<<KnotTheory`

(* Calculations establising the main result for the paper "A remark on Khovanov homology and two-fold branched covers" *)

(* The branch sets associated to the Seifert involutions are given by torus knots *)

(* First check that rkKh(T(5,10 \pm 1))= 65 \pm 8 : *)

Kh[TorusKnot[5,9], Modulus->2][1,1]/2

Kh[TorusKnot[5,11], Modulus->2][1,1]/2

(* Next check that rkKh(T(5,20 \pm 1))= 257 \pm 16 : *)

Kh[TorusKnot[5,19], Modulus->2][1,1]/2

Kh[TorusKnot[5,21], Modulus->2][1,1]/2

(* The branch sets associated to the Montesinos involutions are given by the Montesinos knots in Figure 2 *)

(* These are entered by hand; use TauPlus1 for \tau(1), etc. *)

TauPlus1=PD[
X[2,6,3,5],X[4,9,5,10],X[6,2,7,1],X[7,23,8,22],X[10,3,11,4],X[12,25,13,26],X[14,27,15,28],X[16,29,17,30],X[18,31,19,32],X[20,33,21,34],X[21,1,22,34],X[23,9,24,8],X[24,11,25,12],X[26,13,27,14],X[28,15,29,16],X[30,17,31,18],X[32,19,33,20]
];

KnotDet[TauPlus1]

Kh[TauPlus1, Modulus->2][1,1]/2

TauMinus1=PD[
X[2,6,3,5],X[4,9,5,10],X[6,2,7,1],X[7,23,8,22],X[10,3,11,4],X[12,25,13,26],X[14,27,15,28],X[16,29,17,30],X[18,31,19,32],X[20,33,21,34],X[23,9,24,8],X[24,11,25,12],X[26,13,27,14],X[28,15,29,16],X[30,17,31,18],X[32,19,33,20],X[34,21,1,22]
];

KnotDet[TauMinus1]

Kh[TauMinus1, Modulus->2][1,1]/2

(* This verifies that rkKh(\tau(\pm 1)) = 16 \mp 1 *)

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

KnotDet[TauPlusHALF]

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

KnotDet[TauMinusHALF]

Kh[TauPlus1, Modulus->2][1,1]/2

Kh[TauMinus1, Modulus->2][1,1]/2

(* This verifies that rkKh(\tau(\pm 1)) = 16 \mp 1 *)

Kh[TauPlusHALF, Modulus->2][1,1]/2

Kh[TauMinusHALF, Modulus->2][1,1]/2

(* This verifies that rkKh(\tau(\pm \frac{1}{2})) = 32 \mp 1 *)