Show[BraidPlot[br10132]]

With[{cutoff=5},Series[-((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[-1,b,b,-1,q,x]Rinv[c,-1,-1,c,q,x]R[b,c,b,c,q,x]Rinv[c,-1,c,-1,q,x]Rinv[c,-1,-1,c,q,x]R[a,b,a+b-m,m,q,x]R[a+b-m,m,a+b-n,n,q,x]R[a+b-n,n,a,b,q,x]Rinv[b,-1,-1,b,q,x]Rinv[a,-1,-1,a,q,x](x^(3/2)) (q^(((-3)/2)-a-b-c)),{a,-cutoff,-1},{b,0,cutoff-1},{c,0,cutoff-1},{m,0,cutoff-1},{n,0,cutoff-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Alexander[Knot[10,132]][x]

Series[(((x^(1/2))-(x^(-1/2)))/((x^2)-x+1-(x^(-1))+(x^(-2)))),{x,∞,6}]

(Subscript[10, 133]) knot

br10133=BR[4,{-1,-1,2,-1,-2,3,3,-2,-3,-3,-3}];
Show[BraidPlot[br10133]]

With[{cutoff=4},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[a,-1,a-1-m,m,q,x]R[m,b,-1,b+m+1,q,x]Rinv[a-1-m,-1,-1,a-1-m,q,x]Rinv[a-1-m,b+m+1,a+b-n,n,q,x]R[n,c,c+n-p,p,q,x]R[c+n-p,p,c+n-r,r,q,x]Rinv[a+b-n,c+n-r,a,b+c-r,q,x]Rinv[b+c-r,r,b+c-s,s,q,x]Rinv[b+c-s,s,b+c-u,u,q,x]Rinv[b+c-u,u,b,c,q,x](x^(3/2)) (q^(((-3)/2)-a-b-c)),{a,-cutoff,-1},{b,-cutoff,-1},{c,0,cutoff-1},{m,-cutoff,-1},{n,-cutoff,-1},{p,0,cutoff-1},{r,0,cutoff-1},{s,-cutoff,-1},{u,-cutoff,-1}],{x,∞,5}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^2)+5x-7+5(x^(-1))-(x^(-2)))),{x,∞,5}]

(Subscript[10, 136]) knot

From the braid (Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 3])^2)((Subscript[σ, 2])^(-1))((Subscript[σ, 3])^(-1))(Subscript[σ, 4])((Subscript[σ, 3])^(-1))(Subscript[σ, 4]),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,a,a,0,q,x]Rinv[0,b,0,b,q,x]R[a,0,0,a,q,x]R[b,c,b+c-m,m,q,x]R[b+c-m,m,b+c-n,n,q,x]Rinv[a,b+c-n,a,b+c-n,q,x]Rinv[b+c-n,n,b+c-p,p,q,x]R[p,d,d+p-r,r,q,x]Rinv[b+c-p,d+p-r,b,c+d-r,q,x]R[c+d-r,r,c,d,q,x](x^(4/2)) (q^(((-4)/2)-a-b-c-d)),{a,0,cutoff-1},{b,-cutoff,-1},{c,0,cutoff-1},{d,0,cutoff-1},{m,0,cutoff-1},{n,0,cutoff-1},{p,-cutoff,-1},{r,0,cutoff-1}],{x,∞,5}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^2)+4x-5+4(x^(-1))-(x^(-2)))),{x,∞,5}]

(Subscript[10, 137]) knot

From the braid (Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 3])^(-2))((Subscript[σ, 2])^(-1))(Subscript[σ, 3])((Subscript[σ, 4])^(-1))(Subscript[σ, 3])((Subscript[σ, 4])^(-1)),

With[{cutoff=4},Series[((x^(1/2))-(x^(-1/2)))Sum[R[0,a,a,0,q,x]Rinv[0,b,0,b,q,x]R[a,0,0,a,q,x]Rinv[b,c,b+c-m,m,q,x]Rinv[b+c-m,m,b+c-n,n,q,x]Rinv[a,b+c-n,a,b+c-n,q,x]R[b+c-n,n,b+c-p,p,q,x]Rinv[p,d,d+p-r,r,q,x]R[b+c-p,d+p-r,b,c+d-r,q,x]Rinv[c+d-r,r,c,d,q,x](x^(4/2)) (q^(((-4)/2)-a-b-c-d)),{a,0,cutoff-1},{b,-cutoff,-1},{c,0,cutoff-1},{d,-cutoff,-1},{m,-cutoff,-1},{n,0,cutoff-1},{p,0,cutoff-1},{r,-cutoff,-1}],{x,∞,5}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^2)-6x+11-6(x^(-1))+(x^(-2)))),{x,∞,5}]

(Subscript[10, 140]) knot

From the braid ((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^(-1))((Subscript[σ, 3])^(-1))(Subscript[σ, 2])((Subscript[σ, 1])^3)((Subscript[σ, 3])^(-1))((Subscript[σ, 2])^(-1))((Subscript[σ, 1])^(-2)),

br10140=BR[4,{-1,-2,-3,2,1,1,1,-3,-2,-1,-1}];
Show[BraidPlot[br10140]]

With[{cutoff=4},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[-1,b,b,-1,q,x]Rinv[c,-1,-1,c,q,x]R[b,c,b+c-m,m,q,x]Rinv[m,-1,m-1-c,c,q,x]R[a,b+c-m,a+b+c-m-n,n,q,x]R[a+b+c-m-n,n,a+b+c-m-p,p,q,x]R[a+b+c-m-p,p,a+b+c-m-r,r,q,x]Rinv[r,m-1-c,m+r-1-b-c,b,q,x]Rinv[a+b+c-m-r,m+r-1-b-c,a,-1,q,x]Rinv[a,-1,-1,a,q,x](x^(3/2)) (q^(((-3)/2)-a-b-c)),{a,-cutoff,-1},{b,0,cutoff-1},{c,0,cutoff-1},{m,0,cutoff-1},{n,0,cutoff-1},{p,0,cutoff-1},{r,0,cutoff-1}],{x,∞,5}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^2)-2x+3-2(x^(-1))+(x^(-2)))),{x,∞,5}]

(Subscript[10, 141]) knot

From the braid ((Subscript[σ, 1])^(-2))((Subscript[σ, 2])^(-3))((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^4),

With[{cutoff=4},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[a,-1,a-1-m,m,q,x]Rinv[m,b,b+m-n,n,q,x]Rinv[b+m-n,n,b+m-p,p,q,x]Rinv[b+m-p,p,-1,b+m+1,q,x]Rinv[a-1-m,-1,-1,a-1-m,q,x]R[a-1-m,b+m+1,a+b-r,r,q,x]R[a+b-r,r,a+b-s,s,q,x]R[a+b-s,s,a+b-u,u,q,x]R[a+b-u,u,a,b,q,x](x^(2/2)) (q^(((-2)/2)-a-b)),{a,-cutoff,-1},{b,0,cutoff-1},{m,a,-1},{n,m,-1},{p,-1,-1},{r,0,cutoff-1},{s,0,cutoff-1},{u,0,cutoff-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^3)+3(x^2)-4x+5-4(x^(-1))+3(x^(-2))-(x^(-3)))),{x,∞,6}]

(Subscript[10, 143]) knot

From the braid ((Subscript[σ, 1])^(-2))((Subscript[σ, 2])^3)((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^(-4)),

With[{cutoff=4},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[a,-1,a-1-m,m,q,x]R[m,b,b+m-n,n,q,x]R[b+m-n,n,b+m-p,p,q,x]R[b+m-p,p,-1,b+m+1,q,x]Rinv[a-1-m,-1,-1,a-1-m,q,x]Rinv[a-1-m,b+m+1,a+b-r,r,q,x]Rinv[a+b-r,r,a+b-s,s,q,x]Rinv[a+b-s,s,a+b-u,u,q,x]Rinv[a+b-u,u,a,b,q,x](x^(2/2)) (q^(((-2)/2)-a-b)),{a,-cutoff,-1},{b,0,cutoff-1},{m,a,-1},{n,0,cutoff-1},{p,0,cutoff-1},{r,-cutoff,-1},{s,-cutoff,-1},{u,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^3)-3(x^2)+6x-7+6(x^(-1))-3(x^(-2))+(x^(-3)))),{x,∞,6}]

(Subscript[10, 145]) knot

Already did this in my first R-matrix paper

(Subscript[10, 148]) knot

From the braid ((Subscript[σ, 1])^(-1))(Subscript[σ, 2])((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^2)((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^(-4)),

With[{cutoff=4},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]R[-1,b,b-1-m,m,q,x]Rinv[a,b-1-m,a+b-1-m-n,n,q,x]R[n,m,n+m-p,p,q,x]R[n+m-p,p,-1,n+m+1,q,x]Rinv[a+b-1-m-n,-1,-1,a+b-1-m-n,q,x]Rinv[a+b-1-m-n,n+m+1,a+b-r,r,q,x]Rinv[a+b-r,r,a+b-s,s,q,x]Rinv[a+b-s,s,a+b-u,u,q,x]Rinv[a+b-u,u,a,b,q,x](x^(2/2)) (q^(((-2)/2)-a-b)),{a,-cutoff,-1},{b,0,cutoff-1},{m,0,cutoff-1},{n,a,-1},{p,0,cutoff-1},{r,-cutoff,-1},{s,-cutoff,-1},{u,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^3)-3(x^2)+7x-9+7(x^(-1))-3(x^(-2))+(x^(-3)))),{x,∞,6}]

(Subscript[10, 149]) knot

From the braid ((Subscript[σ, 1])^(-2))(Subscript[σ, 2])((Subscript[σ, 1])^(-1))(Subscript[σ, 2])((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^(-4)),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[a,-1,a-1-m,m,q,x]R[m,b,b+m-n,n,q,x]Rinv[a-1-m,b+m-n,a+b-1-n-p,p,q,x]R[p,n,-1,n+p+1,q,x]Rinv[a+b-1-n-p,-1,-1,a+b-1-n-p,q,x]Rinv[a+b-1-n-p,n+p+1,a+b-r,r,q,x]Rinv[a+b-r,r,a+b-s,s,q,x]Rinv[a+b-s,s,a+b-u,u,q,x]Rinv[a+b-u,u,a,b,q,x](x^(2/2)) (q^(((-2)/2)-a-b)),{a,-cutoff,-1},{b,0,cutoff-1},{m,a,-1},{n,0,cutoff-1},{p,-cutoff,-1},{r,-cutoff,-1},{s,-cutoff,-1},{u,a,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^3)+5(x^2)-9x+11-9(x^(-1))+5(x^(-2))-(x^(-3)))),{x,∞,6}]

(Subscript[10, 150]) knot

br10150=BR[4,{2,{1,-3},-2,{1,3},3,-2,3,3,3}];
Show[BraidPlot[br10150]]

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[a,b,a+b-m,m,q,x]R[0,a+b-m,a+b-m,0,q,x]Rinv[m,c,c+m-n,n,q,x]Rinv[0,c+m-n,0,c+m-n,q,x]R[a+b-m,0,0,a+b-m,q,x]R[c+m-n,n,c+m-p,p,q,x]R[c+m-p,p,c+m-r,r,q,x]Rinv[a+b-m,c+m-r,a,b+c-r,q,x]R[b+c-r,r,b+c-s,s,q,x]R[b+c-s,s,b+c-u,u,q,x]R[b+c-u,u,b,c,q,x](x^(3/2)) (q^(((-3)/2)-a-b-c)),{a,0,cutoff-1},{b,0,cutoff-1},{c,-cutoff,-1},{m,0,a},{n,m,cutoff-1},{p,0,cutoff-1},{r,0,cutoff-1},{s,0,cutoff-1},{u,0,b}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^3)+4(x^2)-6x+7-6(x^(-1))+4(x^(-2))-(x^(-3)))),{x,∞,6}]

(Subscript[10, 151]) knot

br10151=BR[4,{-2,{1,3},-2,{1,-3},-3,2,3,3,3}];
Show[BraidPlot[br10151]]

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[Rinv[a,b,a+b-m,m,q,x]R[0,a+b-m,a+b-m,0,q,x]R[m,c,c+m-n,n,q,x]Rinv[0,c+m-n,0,c+m-n,q,x]R[a+b-m,0,0,a+b-m,q,x]Rinv[c+m-n,n,c+m-p,p,q,x]Rinv[c+m-p,p,c+m-r,r,q,x]R[a+b-m,c+m-r,a,b+c-r,q,x]R[b+c-r,r,b+c-s,s,q,x]R[b+c-s,s,b+c-u,u,q,x]R[b+c-u,u,b,c,q,x](x^(3/2)) (q^(((-3)/2)-a-b-c)),{a,0,cutoff-1},{b,-cutoff,-1},{c,0,cutoff-1},{m,-cutoff,-1},{n,0,cutoff-1},{p,-cutoff,-1},{r,-cutoff,-1},{s,0,Min[cutoff-1,b+c-r]},{u,0,cutoff-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^3)-4(x^2)+10x-13+10(x^(-1))-4(x^(-2))+(x^(-3)))),{x,∞,6}]

(Subscript[10, 153]) knot

From the braid ((Subscript[σ, 1])^(-3))((Subscript[σ, 2])^(-1))((Subscript[σ, 1])^(-2))(Subscript[σ, 3])((Subscript[σ, 2])^3)(Subscript[σ, 3]),

br10153=BR[4,{-1,-1,-1,-2,-1,{-1,3},2,2,2,3}];
Show[BraidPlot[br10153]]

With[{cutoff=4},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[a,-1,a-1-m,m,q,x]Rinv[a-1-m,m,a-1-n,n,q,x]Rinv[n,b,n+b-p,p,q,x]Rinv[a-1-n,n+b-p,a+b-p,-1,q,x]Rinv[a+b-p,-1,-1,a+b-p,q,x]R[p,c,p+c-r,r,q,x]R[a+b-p,p+c-r,a+b+c-r-s,s,q,x]R[a+b+c-r-s,s,a+b+c-r-u,u,q,x]R[a+b+c-r-u,u,a,b+c-r,q,x]R[b+c-r,r,b,c,q,x](x^(3/2)) (q^(((-3)/2)-a-b-c)),{a,-cutoff,-1},{b,-cutoff,-1},{c,0,cutoff-1},{m,-cutoff,-1},{n,-cutoff,-1},{p,-cutoff,-1},{r,-cutoff,-1},{s,0,cutoff-1},{u,0,cutoff-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^3)-(x^2)-x+3-(x^(-1))-(x^(-2))+(x^(-3)))),{x,∞,6}]

(Subscript[10, 154]) knot

Already did this in my first R-matrix paper

(Subscript[10, 155]) knot

From the braid (Subscript[σ, 1])((Subscript[σ, 2])^(-2))(Subscript[σ, 1])((Subscript[σ, 2])^(-2))(Subscript[σ, 1])((Subscript[σ, 2])^3),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,a,a,0,q,x]Rinv[0,b,b-m,m,q,x]Rinv[b-m,m,b-n,n,q,x]R[a,b-n,a+b-n-p,p,q,x]Rinv[p,n,n+p-r,r,q,x]Rinv[n+p-r,r,0,n+p,q,x]R[a+b-n-p,0,0,a+b-n-p,q,x]R[a+b-n-p,n+p,a+b-s,s,q,x]R[a+b-s,s,a+b-u,u,q,x]R[a+b-u,u,a,b,q,x](x^(2/2)) (q^(((-2)/2)-a-b)),{a,0,cutoff-1},{b,-cutoff,-1},{m,-cutoff,-1},{n,-cutoff,-1},{p,0,a},{r,-cutoff,-1},{s,0,cutoff-1},{u,0,a}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^3)+3(x^2)-5x+7-5(x^(-1))+3(x^(-2))-(x^(-3)))),{x,∞,6}]

(Subscript[10, 156]) knot

From the braid ((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^(-1))(Subscript[σ, 3])((Subscript[σ, 2])^(-1))((Subscript[σ, 1])^(-1))((Subscript[σ, 3])^2)(Subscript[σ, 2])((Subscript[σ, 3])^(-3)),

With[{cutoff=4},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[-1,b,b-1-m,m,q,x]R[m,c,c+m-n,n,q,x]Rinv[b-1-m,c+m-n,-1,b+c-n,q,x]Rinv[a,-1,-1,a,q,x]R[b+c-n,n,b+c-p,p,q,x]R[b+c-p,p,b+c-r,r,q,x]R[a,b+c-r,a,b+c-r,q,x]Rinv[b+c-r,r,b+c-s,s,q,x]Rinv[b+c-s,s,b+c-u,u,q,x]Rinv[b+c-u,u,b,c,q,x](x^(3/2)) (q^(((-3)/2)-a-b-c)),{a,-cutoff,-1},{b,-cutoff,-1},{c,0,cutoff-1},{m,-cutoff,-1},{n,0,cutoff-1},{p,0,cutoff-1},{r,-cutoff,-1},{s,-cutoff,-1},{u,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^3)-4(x^2)+8x-9+8(x^(-1))-4(x^(-2))+(x^(-3)))),{x,∞,6}]

(Subscript[10, 157]) knot

From the braid ((Subscript[σ, 1])^2)((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 2])^(-1))((Subscript[σ, 1])^2)((Subscript[σ, 2])^3),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,a,a,0,q,x]R[a,0,a-m,m,q,x]Rinv[m,b,b+m-n,n,q,x]R[a-m,b+m-n,a+b-n-p,p,q,x]Rinv[p,n,n+p-r,r,q,x]R[a+b-n-p,n+p-r,a+b-r,0,q,x]R[a+b-r,0,0,a+b-r,q,x]R[a+b-r,r,a+b-s,s,q,x]R[a+b-s,s,a+b-u,u,q,x]R[a+b-u,u,a,b,q,x](x^(2/2)) (q^(((-2)/2)-a-b)),{a,0,cutoff-1},{b,-cutoff,-1},{m,0,a},{n,-cutoff,-1},{p,0,cutoff-1},{r,-cutoff,-1},{s,0,cutoff-1},{u,0,a}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^3)+6(x^2)-11x+13-11(x^(-1))+6(x^(-2))-(x^(-3)))),{x,∞,6}]

(Subscript[10, 158]) knot

From the braid (Subscript[σ, 1])(Subscript[σ, 2])((Subscript[σ, 3])^(-1))(Subscript[σ, 2])(Subscript[σ, 1])((Subscript[σ, 3])^2)((Subscript[σ, 2])^(-1))((Subscript[σ, 3])^(-3)),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,a,a,0,q,x]R[0,b,b-m,m,q,x]Rinv[m,c,c+m-n,n,q,x]R[b-m,c+m-n,0,b+c-n,q,x]R[a,0,0,a,q,x]R[b+c-n,n,b+c-p,p,q,x]R[b+c-p,p,b+c-r,r,q,x]Rinv[a,b+c-r,a,b+c-r,q,x]Rinv[b+c-r,r,b+c-s,s,q,x]Rinv[b+c-s,s,b+c-u,u,q,x]Rinv[b+c-u,u,b,c,q,x](x^(3/2)) (q^(((-3)/2)-a-b-c)),{a,0,cutoff-1},{b,0,cutoff-1},{c,-cutoff,-1},{m,0,cutoff-1},{n,-cutoff,-1},{p,0,cutoff-1},{r,0,cutoff-1},{s,-cutoff,-1},{u,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^3)+4(x^2)-10x+15-10(x^(-1))+4(x^(-2))-(x^(-3)))),{x,∞,6}]

(Subscript[10, 159]) knot

From the braid ((Subscript[σ, 1])^(-2))((Subscript[σ, 2])^2)((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^(-1))((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^(-3)),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[a,-1,a-1-m,m,q,x]R[m,b,b+m-n,n,q,x]R[b+m-n,n,b+m-p,p,q,x]Rinv[a-1-m,b+m-p,a+b-1-p-r,r,q,x]R[r,p,-1,p+r+1,q,x]Rinv[a+b-1-p-r,-1,-1,a+b-1-p-r,q,x]Rinv[a+b-1-p-r,p+r+1,a+b-s,s,q,x]Rinv[a+b-s,s,a+b-u,u,q,x]Rinv[a+b-u,u,a,b,q,x](x^(2/2)) (q^(((-2)/2)-a-b)),{a,-cutoff,-1},{b,0,cutoff-1},{m,a,-1},{n,0,cutoff-1},{p,0,cutoff-1},{r,-cutoff,-1},{s,-cutoff,-1},{u,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^3)-4(x^2)+9x-11+9(x^(-1))-4(x^(-2))+(x^(-3)))),{x,∞,6}]

(Subscript[10, 160]) knot

From the braid ((Subscript[σ, 1])^(-1))(Subscript[σ, 2])((Subscript[σ, 3])^(-1))(Subscript[σ, 2])((Subscript[σ, 1])^(-1))((Subscript[σ, 3])^2)(Subscript[σ, 2])((Subscript[σ, 3])^3)

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]R[-1,b,b-1-m,m,q,x]Rinv[m,c,c+m-n,n,q,x]R[b-1-m,c+m-n,-1,b+c-n,q,x]Rinv[a,-1,-1,a,q,x]R[b+c-n,n,b+c-p,p,q,x]R[b+c-p,p,b+c-r,r,q,x]R[a,b+c-r,a,b+c-r,q,x]R[b+c-r,r,b+c-s,s,q,x]R[b+c-s,s,b+c-u,u,q,x]R[b+c-u,u,b,c,q,x](x^(3/2)) (q^(((-3)/2)-a-b-c)),{a,-cutoff,-1},{b,0,cutoff-1},{c,0,cutoff-1},{m,0,cutoff-1},{n,0,cutoff-1},{p,0,cutoff-1},{r,0,cutoff-1},{s,0,cutoff-1},{u,0,cutoff-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/(-(x^3)+4(x^2)-4x+3-4(x^(-1))+4(x^(-2))-(x^(-3)))),{x,∞,6}]

(Subscript[10, 161]) knot

From the braid ((Subscript[σ, 1])^(-2))((Subscript[σ, 2])^(-2))((Subscript[σ, 1])^(-1))(Subscript[σ, 2])((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^(-3)),

With[{cutoff=4},Series[((x^(1/2))-(x^(-1/2)))Sum[Rinv[-1,a,a,-1,q,x]Rinv[a,-1,a-1-m,m,q,x]Rinv[m,b,b+m-n,n,q,x]Rinv[b+m-n,n,b+m-p,p,q,x]Rinv[a-1-m,b+m-p,a+b-1-p-r,r,q,x]R[r,p,-1,p+r+1,q,x]Rinv[a+b-1-p-r,-1,-1,a+b-1-p-r,q,x]Rinv[a+b-1-p-r,p+r+1,a+b-s,s,q,x]Rinv[a+b-s,s,a+b-u,u,q,x]Rinv[a+b-u,u,a,b,q,x](x^(2/2)) (q^(((-2)/2)-a-b)),{a,-cutoff,-1},{b,-cutoff,-1},{m,-cutoff,-1},{n,-cutoff,-1},{p,-cutoff,-1},{r,-cutoff,-1},{s,-cutoff,-1},{u,-cutoff,-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^3)-2x+3-2(x^(-1))+(x^(-3)))),{x,∞,6}]

(Subscript[10, 163]) knot

From the braid (Subscript[σ, 1])((Subscript[σ, 2])^2)((Subscript[σ, 3])^(-1))(Subscript[σ, 2])(Subscript[σ, 1])((Subscript[σ, 3])^(-2))((Subscript[σ, 2])^(-1))((Subscript[σ, 3])^2),

With[{cutoff=4},Series[-((x^(1/2))-(x^(-1/2)))Sum[R[0,a,a,0,q,x]R[0,b,b,0,q,x]R[b,0,b-m,m,q,x]Rinv[m,c,0,c+m,q,x]R[b-m,0,0,b-m,q,x]R[a,0,0,a,q,x]Rinv[b-m,c+m,b+c-p,p,q,x]Rinv[b+c-p,p,b+c-r,r,q,x]Rinv[a,b+c-r,a,b+c-r,q,x]R[b+c-r,r,b+c-s,s,q,x]R[b+c-s,s,b,c,q,x](x^(3/2)) (q^(((-3)/2)-a-b-c)),{a,0,cutoff-1},{b,0,cutoff-1},{c,-cutoff,-1},{m,0,cutoff-1},{p,-cutoff,-1},{r,0,cutoff-1},{s,0,cutoff-1}],{x,∞,6}]//Simplify]
%/.{q->1}

Series[(((x^(1/2))-(x^(-1/2)))/((x^3)-5(x^2)+12x-15+12(x^(-1))-5(x^(-2))+(x^(-3)))),{x,∞,6}]

Links

Revisiting old examples

L2a1: Hopf link

Hopf link with a strand (x-strand) open

With[{w=0,cutoff=5},(Series[Sum[Rxy[w,m1,w+m1-m2,m2,q,t x,t y]Rxy[w+m1-m2,m2,w,m1,q,t y,t x](((t y))^(1/2))(q^(((-1)/2)-m1)),{m1,0,cutoff},{m2,0,cutoff}],{t,∞,5}]//Simplify//Normal)/.{t->1}]

This is (1/((x^(1/2))-(x^(-1/2)))). So we should multiply it by ((x^(1/2))-(x^(-1/2))) to get the correct answer, which is 1.

With[{w=0,cutoff=5},(((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[Rxy[w,m1,w+m1-m2,m2,q,t x,t y]Rxy[w+m1-m2,m2,w,m1,q,t y,t x](((t y))^(1/2))(q^(((-1)/2)-m1)),{m1,0,cutoff},{m2,0,cutoff}],{t,∞,5}]//Simplify//Normal)/.{t->1}]

Let’s check for centrality.

With[{w=1,cutoff=5},(((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[Rxy[w,m1,w+m1-m2,m2,q,t x,t y]Rxy[w+m1-m2,m2,w,m1,q,t y,t x](((t y))^(1/2))(q^(((-1)/2)-m1)),{m1,0,cutoff},{m2,0,cutoff}],{t,∞,5}]//Simplify//Normal)/.{t->1}]

With[{w=2,cutoff=5},(((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[Rxy[w,m1,w+m1-m2,m2,q,t x,t y]Rxy[w+m1-m2,m2,w,m1,q,t y,t x](((t y))^(1/2))(q^(((-1)/2)-m1)),{m1,0,cutoff},{m2,0,cutoff}],{t,∞,5}]//Simplify//Normal)/.{t->1}]

With[{w=-1,cutoff=5},(((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[Rxy[w,m1,w+m1-m2,m2,q,t x,t y]Rxy[w+m1-m2,m2,w,m1,q,t y,t x](((t y))^(1/2))(q^(((-1)/2)-m1)),{m1,0,cutoff},{m2,0,cutoff}],{t,∞,5}]//Simplify//Normal)/.{t->1}]

L4a1: T(4,2)

With[{w=0,cutoff=8},((q^(-1/2))((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[Rxy[w,m1,w+m1-m2,m2,q,t x,t y]Rxy[w+m1-m2,m2,w+m1-m3,m3,q,t y,t x]Rxy[w+m1-m3,m3,w+m1-m4,m4,q,t x,t y]Rxy[w+m1-m4,m4,w,m1,q,t y,t x](((t y))^(1/2))(q^(((-1)/2)-m1)),{m1,0,cutoff},{m2,0,cutoff},{m3,0,cutoff},{m4,0,cutoff}],{t,∞,10}]//Simplify)]

As expected!

With[{w=1,cutoff=8},((q^(-1/2))((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[Rxy[w,m1,w+m1-m2,m2,q,t x,t y]Rxy[w+m1-m2,m2,w+m1-m3,m3,q,t y,t x]Rxy[w+m1-m3,m3,w+m1-m4,m4,q,t x,t y]Rxy[w+m1-m4,m4,w,m1,q,t y,t x](((t y))^(1/2))(q^(((-1)/2)-m1)),{m1,0,cutoff},{m2,0,cutoff},{m3,0,cutoff},{m4,0,cutoff}],{t,∞,10}]//Simplify)]

With[{w=-1,cutoff=8},((q^(-1/2))((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[Rxy[w,m1,w+m1-m2,m2,q,t x,t y]Rxy[w+m1-m2,m2,w+m1-m3,m3,q,t y,t x]Rxy[w+m1-m3,m3,w+m1-m4,m4,q,t x,t y]Rxy[w+m1-m4,m4,w,m1,q,t y,t x](((t y))^(1/2))(q^(((-1)/2)-m1)),{m1,0,cutoff},{m2,0,cutoff},{m3,0,cutoff},{m4,0,cutoff}],{t,∞,10}]//Simplify)]

Multivariable Alexander polynomial of T(4,2):

(1/(t1-(t1^(-1))))Sum[t2(((-1))^j)RAlex[0,j,i1,j1,t1,t2]RAlex[i1,j1,i2,j2,t2,t1]RAlex[i2,j2,i3,j3,t1,t2]RAlex[i3,j3,0,j,t2,t1],{j,0,1},{i1,0,1},{j1,0,1},{i2,0,1},{j2,0,1},{i3,0,1},{j3,0,1}]//Simplify
%/.{t1->(x^(1/2)),t2->(y^(1/2))}

Series[(1/((1/(t1 t2))+t1 t2)),{t1,∞,10}]

L5a1: Whitehead link

Take the braid (Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 1]).

With[{cutoff=8},((q^(-1/2))((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[Rxy[0,m1,m1,0,q,t x,t y]Rxyinv[0,k1,k1-k2,k2,q,t x,t y](q^(1/4))Rxy[m1,k1-k2,m1,k1-k2,q,t y,t y]Rxyinv[k1-k2,k2,0,k1,q,t y,t x]Rxy[m1,0,0,m1,q,t y,t x](t y)(q^(-1-m1-k1)),{m1,0,cutoff},{k1,-cutoff,-1},{k2,-cutoff,-1}],{t,∞,8}]//Simplify)]
Table[Coefficient[(%//Normal)/.{t->1}//Expand,(x^(-i-(1/2)))(y^(-j-(1/2)))],{i,0,3},{j,0,3}]//MatrixForm

This matches perfectly with the results in Sec. 5.2.3. of arXiv:2004.02087. (This is the mirror of the one used in that paper.)

L6a4: Borromean rings

Take the braid (Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 2])^(-1)).

With[{cutoff=4,tdegcutoff=8},-(((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[If[2k2+k3-m1+3/2≥-tdegcutoff,Rxy[0,m1,m1,0,q,t x,t y]Rxyinv[0,k1,k1-k2,k2,q,t x,t z]Rxy[m1,k1-k2,m1+k1-k3,k3-k2,q,t y,t z]Rxyinv[k3-k2,k2,0,k3,q,t y,t x]Rxy[m1+k1-k3,0,0,m1+k1-k3,q,t z,t x]Rxyinv[m1+k1-k3,k3,m1,k1,q,t z,t y](((t y))^(1/2))(((t z))^(1/2))(q^(-1-m1-k1)),0],{k1,-cutoff,-1},{k2,-cutoff,k1},{k3,k2,-1},{m1,Max[0,k3-k1],cutoff}],{t,∞,tdegcutoff}]//Simplify)];
Simplify[(%//Normal)/.{t->1,x->(X^2),y->(Y^2),z->(Z^2)},Assumptions->X>0&&Y>0&&Z>0]//Expand;
Table[Coefficient[%,(X^(-1))(Y^(-1-2i))(Z^(-1-2j))],{i,0,3},{j,0,3}]//MatrixForm
Table[Coefficient[%%,(X^(-3))(Y^(-1-2i))(Z^(-1-2j))],{i,0,3},{j,0,3}]//MatrixForm
Table[Coefficient[%%%,(X^(-5))(Y^(-1-2i))(Z^(-1-2j))],{i,0,3},{j,0,3}]//MatrixForm

Comparing with what we already had from inverted Habiro series, we see perfect agreement, as expected!

Series[(X-(X^(-1)))(Y-(Y^(-1)))(Z-(Z^(-1)))Sum[(((((-1))^n)(q^(-((3(n^2)+n)/2)))(((Product[1-(q^(n+l)),{l,1,n}]))^2))/(Product[((X^2)+(X^(-2))-(q^j)-(q^(-j)))((Y^2)+(Y^(-2))-(q^j)-(q^(-j)))((Z^2)+(Z^(-2))-(q^j)-(q^(-j))),{j,0,n}])),{n,0,4}],{X,∞,7},{Y,∞,7},{Z,∞,7}]//Normal//Expand;
Table[Coefficient[%,(X^(-1))(Y^(-1-2i))(Z^(-1-2j))],{i,0,3},{j,0,3}]//MatrixForm
Table[Coefficient[%%,(X^(-3))(Y^(-1-2i))(Z^(-1-2j))],{i,0,3},{j,0,3}]//MatrixForm
Table[Coefficient[%%%,(X^(-5))(Y^(-1-2i))(Z^(-1-2j))],{i,0,3},{j,0,3}]//MatrixForm
Table[Coefficient[%%%%,(X^(-7))(Y^(-1-2i))(Z^(-1-2j))],{i,0,3},{j,0,3}]//MatrixForm

New examples

L7a1

Take the braid ((Subscript[σ, 1])^2)((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 2])^(-1)).

With[{cutoff=4,tdegcutoff=7},-(q^((-1)/2))(((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[If[2k2+k3-m1-m2+1≥-tdegcutoff,R[0,m1,m1,0,q,t x]R[m1,0,m1-m2,m2,q,t x]Rxyinv[m2,k1,m2+k1-k2,k2,q,t x,t y]Rxy[m1-m2,m2+k1-k2,m1+k1-k3,k3-k2,q,t x,t y]Rinv[k3-k2,k2,0,k3,q,t x]Rxy[m1+k1-k3,0,0,m1+k1-k3,q,t y,t x]Rxyinv[m1+k1-k3,k3,m1,k1,q,t y,t x](((t x))^(1/2))(((t y))^(1/2))(q^(-1-m1-k1)),0],{k1,-cutoff,-1},{k2,-cutoff,-1},{k3,k2,-1},{m1,Max[0,k3-k1],cutoff},{m2,Max[0,k2-k1],m1}],{t,∞,tdegcutoff}]//Simplify)]
Simplify[(%//Normal)/.{t->1,x->(X^2),y->(Y^2)},Assumptions->X>0&&Y>0]//Expand;
Table[Coefficient[%,(X^(-2i-1))(Y^(-2j-1))],{i,0,5},{j,0,5}]//MatrixForm

With[{cutoff=4,tdegcutoff=8},-(q^((-1)/2))(((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[If[2k2+k3-m1-m2+1≥-tdegcutoff,R[0,m1,m1,0,q,t x]R[m1,0,m1-m2,m2,q,t x]Rxyinv[m2,k1,m2+k1-k2,k2,q,t x,t y]Rxy[m1-m2,m2+k1-k2,m1+k1-k3,k3-k2,q,t x,t y]Rinv[k3-k2,k2,0,k3,q,t x]Rxy[m1+k1-k3,0,0,m1+k1-k3,q,t y,t x]Rxyinv[m1+k1-k3,k3,m1,k1,q,t y,t x](((t x))^(1/2))(((t y))^(1/2))(q^(-1-m1-k1)),0],{k1,-cutoff,-1},{k2,-cutoff,-1},{k3,k2,-1},{m1,Max[0,k3-k1],cutoff},{m2,Max[0,k2-k1],m1}],{t,∞,tdegcutoff}]//Simplify)]
Simplify[(%//Normal)/.{t->1,x->(X^2),y->(Y^2)},Assumptions->X>0&&Y>0]//Expand;
Table[Coefficient[%,(X^(-2i-1))(Y^(-2j-1))],{i,0,5},{j,0,5}]//MatrixForm
%/.{q->1}//MatrixForm

The classical limit agrees with the inverse Alexander polynomial:

Simplify[(Series[(1/((x^(-3/2))(y^(-1/2))(x-1)((x^2)-x+1)(y-1))),{x,∞,14},{y,∞,14}]//Normal)/.{x->(X^2),y->(Y^2)},Assumptions->X>0&&Y>0]//Expand;
Table[Coefficient[(%//Normal)/.{x->(X^2),y->(Y^2)},(X^(-2i-1))(Y^(-2j-1))],{i,0,13},{j,0,13}]//MatrixForm

-1-surgery on the y-strand gives (Subscript[6, 3]) knot

With[{cutoff=4,tdegcutoff=8},-(q^((-1)/2))(((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[If[2k2+k3-m1-m2+1≥-tdegcutoff,R[0,m1,m1,0,q,t x]R[m1,0,m1-m2,m2,q,t x]Rxyinv[m2,k1,m2+k1-k2,k2,q,t x,t y]Rxy[m1-m2,m2+k1-k2,m1+k1-k3,k3-k2,q,t x,t y]Rinv[k3-k2,k2,0,k3,q,t x]Rxy[m1+k1-k3,0,0,m1+k1-k3,q,t y,t x]Rxyinv[m1+k1-k3,k3,m1,k1,q,t y,t x](((t x))^(1/2))(((t y))^(1/2))(q^(-1-m1-k1)),0],{k1,-cutoff,-1},{k2,-cutoff,-1},{k3,k2,-1},{m1,Max[0,k3-k1],cutoff},{m2,Max[0,k2-k1],m1}],{t,∞,tdegcutoff}]//Simplify)];
(Simplify[-(%//Normal)/.{t->1,x->(X^2),y->(Y^2)},Assumptions->X>0&&Y>0]//Expand)/.{(Y^(u_))->(q^((((u-1))^2)/4))-(q^((((u+1))^2)/4))}//Expand;
Series[Table[Coefficient[%,(X^(-2i-1))],{i,0,3}],{q,0,10}]//MatrixForm

+1-surgery on the y-strand gives (Subscript[6, 2]) knot

With[{cutoff=4,tdegcutoff=8},-(q^((-1)/2))(((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[If[2k2+k3-m1-m2+1≥-tdegcutoff,R[0,m1,m1,0,q,t x]R[m1,0,m1-m2,m2,q,t x]Rxyinv[m2,k1,m2+k1-k2,k2,q,t x,t y]Rxy[m1-m2,m2+k1-k2,m1+k1-k3,k3-k2,q,t x,t y]Rinv[k3-k2,k2,0,k3,q,t x]Rxy[m1+k1-k3,0,0,m1+k1-k3,q,t y,t x]Rxyinv[m1+k1-k3,k3,m1,k1,q,t y,t x](((t x))^(1/2))(((t y))^(1/2))(q^(-1-m1-k1)),0],{k1,-cutoff,-1},{k2,-cutoff,-1},{k3,k2,-1},{m1,Max[0,k3-k1],cutoff},{m2,Max[0,k2-k1],m1}],{t,∞,tdegcutoff}]//Simplify)];
(Simplify[-(%//Normal)/.{t->1,x->(X^2),y->(Y^2)},Assumptions->X>0&&Y>0]//Expand)/.{(Y^(u_))->((q^(-((((u-1))^2)/4)))-(q^(-((((u+1))^2)/4))))(1-((Sum[(((-1))^k)(q^(((3k+1)k)/2)),{k,-((Abs[u]-1)/2),((Abs[u]-1)/2)}])/(Sum[(((-1))^k)(q^(((3k+1)k)/2)),{k,-5,5}])))};
Series[Table[q Coefficient[%//Expand,(X^(-2i-1))],{i,0,3}],{q,0,10}]//MatrixForm

The last entry will converge to 1+3q-(q^2) if we increase the cutoff in our computation.

((-1)/2)-surgery on the y-strand gives (Subscript[8, 8]) knot

With[{cutoff=4,tdegcutoff=8},-(q^((-1)/2))(((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[If[2k2+k3-m1-m2+1≥-tdegcutoff,R[0,m1,m1,0,q,t x]R[m1,0,m1-m2,m2,q,t x]Rxyinv[m2,k1,m2+k1-k2,k2,q,t x,t y]Rxy[m1-m2,m2+k1-k2,m1+k1-k3,k3-k2,q,t x,t y]Rinv[k3-k2,k2,0,k3,q,t x]Rxy[m1+k1-k3,0,0,m1+k1-k3,q,t y,t x]Rxyinv[m1+k1-k3,k3,m1,k1,q,t y,t x](((t x))^(1/2))(((t y))^(1/2))(q^(-1-m1-k1)),0],{k1,-cutoff,-1},{k2,-cutoff,-1},{k3,k2,-1},{m1,Max[0,k3-k1],cutoff},{m2,Max[0,k2-k1],m1}],{t,∞,tdegcutoff}]//Simplify)];
(Simplify[-(q^((-1)/8))(%//Normal)/.{t->1,x->(X^2),y->(Y^2)},Assumptions->X>0&&Y>0]//Expand)/.{(Y^(u_))->(q^(2((((u/2)-(1/4)))^2)))-(q^(2((((u/2)+(1/4)))^2)))}//Expand;
Series[Table[Coefficient[%,(X^(-2i-1))],{i,0,3}],{q,0,10}]//MatrixForm

L7a3

Take the braid (Subscript[σ, 2])((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^3)((Subscript[σ, 1])^(-1))(Subscript[σ, 2]).

With[{cutoff=8,tdegcutoff=8},(q^(-3/2))(((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[If[m-3k1-k2-k3-1≥-tdegcutoff,Rxy[m,k,m+k-k1,k1,q,t x,t y]Rxyinv[-1,m+k-k1,m+k-k1,-1,q,t x,t y]Rxyinv[m+k-k1,-1,-1,m+k-k1,q,t y,t x]Rxy[m+k-k1,k1,m,k,q,t y,t x]R[-1,k1,-1+k1-k2,k2,q,t x]R[-1+k1-k2,k2,-1+k1-k3,k3,q,t x]R[-1+k1-k3,k3,-1,k1,q,t x](((t x))^(1/2))(((t y))^(1/2))(q^(-1-m-k)),0],{m,-cutoff,-1},{k,0,cutoff},{k1,Max[0,m+k+1],cutoff},{k2,0,cutoff},{k3,0,cutoff}],{t,∞,tdegcutoff}]//Simplify)]
Simplify[(%//Normal)/.{t->1,x->(X^2),y->(Y^2)},Assumptions->X>0&&Y>0&&q>0]//Expand;
Table[Coefficient[%,(X^(-2i-1))(Y^(-2j-1))],{i,0,6},{j,0,6}]//MatrixForm
%/.{q->1}//MatrixForm

The multivariable Alexander polynomial:

(1/(t1-(t1^(-1))))Sum[t1 t2(((-1))^(j+k))RAlex[j,k,j1,k1,t1,t2]RAlexinv[0,j1,i1,j2,t1,t2]RAlex[j2,k1,j3,j2+k1-j3,t1,t1]RAlex[j3,j2+k1-j3,j4,j2+k1-j4,t1,t1]RAlex[j4,j2+k1-j4,j5,j2+k1-j5,t1,t1]RAlexinv[i1,j5,0,j6,t2,t1]RAlex[j6,j2+k1-j5,j,k,t2,t1],{j,0,1},{k,0,1},{i1,0,1},{j1,0,1},{j2,0,1},{j3,0,1},{j4,0,1},{j5,0,1},{j6,0,1},{k1,0,1}]//Simplify
%/.{t1->(x^(1/2)),t2->(y^(1/2))}//Simplify

Series[(1/(-(((-1+x) (1+(x^2)) (-1+y))/(Sqrt[(x^3) y])))),{x,∞,11},{y,∞,11}];
Simplify[(%//Normal)/.{x->(X^2),y->(Y^2)},Assumptions->X>0&&Y>0]//Expand;
Table[Coefficient[%,(X^(-2i-1))(Y^(-2j-1))],{i,0,10},{j,0,10}]//MatrixForm

L7a6

Take the braid (Subscript[σ, 2])((Subscript[σ, 1])^(-1))(Subscript[σ, 2])((Subscript[σ, 1])^(-1))((Subscript[σ, 2])^3).

With[{cutoff=4,tdegcutoff=7},(q^(-1))(((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[If[m-3k1-k2-k3-1≥-tdegcutoff,Rxy[m,k,m+k-k1,k1,q,t x,t y]Rxyinv[-1,m+k-k1,m+k-k1,-1,q,t x,t y]Rxyinv[m+k-k1,-1,-1,m+k-k1,q,t y,t x]Rxy[m+k-k1,k1,m+k-k2,k2,q,t y,t x]R[-1,k1,-1,k1,q,t x]Rxy[m+k-k2,k2,m+k-k3,k3,q,t x,t y]Rxy[m+k-k3,k3,m,k,q,t y,t x](((t x))^(1/2))(((t y))^(1/2))(q^(-1-m-k)),0],{m,-cutoff,-1},{k,0,cutoff},{k1,Max[0,m+k+1],cutoff},{k2,0,cutoff},{k3,0,cutoff}],{t,∞,tdegcutoff}]//Simplify)]
Simplify[(%//Normal)/.{t->1,x->(X^2),y->(Y^2)},Assumptions->X>0&&Y>0&&q>0]//Expand;
Table[Coefficient[%,(X^(-2i-2))(Y^(-2j-2))],{i,0,4},{j,0,4}]//MatrixForm
%/.{q->1}//MatrixForm

The multivariable Alexander polynomial:

(1/(t1-(t1^(-1))))Sum[t1 t2(((-1))^(j+k))RAlex[j,k,j1,k1,t1,t2]RAlexinv[0,j1,i1,j2,t1,t2]RAlex[j2,k1,j3,j2+k1-j3,t1,t1]RAlexinv[i1,j3,0,i1+j3,t2,t1]RAlex[i1+j3,j2+k1-j3,j4,i1+j2+k1-j4,t2,t1]RAlex[j4,i1+j2+k1-j4,j5,i1+j2+k1-j5,t1,t2]RAlex[j5,i1+j2+k1-j5,j,k,t2,t1],{j,0,1},{k,0,1},{i1,0,1},{j1,0,1},{j2,0,1},{j3,0,1},{j4,0,1},{j5,0,1},{k1,0,1}]//Simplify
%/.{t1->(x^(1/2)),t2->(y^(1/2))}//Simplify

Series[(1/(-1+x+(1/y)+((-1+y)/(x y))+y-x y)),{x,∞,11},{y,∞,11}];
Simplify[(%//Normal)/.{x->(X^2),y->(Y^2)},Assumptions->X>0&&Y>0]//Expand;
Table[Coefficient[%,(X^(-2i-2))(Y^(-2j-2))],{i,0,10},{j,0,10}]//MatrixForm

L8n5: similar to Borromean rings, but with a -1/2 twist

We use the same link as in the planar diagram in Knot Atlas, which is the mirror of the braid presentation given there.

Take the braid (Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 2])^(-1))(Subscript[σ, 1])((Subscript[σ, 2])^3).

With[{cutoff=2,tdegcutoff=8},-(((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[If[-m1-m3-m4-k1+2k2-1/2≥-tdegcutoff,Rxy[0,m1,m1,0,q,t x,t y]Rxyinv[0,k1,k1-k2,k2,q,t x,t z]Rxy[m1,k1-k2,m1+k1-k2-m2,m2,q,t y,t z]Rxyinv[m2,k2,0,m2+k2,q,t y,t x]Rxy[m1+k1-k2-m2,0,0,m1+k1-k2-m2,q,t z,t x]Rxy[m1+k1-k2-m2,m2+k2,m1+k1-m3,m3,q,t z,t y]Rxy[m1+k1-m3,m3,m1+k1-m4,m4,q,t y,t z]Rxy[m1+k1-m4,m4,m1,k1,q,t z,t y](((t y))^(1/2))(((t z))^(1/2))(q^(-1-m1-k1)),0],{k1,-cutoff,-1},{k2,-cutoff,k1},{m1,0,cutoff-1},{m2,0,Min[cutoff-1,-k2-1,m1+k1-k2]},{m3,0,cutoff-1},{m4,0,cutoff-1}],{t,∞,tdegcutoff}]//Simplify)];
Simplify[(%//Normal)/.{t->1,x->(X^2),y->(Y^2),z->(Z^2)},Assumptions->X>0&&Y>0&&Z>0]//Expand;
Table[Coefficient[%,(X^(-1))(Y^(-1-2i))(Z^(-1-2j))],{i,0,3},{j,0,3}]//MatrixForm
Table[Coefficient[%%,(X^(-3))(Y^(-1-2i))(Z^(-1-2j))],{i,0,3},{j,0,3}]//MatrixForm
Table[Coefficient[%%%,(X^(-5))(Y^(-1-2i))(Z^(-1-2j))],{i,0,3},{j,0,3}]//MatrixForm

With[{cutoff=3,tdegcutoff=8},-(((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[If[-m1-m3-m4-k1+2k2-1/2≥-tdegcutoff,Rxy[0,m1,m1,0,q,t x,t y]Rxyinv[0,k1,k1-k2,k2,q,t x,t z]Rxy[m1,k1-k2,m1+k1-k2-m2,m2,q,t y,t z]Rxyinv[m2,k2,0,m2+k2,q,t y,t x]Rxy[m1+k1-k2-m2,0,0,m1+k1-k2-m2,q,t z,t x]Rxy[m1+k1-k2-m2,m2+k2,m1+k1-m3,m3,q,t z,t y]Rxy[m1+k1-m3,m3,m1+k1-m4,m4,q,t y,t z]Rxy[m1+k1-m4,m4,m1,k1,q,t z,t y](((t y))^(1/2))(((t z))^(1/2))(q^(-1-m1-k1)),0],{k1,-cutoff,-1},{k2,-cutoff,k1},{m1,0,cutoff-1},{m2,0,Min[cutoff-1,-k2-1,m1+k1-k2]},{m3,0,cutoff-1},{m4,0,cutoff-1}],{t,∞,tdegcutoff}]//Simplify)];
Simplify[(%//Normal)/.{t->1,x->(X^2),y->(Y^2),z->(Z^2)},Assumptions->X>0&&Y>0&&Z>0]//Expand;
Table[Coefficient[%,(X^(-1))(Y^(-1-2i))(Z^(-1-2j))],{i,0,3},{j,0,3}]//MatrixForm
Table[Coefficient[%%,(X^(-3))(Y^(-1-2i))(Z^(-1-2j))],{i,0,3},{j,0,3}]//MatrixForm
Table[Coefficient[%%%,(X^(-5))(Y^(-1-2i))(Z^(-1-2j))],{i,0,3},{j,0,3}]//MatrixForm

As expected, this is symmetric in y and z. 
In the classical limit q=1, all the entries become -1, which is consistent with its Alexander-Conway polynomial. 
Note, +1 surgery on either y or z component gives the Whitehead link obtained by a -1 surgery on a component of the Borromean rings. 
What is more, -1 surgery on either y or z component gives L7n2, in the same orientation as the planar diagram in KnotAtlas.

-1 surgery on z-component = L7n2

This is essentially a framing change on x-component from the Whitehead link obtained by +1 surgery on a component of the Borromean rings. (See Example 4 in arXiv:2106.03942)

Note, the framing change on the x-component we need to make L7n2 into the Whitehead link is essentially (x^u)->(q^(u^2))(x^u), where u is a half-integer. The computation below is in perfect agreement.

With[{cutoff=3,tdegcutoff=8},-(((((t x))^(1/2))-(((t x))^(-1/2)))Series[Sum[If[-m1-m3-m4-k1+2k2-1/2≥-tdegcutoff,Rxy[0,m1,m1,0,q,t x,t y]Rxyinv[0,k1,k1-k2,k2,q,t x,t z]Rxy[m1,k1-k2,m1+k1-k2-m2,m2,q,t y,t z]Rxyinv[m2,k2,0,m2+k2,q,t y,t x]Rxy[m1+k1-k2-m2,0,0,m1+k1-k2-m2,q,t z,t x]Rxy[m1+k1-k2-m2,m2+k2,m1+k1-m3,m3,q,t z,t y]Rxy[m1+k1-m3,m3,m1+k1-m4,m4,q,t y,t z]Rxy[m1+k1-m4,m4,m1,k1,q,t z,t y](((t y))^(1/2))(((t z))^(1/2))(q^(-1-m1-k1)),0],{k1,-cutoff,-1},{k2,-cutoff,k1},{m1,0,cutoff-1},{m2,0,Min[cutoff-1,-k2-1,m1+k1-k2]},{m3,0,cutoff-1},{m4,0,cutoff-1}],{t,∞,tdegcutoff}]//Simplify)];
Simplify[(%//Normal)/.{t->1,x->(X^2),y->(Y^2),z->(Z^2)},Assumptions->X>0&&Y>0&&Z>0]//Expand;
(-%//Normal)/.{(Z^(u_))->((q^(((((u+1)/2)))^2))(Y^(2(u+1)))-(q^(((((u-1)/2)))^2))(Y^(2(u-1))))}//Expand;
Table[Coefficient[(q^(i(i+1)))%//Expand,(X^(-1-2i))(Y^(-1-2j))],{i,0,2},{j,0,2}]//MatrixForm