<<knottheory`

(*

This file produces some rough calculations used in the paper: 

"Khovanov homology and the symmetry group of a knot" 
<http://arxiv.org/abs/1311.1085>

All references to sections, etc. refer to this paper.

k will denote the invariant $\varkappa$ throughout
*)

(* Consider the trefoil in order to set conventions. *)

TrefoilTest=PD[
X[1,5,2,4],X[3,1,4,6],X[5,3,6,2]
];

KnotDet[TrefoilTest]

(* d represents twice the diagonal grading $\delta=u-(q/2)$ in Section 2. *)

(* bigraded reduced Khovanov homology over Z/2Z *)

Expand[PolynomialQuotient[Kh[TrefoilTest,Modulus->2][q,u],q^(-1)+q,q]]

(* u-graded, reduced Khovanov homology *)

Expand[Kh[TrefoilTest,Modulus->2][1,u]/2]

(* d-graded, reduced Khovanov homology *)

Expand[PolynomialQuotient[Kh[TrefoilTest,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]

(* comparing the torus knots T_(2,5) and T_(3,4) as in Section 4.1 *)

Torus25plus9=PD[
X[1,12,2,13],X[2,9,3,10],X[4,12,5,11],X[5,1,6,14],X[8,3,9,4],X[10,8,11,7],X[13,7,14,6]
];

(* This represents the branch set coresponding to +9 surgery *)

KnotDet[Torus25plus9]

(* This is a two-birge knot and has thin homology *)

Expand[PolynomialQuotient[Kh[Torus25plus9,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]

Expand[PolynomialQuotient[Kh[Torus25plus9,Modulus->2][q,u],q^(-1)+q,q]]

(* hence k has dimension at most 9 *)

Torus25plus7=PD[
X[1,14,2,15],X[3,16,4,17],X[4,11,5,12],X[6,14,7,13],X[7,1,8,18],X[10,5,11,6],X[12,10,13,9],X[15,2,16,3],X[17,9,18,8]
];

(* This represents the branch set coresponding to +7 surgery *)

KnotDet[Torus25plus7]

Expand[PolynomialQuotient[Kh[Torus25plus7,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]

Expand[PolynomialQuotient[Kh[Torus25plus7,Modulus->2][q,u],q^(-1)+q,q]]

1+(1/(q^2))+(1/((q^6) (u^2)))+(1/((q^4) u))+2 u+(q^2) (u^2)+(q^4) (u^3)+(q^6) (u^4)

(* kappa is of dimension 8 and consists of all but the element in graging (u,q)=(0,0) in the above homology

The following presents the u-graded vectors space k(T_(2,5)) (this is admitedly ad hoc) *)

Expand[Kh[Torus25plus7,Modulus->2][1,u]/2] - 1

Torus34plus7=PD[
X[1,10,2,11],X[3,12,4,13],X[5,14,6,15],
X[7,21,8,20],X[9,19,10,18],
X[11,2,12,3],X[13,4,14,5],
X[16,24,17,23],X[19,9,20,8],X[21,7,22,6],X[22,16,23,15],X[24,18,1,17]
];

KnotDet[Torus34plus7]

(* branch set for +7 surgery on T_(3,4), as above *)

Torus34plus9=PD[
X[1,10,2,11],X[3,12,4,13],
X[7,21,8,20],X[9,19,10,18],
X[11,2,12,3],X[13,4,14,5],X[14,6,15,5],
X[16,24,17,23],X[19,9,20,8],X[21,7,22,6],X[22,16,23,15],X[24,18,1,17]
];

KnotDet[Torus34plus9]

(* branch set for +9 surgery on T_(3,4), as above *)

Expand[PolynomialQuotient[Kh[Torus34plus9,Modulus->2][q,u],q^(-1)+q,q]]

Expand[PolynomialQuotient[Kh[Torus34plus9,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]

Expand[PolynomialQuotient[Kh[Torus34plus7,Modulus->2][q,u],q^(-1)+q,q]]

Expand[PolynomialQuotient[Kh[Torus34plus7,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]

(* Comparing these k(T_(3,4)) just drops the single off diagonal element, in this case (as in the previous case, and again, add hoc) *)

Expand[Kh[Torus34plus7,Modulus->2][1,u]/2] -1

(* Two inversions on the knot 9_9 (Section 4.2) *)

(* The inversion h_1 and h_2 will be denoted here A and B, respectively. *)

TApos=PD[
X[1,15,2,14],
X[3,41,4,40],X[4,12,5,11],X[6,27,7,28],X[8,23,9,24],X[10,20,11,19],X[13,31,14,30],
X[15,3,16,2],
X[17,39,18,38],X[18,10,19,9],X[21,17,22,16],X[24,7,25,8],X[25,36,26,37],X[28,5,29,6],X[29,34,30,35],X[32,42,33,41],X[33,13,34,12],X[35,26,36,27],X[37,22,38,23],X[39,21,40,20],X[42,32,1,31]
];
TAzero=PD[
X[1,15,2,14],X[2,15,3,16],
X[3,41,4,40],X[4,12,5,11],X[6,27,7,28],X[8,23,9,24],X[10,20,11,19],X[13,31,14,30],

X[17,39,18,38],X[18,10,19,9],X[21,17,22,16],X[24,7,25,8],X[25,36,26,37],X[28,5,29,6],X[29,34,30,35],X[32,42,33,41],X[33,13,34,12],X[35,26,36,27],X[37,22,38,23],X[39,21,40,20],X[42,32,1,31]
];

TAneg=PD[
X[2,15,3,16],
X[3,41,4,40],X[4,12,5,11],X[6,27,7,28],X[8,23,9,24],X[10,20,11,19],X[13,31,14,30],
X[14,1,15,2],
X[17,39,18,38],X[18,10,19,9],X[21,17,22,16],X[24,7,25,8],X[25,36,26,37],X[28,5,29,6],X[29,34,30,35],X[32,42,33,41],X[33,13,34,12],X[35,26,36,27],X[37,22,38,23],X[39,21,40,20],X[42,32,1,31]
];

KnotDet[TAneg]
KnotDet[TAzero]
KnotDet[TApos]

(*
 From this we confirm: 
	TAneg <-> branch set for -17 surgery
TAzero <-> branch set for -15 surgery
TApos <-> branch set for -13 surgery
*)

Expand[PolynomialQuotient[Kh[TApos,Modulus->2][q,u],q^(-1)+q,q]]
Expand[PolynomialQuotient[Kh[TAzero,Modulus->2][q,u],q^(-1)+q,q]]
Expand[PolynomialQuotient[Kh[TAneg,Modulus->2][q,u],q^(-1)+q,q]]

Expand[PolynomialQuotient[Kh[TApos,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]
Expand[PolynomialQuotient[Kh[TAzero,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]
Expand[PolynomialQuotient[Kh[TAneg,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]

(* There is still some ambiguity here with the right-most diagonal... we need to calculate the homlogy of the branch set associated with -7 surgery (say) resulting from 3 positive full-twists (6 half-twsits) added to TApos *)

TAminus7 = PD[
X[1,21,2,20],X[3,23,4,22],X[5,25,6,24],X[7,27,8,26],

X[9,53,10,52],X[10,18,11,17],X[12,39,13,40],X[14,35,15,36],X[16,32,17,31],X[19,43,20,42],

X[21,3,22,2],X[23,5,24,4],X[25,7,26,6],X[27,9,28,8],

X[29,51,30,50],X[30,16,31,15],X[33,29,34,28],X[36,13,37,14],X[37,48,38,49],X[40,11,41,12],X[41,46,42,47],X[44,54,45,53],X[45,19,46,18],X[47,38,48,39],X[49,34,50,35],X[51,33,52,32],X[54,44,1,43]
 ];

(* check that nothing went wrong... *)

KnotDet[TAminus7]

(* Looks ok *)

Expand[PolynomialQuotient[Kh[TAminus7,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]

(* This new diagonal was expected. Things should be stable now. *)

Expand[PolynomialQuotient[Kh[TAminus7,Modulus->2][q,u],q^(-1)+q,q]]

(* in fact, K(9_9,h_1) results from dropping this new diagonal. *)

Expand[Kh[TAminus7,Modulus->2][1,u]/2] -u^9-u^8-u^7-u^6-u^4

(* Now the second inversion *)

TBpos=PD[
X[1,19,2,18],
X[5,12,6,13],X[6,45,7,46],X[7,21,8,20],X[9,28,10,29],X[13,4,14,5],X[14,37,15,38],X[15,31,16,30],X[17,35,18,34],
X[19,3,20,2],
X[22,42,23,41],X[23,9,24,8],X[25,10,26,11],X[26,43,27,44],X[29,25,30,24],X[32,50,33,49],X[33,17,34,16],X[38,11,39,12],X[39,44,40,45],X[40,22,41,21],X[42,27,43,28],X[46,3,47,4],X[47,36,48,37],X[48,32,49,31],X[50,36,1,35]
];
TBzero=PD[
X[1,19,2,18],X[2,19,3,20],
X[5,12,6,13],X[6,45,7,46],X[7,21,8,20],X[9,28,10,29],X[13,4,14,5],X[14,37,15,38],X[15,31,16,30],X[17,35,18,34],

X[22,42,23,41],X[23,9,24,8],X[25,10,26,11],X[26,43,27,44],X[29,25,30,24],X[32,50,33,49],X[33,17,34,16],X[38,11,39,12],X[39,44,40,45],X[40,22,41,21],X[42,27,43,28],X[46,3,47,4],X[47,36,48,37],X[48,32,49,31],X[50,36,1,35]
];
TBneg=PD[
X[2,19,3,20],
X[5,12,6,13],X[6,45,7,46],X[7,21,8,20],X[9,28,10,29],X[13,4,14,5],X[14,37,15,38],X[15,31,16,30],X[17,35,18,34],
X[18,1,19,2],
X[22,42,23,41],X[23,9,24,8],X[25,10,26,11],X[26,43,27,44],X[29,25,30,24],X[32,50,33,49],X[33,17,34,16],X[38,11,39,12],X[39,44,40,45],X[40,22,41,21],X[42,27,43,28],X[46,3,47,4],X[47,36,48,37],X[48,32,49,31],X[50,36,1,35]
];

KnotDet[TBpos]
KnotDet[TBzero]
KnotDet[TBneg]

(*
 From this we confirm: 
	TBneg <-> branch set for -19 surgery
TBzero <-> branch set for -17 surgery
TBpos <-> branch set for -15 surgery
*)

(* We'll need -9 surgery too... *)

TBminus9 = PD[
X[1,25,2,24],X[3,27,4,26],X[5,29,6,28],X[7,31,8,30],

X[11,18,12,19],X[12,57,13,58],X[13,33,14,32],X[15,40,16,41],X[19,10,20,11],X[20,49,21,50],X[21,43,22,42],X[23,47,24,46],

X[25,3,26,2],X[27,5,28,4],X[29,7,30,6],X[31,9,32,8],

X[34,54,35,53],X[35,15,36,14],X[37,16,38,17],X[38,55,39,56],X[41,37,42,36],X[44,62,45,61],X[45,23,46,22],X[50,17,51,18],X[51,56,52,57],X[52,34,53,33],X[54,39,55,40],X[58,9,59,10],X[59,48,60,49],X[60,44,61,43],X[62,48,1,47]
];

(* check... *)

KnotDet[TBminus9]

(* Now compare the homomolgies for the -15 and -9 branch sets *)

Expand[PolynomialQuotient[Kh[TBpos,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]

Expand[PolynomialQuotient[Kh[TBminus9,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]

Expand[PolynomialQuotient[Kh[TBpos,Modulus->2][q,u],q^(-1)+q,q]]

Expand[PolynomialQuotient[Kh[TBminus9,Modulus->2][q,u],q^(-1)+q,q]]

(* k(9_9, h_2) *)

Expand[Kh[TBminus9,Modulus->2][1,u]/2] -u^7-u^6-u^4

(* Non-Amphicheirality (Section 5) *)

(* warmup example: the knot 8_20 *)

Br820plus=PD[
X[1,7,2,6],X[2,12,3,11], X[5,16,6,17],X[7,20,8,21],
X[9,23,10,22],X[10,4,11,3],X[13,8,14,9],X[14,20,15,19],X[17,4,18,5],X[18,23,19,24],X[21,13,22,12],X[24,15,1,16]
];

Br820minus=PD[

X[2,12,3,11], X[5,16,6,17],
X[6,1,7,2],
X[7,20,8,21],X[9,23,10,22],X[10,4,11,3],X[13,8,14,9],X[14,20,15,19],X[17,4,18,5],X[18,23,19,24],X[21,13,22,12],X[24,15,1,16]
];

(* 

Branch sets for +1 and -1 surgery on 8_20, respectively. 
Note that these happen to be the knots 11n49 and 11n116, respectively. 
*)

KnotDet[Br820plus]
KnotDet[Br820minus]

Expand[PolynomialQuotient[Kh[Br820minus,Modulus->2][q,u],q^(-1)+q,q]]

Expand[PolynomialQuotient[Kh[Br820plus,Modulus->2][q,u],q^(-1)+q,q]]

(* Both homologies are supported in two diagonals *)

Expand[PolynomialQuotient[Kh[Br820minus,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]

Expand[PolynomialQuotient[Kh[Br820plus,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]

9+(8/(d^2))

(* analysing/comparing the (u,d)-graded vector spaces above it must be that the branch sets for surgeries larger than +1 stabilize -- that is, the connecting homomorphisms computing +n from +n-1 always vanish for grading reasons *)

(* k(8_20,h) is described *)

Expand[Kh[Br820minus,Modulus->2][1,u]/2] -1

Expand[Kh[Br820plus,Modulus->2][1,u]/2] -1

(* while k(8_20^*,h) is described *)

Expand[Kh[Br820plus,Modulus->2][1,u^-1]/2] -1

(* The knot 10_48 *)

Br1048plus=PD[
X[1,31,2,30],
X[2,11,3,12],X[5,19,6,18],X[6,26,7,25],X[8,29,9,30],X[9,38,10,1],X[12,3,13,4],X[13,32,14,33],X[14,22,15,21],X[16,34,17,33],X[17,5,18,4],X[20,16,21,15],X[23,37,24,36],X[24,8,25,7],X[27,22,28,23],

X[31,10,32,11],X[34,20,35,19],X[35,27,36,26],X[37,28,38,29]
];

Br1048minus=PD[

X[2,11,3,12],X[5,19,6,18],X[6,26,7,25],X[8,29,9,30],X[9,38,10,1],X[12,3,13,4],X[13,32,14,33],X[14,22,15,21],X[16,34,17,33],X[17,5,18,4],X[20,16,21,15],X[23,37,24,36],X[24,8,25,7],X[27,22,28,23],
X[30,1,31,2],
X[31,10,32,11],X[34,20,35,19],X[35,27,36,26],X[37,28,38,29]
];

KnotDet[Br1048plus]
KnotDet[Br1048minus]

(* Conclude: Br1048plus is the branch set for +1 surgery on 10_48; Br1048minus is the branch set for -1 surgery on 10_48;*)

Expand[PolynomialQuotient[Kh[Br1048minus,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]
Expand[PolynomialQuotient[Kh[Br1048plus,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]

Expand[PolynomialQuotient[Kh[Br1048plus,Modulus->2][q,u],q^(-1)+q,q]]

(* We'll need +5 surgery as well... *)

Br1048plus5 = PD[
X[1,35,2,34],X[3,37,4,36],X[5,39,6,38],

X[6,15,7,16],X[9,23,10,22],X[10,30,11,29],X[12,33,13,34],X[13,46,14,1],X[16,7,17,8],X[17,40,18,41],X[18,26,19,25],X[20,42,21,41],X[21,9,22,8],X[24,20,25,19],X[27,45,28,44],X[28,12,29,11],X[31,26,32,27],

X[35,3,36,2],X[37,5,38,4],

X[39,14,40,15],X[42,24,43,23],X[43,31,44,30],X[45,32,46,33]
];

(* check... *)

KnotDet[Br1048plus5]

Expand[PolynomialQuotient[Kh[Br1048plus5,Modulus->2][q,u],q^(-1)+q,q]]
Expand[PolynomialQuotient[Kh[Br1048plus5,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]

Expand[Kh[Br1048plus5,Modulus->2][1,u]/2]-u^5-u^4-u^3-u^2-1

(* Conjecture 28 *)

(10+1+3+4+5+8+10+11 +11+9+8 +7 +4 +2 +2 +1)/4

(* The knot 10_71 *)

Br1071plus=PD[
X[1,37,2,36],
X[2,15,3,16],X[4,26,5,25],X[6,22,7,21],X[8,33,9,34],X[10,29,11,30],X[12,35,13,36],X[13,46,14,1],X[16,3,17,4],X[17,38,18,39],X[19,43,20,42],X[20,8,21,7],X[23,41,24,40],X[24,6,25,5],X[27,18,28,19],X[30,9,31,10],X[31,44,32,45],X[34,11,35,12],

X[37,14,38,15],X[39,27,40,26],X[41,23,42,22],X[43,32,44,33],X[45,28,46,29]
];
Br1071minus=PD[

X[2,15,3,16],X[4,26,5,25],X[6,22,7,21],X[8,33,9,34],X[10,29,11,30],X[12,35,13,36],X[13,46,14,1],X[16,3,17,4],X[17,38,18,39],X[19,43,20,42],X[20,8,21,7],X[23,41,24,40],X[24,6,25,5],X[27,18,28,19],X[30,9,31,10],X[31,44,32,45],X[34,11,35,12],
X[36,1,37,2],
X[37,14,38,15],X[39,27,40,26],X[41,23,42,22],X[43,32,44,33],X[45,28,46,29]
];

KnotDet[Br1071minus]
KnotDet[Br1071plus]

Expand[PolynomialQuotient[Kh[Br1071minus,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]
Expand[PolynomialQuotient[Kh[Br1071plus,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]

Expand[PolynomialQuotient[Kh[Br1071minus,Modulus->2][q,u],q^(-1)+q,q]]
Expand[PolynomialQuotient[Kh[Br1071plus,Modulus->2][q,u],q^(-1)+q,q]]

Expand[-Expand[PolynomialQuotient[Kh[Br1071minus,Modulus->2][q,u],q^(-1)+q,q]]
+q^-2*Expand[PolynomialQuotient[Kh[Br1071plus,Modulus->2][q,u],q^(-1)+q,q]]]

(*
Br1071minus <-> +5 surgery 
Br1071plus <-> +7 surgery 
*)

(* Look at the branch set for -1 surgery (say) *)

(* Br1071plus1=PD[
X[7,28,8,29],X[9,30,10,31],X[11,32,12,33],

X[12,45,13,46],X[14,58,15,57],X[16,54,17,53],X[18,63,19,64],X[20,59,21,60],

X[22,2,23,1],X[23,2,24,3],X[24,4,25,3],X[25,4,26,5],X[26,6,27,5],X[27,6,28,7],X[29,8,30,9],X[31,10,32,11],

X[33,46,34,47],X[35,57,36,56],X[37,53,38,52],X[39,64,40,65],X[41,60,42,61],X[43,66,44,1],X[44,21,45,22],X[47,34,48,35],X[48,13,49,14],X[50,18,51,17],X[51,39,52,38],X[54,16,55,15],X[55,37,56,36],X[58,49,59,50],X[61,40,62,41],X[62,19,63,20],X[65,43,66,42]
];*)
Br1071plus1=PD[
X[3,20,4,21],X[5,22,6,23],X[7,24,8,25],

X[8,37,9,38],X[10,50,11,49],X[12,46,13,45],X[14,55,15,56],X[16,51,17,52],

X[18,2,19,1],X[19,2,20,3],X[21,4,22,5],X[23,6,24,7],

X[25,38,26,39],X[27,49,28,48],X[29,45,30,44],X[31,56,32,57],X[33,52,34,53],X[35,58,36,1],X[36,17,37,18],X[39,26,40,27],X[40,9,41,10],X[42,14,43,13],X[43,31,44,30],X[46,12,47,11],X[47,29,48,28],X[50,41,51,42],X[53,32,54,33],X[54,15,55,16],X[57,34,58,35]
];
KnotDet[Br1071plus1]

Jones[Br1071plus1][t]

Expand[PolynomialQuotient[Kh[Br1071plus1,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]
Expand[PolynomialQuotient[Kh[Br1071plus1,Modulus->2][q,u],q^(-1)+q,q]]

Expand[Kh[Br1071plus1,Modulus->2][1,u]/2]-u^-8

(* Conjecture 28 *)
(11+1+2+2+3+6+8+7+10+12+12+12+13+12+10+9+6 +5 +5 +3 +1+1+1)/4

(* The knot 10_104 *)

Br10104plus=PD[
X[1,23,2,22],
X[4,40,5,39],X[5,17,6,16],X[8,42,9,41],X[9,19,10,18],X[12,27,13,28],X[14,30,15,29],X[15,7,16,6],X[17,3,18,2],X[20,34,21,33],X[21,11,22,10],

X[24,13,25,14],X[25,36,26,37],X[28,24,29,23],X[31,43,32,42],X[32,20,33,19],X[35,26,36,27],X[37,31,38,30],X[38,8,39,7],X[40,4,41,3],X[43,35,44,34],X[44,12,1,11]
];
KnotDet[Br10104plus]

(* -1 surgery *)

Br10104minus=PD[

X[4,40,5,39],X[5,17,6,16],X[8,42,9,41],X[9,19,10,18],X[12,27,13,28],X[14,30,15,29],X[15,7,16,6],X[17,3,18,2],X[20,34,21,33],X[21,11,22,10],
X[22,1,23,2],
X[24,13,25,14],X[25,36,26,37],X[28,24,29,23],X[31,43,32,42],X[32,20,33,19],X[35,26,36,27],X[37,31,38,30],X[38,8,39,7],X[40,4,41,3],X[43,35,44,34],X[44,12,1,11]
];
KnotDet[Br10104minus]

(* -3 surgery *)

Expand[PolynomialQuotient[Kh[Br10104minus,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]
Expand[PolynomialQuotient[Kh[Br10104plus,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]

Br10104minus7=PD[
X[1,28,2,29],X[3,30,4,31],X[5,32,6,33],

X[7,44,8,45],X[8,19,9,20],X[11,7,12,6],X[14,26,15,25],X[15,51,16,50],X[18,9,19,10],X[20,14,21,13],X[21,39,22,38],X[23,35,24,34],X[26,18,27,17],X[27,43,28,42],

X[29,2,30,3],X[31,4,32,5],

X[35,23,36,22],X[36,48,37,47],X[39,25,40,24],X[40,50,41,49],X[43,10,44,11],X[45,13,46,12],X[46,38,47,37],X[48,34,49,33],X[51,17,52,16],X[52,42,1,41]
];
KnotDet[Br10104minus7]

(* -7 surgery *)

Expand[PolynomialQuotient[Kh[Br10104minus7,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]
Expand[PolynomialQuotient[Kh[Br10104minus,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]
Expand[PolynomialQuotient[Kh[Br10104plus,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]

Expand[PolynomialQuotient[Kh[Br10104minus7,Modulus->2][q,u],q^(-1)+q,q]]

Br10104plus3=PD[
X[2,30,3,29],X[4,32,5,31],

X[7,44,8,45],X[8,19,9,20],X[11,7,12,6],X[14,26,15,25],X[15,51,16,50],X[18,9,19,10],X[20,14,21,13],X[21,39,22,38],X[23,35,24,34],X[26,18,27,17],X[27,43,28,42],

X[28,2,29,1],X[30,4,31,3],X[32,6,33,5],

X[35,23,36,22],X[36,48,37,47],X[39,25,40,24],X[40,50,41,49],X[43,10,44,11],X[45,13,46,12],X[46,38,47,37],X[48,34,49,33],X[51,17,52,16],X[52,42,1,41]
];
KnotDet[Br10104plus3]

Expand[PolynomialQuotient[Kh[Br10104plus3,Modulus->2][d^(-1),d^2],d^(-1)+d,d]]
Expand[PolynomialQuotient[Kh[Br10104plus3,Modulus->2][q,u],q^(-1)+q,q]]

Expand[Kh[Br10104plus3,Modulus->2][1,u]/2]-u^11-u^10-u^8

(* Conjecture 28 *)

(5+1+1+1+3+4 +7 +10 +10 +11 +14 +14 +12 +14 +12+9 +8 +7 +4+2 +2 +1)/4