Computing generators of the KB ideal of the
genus-1 tangle F
The Idea:
Since the g_{i,e}’s are a basis, we have that F is a linear combination of the g_{i,e}’s. We let Coeff[i,e] denote the coefficient of
g_{i,e} in this linear combination (Coeff[i,e] is denoted by c_{i,e} in the paper).
Since the graph basis is orthogonal with respect to the doubling pairing < , >_D, we have that Coeff[i,e] = <F, g_{i,e}>_D/
<g_{i,e}, g_{i,e}>_D. To compute Coeff[i,e], we first compute <F, g_{i,e}>_D (denoted by Dpairing[i,e] in this notebook) and
then divide by <g_{i,e}, g_{i,e}>_D.
A formula for <g_{i,e}, g_{i,e}>_D is given in Theorem 2.4 of the paper.
The explicit computation of the formula for we use for Dpairing[i,e] is given in Appendix A of the paper. We have from the
second line of this computation, using admissibility, that Dpairing[i,e] = 0 unless i = 1 or i = 3. So in this notebook, we need
only compute Coeff[1,0], Coeff[1,2], Coeff[3,2], and Coeff[3,4]. Explicit formulas for these are given below.
Preliminaries- Banded colored TrivalentGraph coefficients
Most of the code in this section (all except the last four commands) was written by John M. Harris. It is available in Appendix
B of his dissertation, which can be found at http://etd.lsu.edu/docs/available/etd - 0701103 - 164728/. We greatly appreciate
his allowing us to use it.
oddq and evenq evaluate whether a product or sum of two integers is odd or even, respectively.
oddq@a_ b_ ; oddq@aD && oddq@bDD := True;
oddq@a_ + b_ ; Hoddq@aD && evenq@bDL ÈÈ Hevenq@aD && oddq@bDLD := True;
oddq@a_D := OddQ@aD;
evenq@a_ b_ ; Hevenq@aD && IntegerQ@bDL ÈÈ Hevenq@bD && IntegerQ@aDLD := True;
evenq@a_ + b_ ; Hevenq@aD && evenq@bDL ÈÈ Hoddq@aD && oddq@bDLD := True;
evenq@a_D := EvenQ@aD;
qi[n] is the nth quantum integer; qif[n] is qi[n]! (factorial). Quantum integers and their factorials are left unevaluated.
delta[n] is the nth Chebyshev polynomial.
adm[a1,b1,c1] returns “ True”if the triple (a1,b1,c1) is admissible, and “ False”if the triple is not admissible.
Lambda[a,b,c] is the coefficient that comes from removing a twist as in Formula 2.2 in the paper.
theta[a,b,c] is the evaluation of a theta graph labelled (a,b,c).
(Lambdas, thetas, and tets are evaluated as in Kauffman-Lins.)
2
computinggeneratorsforKBidealofF.nb
qi@0D = 0; qi@1D = 1;
qif@0D = 1; qif@n_ ; n ³ 1D := qif@n - 1D qi@nD;
qif@n_ + x_ ; n ³ 1D := qif@n + x - 1D qi@n + xD;
qi@n_D := Sum@A ^ i, 8i, 2 - 2 n, 2 n - 2, 4<D;
delta@n_D := H- 1L ^ n qi@n + 1D ;
adm@a1_, b1_, c1_D := Module@8a = Simplify@a1D, b = Simplify@b1D, c = Simplify@c1D<,
Simplify@a ³ 0 && b ³ 0 && c ³ 0 && Abs@a - bD £ c && c £ a + b, givenD && evenq@a + b + cDD;
lambda@a_, b_, c_D :=
H- 1L ^ HHa + b - cL  2L A ^ HHa Ha + 2L + b Hb + 2L - c Hc + 2LL  2L  Simplify  Expand;
theta@a_, b_, c_D := Module@8
m = Ha + b - cL  2  Simplify,
n = Hb + c - aL  2  Simplify,
p = Ha + c - bL  2  Simplify
<,
If@adm@a, b, cD,
H- 1L ^ Hm + n + pL qif@m + n + p + 1D qif@mD
qif@nD qif@pD  qif@m + nD  qif@n + pD  qif@m + pD  Simplify,
0D
D;
admtet[a,b,c,d,e,f] tests to see if (a,b,c,d,e,f) is an admissible coloring of a tet graph. The colors correspond to the labelling of
the tet graph pictured in Section 2 of the paper.
tet[a,b,c,d,e,f] is the evaluation of a tet graph colored by (a,b,c,d,e,f) as in the paper. In the notation of the paper,
tet[a,b,c,d,e,f] corresponds to the Tet coefficient with first row a, b, e and second row c, d, f.
computinggeneratorsforKBidealofF.nb
3
admtet@a_, b_, c_, d_, e_, f_D := adm@a, d, eD && adm@b, c, eD && adm@a, b, fD && adm@c, d, fD;
tet@a_, b_, c_, d_, e_, f_D := Module@8
a1 = Ha + d + eL  2  Simplify,
a2 = Hb + c + eL  2  Simplify,
a3 = Ha + b + fL  2  Simplify,
a4 = Hc + d + fL  2  Simplify,
av,
b1 = Hb + d + e + fL  2  Simplify,
b2 = Ha + c + e + fL  2  Simplify,
b3 = Ha + b + c + dL  2  Simplify,
bv,
m, M, cv, s
<,
av = 8a1, a2, a3, a4<; bv = 8b1, b2, b3<;
m = Max@a1, a2, a3, a4D; M = Min@b1, b2, b3D;
If@admtet@a, b, c, d, e, fD,
intfac = Product@qif@bv@@jDD - av@@iDDD, 8i, 1, 4<, 8j, 1, 3<D;
extfac = qif@aD qif@bD qif@cD qif@dD qif@eD qif@fD;
cv = Intersection@av, bvD;
Hintfac  extfacL If@Length@cvD > 0, s = cv@@1DD; H- 1L ^ s
qif@s + 1D  Product@qif@s - av@@iDDD, 8i, 1, 4<D  Product@qif@bv@@jDD - sD, 8j, 1, 3<D,
Sum@H- 1L ^ s qif@s + 1D  Product@qif@s - av@@iDDD, 8i, 1, 4<D 
Product@qif@bv@@jDD - sD, 8j, 1, 3<D, 8s, m, M<DD  Simplify,
0D
D;
Fusion[c] is the coefficient that comes from performing fusion on two strands both colored 1, where c is the new color you
get from fusion. (Theorem 2.2 in the paper)
Fusion@c_D := If@adm@1, 1, cD, delta@cD  theta@1, 1, cD, 0D
TetR[a,b,c,d,e,f] is the coefficient that comes from reducing a tetrahedron in a graph, where a, d, and e are the colors of the
edges leading into the tet. (Formula 2.3 in the paper)
TetR@a_, b_, c_, d_, e_, f_D :=
If@admtet@a, b, c, d, e, fD, tet@a, b, c, d, e, fD  theta@a, d, eD, 0D
SixJ[a,b,c,d,e,f] is the 6j-coefficient that comes from changing an “ H”in a graph to an “ I.”(Formula 2.4 in the paper)
SixJ@a_, b_, c_, d_, e_, f_D :=
If@admtet@a, b, c, d, e, fD, tet@a, b, c, d, e, fD delta@eD  theta@a, d, eD  theta@b, c, eD, 0D
Phi[i] is the coefficient that comes from removing a loop from a strand colored i.
Phi@i_D := - A ^ H2 i + 2L - A ^ H- 2 i - 2L
4
computinggeneratorsforKBidealofF.nb
Formulas for the Dpairing[i,e] and coefficientsof the linear combination
Dpairing[i,e] gives the evaluation of the doubling pairing of the genus-1 tangle F with the graph basis element g_{i,e}. (See
Appendix A of the paper for computation of this formula.)
Dpairing@i_, e_D := Sum@
If@adm@1, i, jD && adm@1, e, rD && adm@q, r, iD, Hdelta@jD  theta@1, 1, jD  theta@1, i, jDL *
lambda@1, 1, jD ^ 2 Fusion@kD TetR@j, i, 1, k, e, 1D lambda@1, 1, kD ^ - 3 Fusion@lD
lambda@1, 1, lD ^ - 2 TetR@i, j, 1, l, 1, 1D Fusion@mD lambda@1, 1, mD TetR@1, l, 1, m, i, 1D
Fusion@nD lambda@1, 1, nD ^ - 1 TetR@1, 1, 1, i, n, mD Fusion@pD lambda@1, 1, pD ^ - 1
Fusion@qD lambda@1, 1, qD ^ - 1 TetR@n, 1, 1, q, p, 1D SixJ@q, 1, e, i, r, 1D
TetR@1, n, q, r, p, iD TetR@1, k, e, r, j, 1D tet@1, p, 1, j, 1, rD, 0D,
8j, 0, 2, 2<, 8k, 0, 2, 2<, 8l, 0, 2, 2<, 8m, 0, 2, 2<, 8n, 0, 2, 2<,
8p, 0, 2, 2<, 8q, 0, 2, 2<, 8r, i - 2, i + 2<D  Simplify
Coeff[i,e] gives the coefficient of g_{i,e} in the linear combination. Coeff[i,e] = <F, g_{i,e}>_D/ <g_{i,e}, g_{i,e}>_D. (See
Theorem 2.4 for the formula for <g_{i,e}, g_{i,e}>_D.)
Coeff@i_, e_D := Dpairing@i, eD  Htheta@i, 1, eD ^ 2  delta@iD  delta@eDL  Simplify
Computingcoefficientsof the linear combination
Coeff@1, 0D
0
Coeff@1, 2D
I- 1 + 2 A 4 - 4 A 8 + 4 A 12 - 3 A 16 + 2 A 20 + A 24 - 4 A 28 + 4 A 32 - 4 A 36 + 2 A 40 - A 44 M ‘ IA 21 I1 + A 4 + A 8 MM
Coeff@3, 2D
1 - A 4 + A 8 - A 16 + A 20
A 7 + A 11 + A 15 + A 19
Coeff@3, 4D
A3
Computinggeneratorsof the KB ideal
Generators are: <F,x_i>/delta = <Coeff[1,2] g_{1,2} + Coeff[3,2] g_{3,2} + Coeff[3,4] g_{3,4},x_i>/delta =
(Coeff[1,2]<g_{1,2},x_i> + Coeff[3,2]<g_{3,2},x_i> + Coeff[3,4]<g_{3,4},x_i>)/delta; and <F,y_i>/delta. Here, i is at most 3, and
the pairing is the relative Hopf pairing. Formulas for <g_{j,e},x_i> and <g_{j,e},y_i> are given in Proposition 4.2 in the paper.
The generators are rescaled by the minimum power of A necessary to make the lowest degree term a constant. Before
rescaling, they’re labelled as follows:
g1hat = <F,x_0>/delta
g2hat = <F,x_1>/delta
g3hat = <F,x_2>/delta
g4hat = <F,x_3>/delta
Generators are: <F,x_i>/delta = <Coeff[1,2] g_{1,2} + Coeff[3,2] g_{3,2} + Coeff[3,4] g_{3,4},x_i>/delta =
computinggeneratorsforKBidealofF.nb
5
(Coeff[1,2]<g_{1,2},x_i> + Coeff[3,2]<g_{3,2},x_i> + Coeff[3,4]<g_{3,4},x_i>)/delta; and <F,y_i>/delta. Here, i is at most 3, and
the pairing is the relative Hopf pairing. Formulas for <g_{j,e},x_i> and <g_{j,e},y_i> are given in Proposition 4.2 in the paper.
The generators are rescaled by the minimum power of A necessary to make the lowest degree term a constant. Before
rescaling, they’re labelled as follows:
g1hat = <F,x_0>/delta
g2hat = <F,x_1>/delta
g3hat = <F,x_2>/delta
g4hat = <F,x_3>/delta
g5hat = <F,y_0>/delta
g6hat = <F,y_1>/delta
g7hat = <F,y_2>/delta
g8hat = <F,y_3>/delta
After rescaling, they’re labelled g1,...,g8.
del := - A ^ 2 - A ^ - 2
g1hat =
HCoeff@1, 2D theta@1, 2, 1D + Coeff@3, 2D theta@1, 2, 3D + Coeff@3, 4D theta@1, 4, 3DL  del 
Simplify  Expand
1
3
-
A 23
7
+
A 19
-
A 15
10
+
A 11
12
-
A7
14
+ 12 A - 10 A 5 + 6 A 9 - 3 A 13 + A 17
A3
g1 = g1hat * A ^ 23  Simplify
1 - 3 A 4 + 7 A 8 - 10 A 12 + 12 A 16 - 14 A 20 + 12 A 24 - 10 A 28 + 6 A 32 - 3 A 36 + A 40
g2hat =
HCoeff@1, 2D theta@1, 2, 1D HPhi@1D - Phi@0DL + Coeff@3, 2D theta@1, 2, 3D HPhi@3D - Phi@0DL +
Coeff@3, 4D theta@1, 4, 3D HPhi@3D - Phi@0DLL  del  Simplify  Expand
-
1
1
+
3
+
2
-
9
-
4
+
+
16
3
-
-
23
+
2
+
30
-
2
A 27 A 25 A 23 A 21 A 19 A 17 A 15 A 13 A 11 A 9 A 7 A 5
31 2
- + 30 A + 2 A 3 - 24 A 5 - 4 A 7 + 17 A 9 + 3 A 11 - 9 A 13 - 2 A 15 + 4 A 17 + A 19 - A 21
3
A
A
g2 = g2hat * A ^ 27  Simplify  Expand
- 1 + A 2 + 3 A 4 - 2 A 6 - 9 A 8 + 4 A 10 + 16 A 12 - 3 A 14 - 23 A 16 + 2 A 18 + 30 A 20 - 2 A 22 31 A 24 - 2 A 26 + 30 A 28 + 2 A 30 - 24 A 32 - 4 A 34 + 17 A 36 + 3 A 38 - 9 A 40 - 2 A 42 + 4 A 44 + A 46 - A 48
g3hat =
HCoeff@3, 2D theta@1, 2, 3D HPhi@3D - Phi@0DL HPhi@3D - Phi@1DL + Coeff@3, 4D theta@1, 4, 3D
HPhi@3D - Phi@0DL HPhi@3D - Phi@1DLL  del  Simplify  Expand
1
-
A 27
10
A3
3
-
A 23
-
1
1
A 21
+
4
A 19
+
2
A 17
-
6
A 15
-
1
A 13
8
+
A 11
+
2
A9
-
9
A7
-
1
+
A5
- 10 A + 9 A 5 - 2 A 7 - 7 A 9 + 2 A 11 + 6 A 13 - A 15 - 4 A 17 + A 19 + 2 A 21 - A 25
A
g3 = g3hat * A ^ 27  Simplify  Expand
1 - 3 A 4 - A 6 + 4 A 8 + 2 A 10 - 6 A 12 - A 14 + 8 A 16 + 2 A 18 - 9 A 20 - A 22 + 10 A 24 A 26 - 10 A 28 + 9 A 32 - 2 A 34 - 7 A 36 + 2 A 38 + 6 A 40 - A 42 - 4 A 44 + A 46 + 2 A 48 - A 52
6
computinggeneratorsforKBidealofF.nb
g4hat = HCoeff@3, 2D theta@1, 2, 3D HPhi@3D - Phi@0DL HPhi@3D - Phi@1DL HPhi@3D - Phi@2DL +
Coeff@3, 4D theta@1, 4, 3D HPhi@3D - Phi@0DL HPhi@3D - Phi@1DL
HPhi@3D - Phi@2DLL  del  Simplify  Expand
-
1
1
+
3
2
-
5
-
2
+
8
+
4
-
-
10
3
+
+
13
A 31 A 29 A 27 A 25 A 23 A 21 A 19 A 17 A 15
3
14
1
15
+
+
+ 13 A + 2 A 3 - 15 A 5 - 2 A 7 + 13 A 9 + 4 A 11 13
11
9
7
5
A
A
A
A
A
A3
12 A 13 - 3 A 15 + 10 A 17 + 3 A 19 - 7 A 21 - 3 A 23 + 5 A 25 + A 27 - 2 A 29 - A 31 + A 33
A 35
3
+
A 33
13
g4 = g4hat * A ^ 35  Simplify  Expand
- 1 + A 2 + 3 A 4 - 2 A 6 - 5 A 8 + 2 A 10 + 8 A 12 - 4 A 14 - 10 A 16 + 3 A 18 + 13 A 20 3 A 22 - 13 A 24 + 3 A 26 + 14 A 28 - A 30 - 15 A 32 + 13 A 36 + 2 A 38 - 15 A 40 - 2 A 42 + 13 A 44 +
4 A 46 - 12 A 48 - 3 A 50 + 10 A 52 + 3 A 54 - 7 A 56 - 3 A 58 + 5 A 60 + A 62 - 2 A 64 - A 66 + A 68
g5hat = HCoeff@1, 2D lambda@1, 1, 2D ^ - 2 theta@1, 2, 1D +
Coeff@3, 2D lambda@1, 3, 2D ^ - 1 lambda@3, 1, 2D ^ - 1 theta@1, 2, 3D + Coeff@3, 4D
lambda@1, 3, 4D ^ - 1 lambda@3, 1, 4D ^ - 1 theta@1, 4, 3DL  del  Simplify  Expand
2
5
-
A 21
+
A 17
10
-
A 13
14
+
A9
16
-
17
A5
+ 15 A 3 - 12 A 7 + 7 A 11 - 4 A 15 + A 19
A
g5 = g5hat * A ^ 21  Simplify  Expand
2 - 5 A 4 + 10 A 8 - 14 A 12 + 16 A 16 - 17 A 20 + 15 A 24 - 12 A 28 + 7 A 32 - 4 A 36 + A 40
g6hat =
HCoeff@1, 2D lambda@1, 1, 2D ^ - 2 theta@1, 2, 1D HPhi@1D - Phi@0DL + Coeff@3, 2D lambda@1, 3, 2D ^
- 1 lambda@3, 1, 2D ^ - 1 theta@1, 2, 3D HPhi@3D - Phi@0DL + Coeff@3, 4D lambda@1, 3, 4D ^
- 1 lambda@3, 1, 4D ^ - 1 theta@1, 4, 3D HPhi@3D - Phi@0DLL  del  Simplify  Expand
1
-
1
+
A 29
25
2
+
A 25
3
-
A 23
5
-
A 19
5
+
A 17
+
A 15
11
4
-
A 13
-
A 11
18
+
A9
2
+
A7
24
1
-
A5
-
A3
- 2 A + 25 A 3 + 3 A 5 - 20 A 7 - 5 A 9 + 14 A 11 + 3 A 13 - 7 A 15 - 3 A 17 + 3 A 19 + A 21
A
g6 = g6hat * A ^ 29  Simplify  Expand
- 1 + A 4 + 2 A 6 - 3 A 10 - 5 A 12 + 5 A 14 + 11 A 16 - 4 A 18 - 18 A 20 + 2 A 22 + 24 A 24 - A 26 25 A 28 - 2 A 30 + 25 A 32 + 3 A 34 - 20 A 36 - 5 A 38 + 14 A 40 + 3 A 42 - 7 A 44 - 3 A 46 + 3 A 48 + A 50
g7hat = HCoeff@3, 2D lambda@1, 3, 2D ^ - 1 lambda@3, 1, 2D ^ - 1 theta@1, 2, 3D HPhi@3D - Phi@0DL
HPhi@3D - Phi@1DL + Coeff@3, 4D lambda@1, 3, 4D ^ - 1 lambda@3, 1, 4D ^ - 1
theta@1, 4, 3D HPhi@3D - Phi@0DL HPhi@3D - Phi@1DLL  del  Simplify  Expand
1
-
A 37
12
A5
3
-
A 33
-
1
A3
1
+
A 31
-
10
5
A 29
+
2
A 27
-
7
A 25
-
2
A 23
+
9
A 21
+
2
A 19
-
11
A 17
-
1
A 15
+
12
A 13
-
13
A9
+
1
+
A7
+ 2 A + 9 A 3 - 2 A 5 - 7 A 7 + 5 A 11 - A 13 - 3 A 15 + 2 A 19 - A 23 + A 25 + A 27 - A 31
A
g7 = g7hat * A ^ 37  Simplify  Expand
1 - 3 A 4 - A 6 + 5 A 8 + 2 A 10 - 7 A 12 - 2 A 14 + 9 A 16 + 2 A 18 - 11 A 20 - A 22 + 12 A 24 - 13 A 28 + A 30 +
12 A 32 - A 34 - 10 A 36 + 2 A 38 + 9 A 40 - 2 A 42 - 7 A 44 + 5 A 48 - A 50 - 3 A 52 + 2 A 56 - A 60 + A 62 + A 64 - A 68
computinggeneratorsforKBidealofF.nb
7
g8hat = HCoeff@3, 2D lambda@1, 3, 2D ^ - 1 lambda@3, 1, 2D ^ - 1
theta@1, 2, 3D HPhi@3D - Phi@0DL HPhi@3D - Phi@1DL HPhi@3D - Phi@2DL +
Coeff@3, 4D lambda@1, 3, 4D ^ - 1 lambda@3, 1, 4D ^ - 1 theta@1, 4, 3D
HPhi@3D - Phi@0DL HPhi@3D - Phi@1DL HPhi@3D - Phi@2DLL  del  Simplify  Expand
-
1
+
A 45
3
1
+
A 43
18
3
-
A 41
3
2
-
A 39
18
6
19
3
+
A 37
1
9
+
A 35
-
A 33
19
4
-
A 31
2
12
A 29
+
4
A 27
+
15
A 25
-
4
A 23
-
16
+
A 21
19
+ 4 A - 18 A 3 - 3 A 5 + 14 A 7 + 4 A 9 A
A 19 A 17 A 15 A 13 A 9 A 7 A 5 A 3
13 A 11 - 3 A 13 + 8 A 15 + 3 A 17 - 6 A 19 - A 21 + 4 A 23 + A 25 - 2 A 27 - A 29 + 2 A 31 - A 35 - A 37 + A 39
+
-
-
+
+
-
-
+
g8 = g8hat * A ^ 45  Simplify  Expand
- 1 + A 2 + 3 A 4 - 2 A 6 - 6 A 8 + 3 A 10 + 9 A 12 - 4 A 14 - 12 A 16 + 4 A 18 + 15 A 20 - 4 A 22 - 16 A 24 + 3 A 26 +
18 A 28 - 3 A 30 - 18 A 32 + 19 A 36 + A 38 - 19 A 40 - 2 A 42 + 19 A 44 + 4 A 46 - 18 A 48 - 3 A 50 + 14 A 52 +
4 A 54 - 13 A 56 - 3 A 58 + 8 A 60 + 3 A 62 - 6 A 64 - A 66 + 4 A 68 + A 70 - 2 A 72 - A 74 + 2 A 76 - A 80 - A 82 + A 84
The ideal of Z[A] generatedby g1,...,g8
We use the GroebnerBasis command to find an easier-to-analyze generating set for the ideal over Z[A] generated by g1,...,g8.
GroebnerBasis@8g1, g2, g3, g4, g5, g6, g7, g8<, A, CoefficientDomain ® IntegersD
911, 4 - A 4 =
We prove in the paper that I_F = <11,4-A^4> as ideals in Z[A,A^-1] and that this ideal is non-trivial. (Theorem 1.3, proved in
Section 5)