Pensieve Header: Computing the Alexander polynomial in Β-calculus.

In[1]:= SetDirectory@"C:\\drorbn\\AcademicPensieve\\2012-01"D;

<< betaCalculus.m

<< KnotTheory`

GC@K_D := GC �� HPD@KD �. X@i_, j_, k_, l_D ¦ If@PositiveQ@X@i, j, k, lDD,

Ar@l, i, +1D, Ar@j, i, -1DD

LLoading KnotTheory` version of August 22, 2010, 13:36:57.55.

Read more at http:��katlas.org�wiki�KnotTheory.

The Program and a Test RunIn[5]:= 8K = Knot@8, 17D;

Alexander@KD@XD, GC@KD<

KnotTheory::loading : Loading precomputed data in PD4Knots`.

Out[5]= :11 -

1

X3+

4

X2-

8

X- 8 X + 4 X2

- X3, GC@Ar@1, 6, 1D, Ar@7, 14, 1D, Ar@3, 8, -1D,

Ar@13, 2, -1D, Ar@5, 12, -1D, Ar@9, 4, -1D, Ar@11, 16, 1D, Ar@15, 10, 1DD>

ΒForm@Plus �� HGC@KD �. 8Ar@i_, j_, +1D ¦ R@i, jD, Ar@i_, j_, -1D ¦ RInv@i, jD<LD;

ΒAlex@K_D := Module@8gc, Β<,

gc = GC@KD;

Β =

ΒCollect@Plus �� Hgc �. 8Ar@i_, j_, +1D ¦ R@i, jD, Ar@i_, j_, -1D ¦ RInv@i, jD<LD;

Do@Β = dm@1, k, 1D@ΒD, 8k, 2, 2 Length@gcD<D;

Expand@Β �. h@1D ® 0 �. E^Hn_. c@1DL ¦ X^n �. W@a_D ¦ aDD

ΒAlex@KD

-8 -

1

X2+

4

X+ 11 X - 8 X2

+ 4 X3- X4

Testing the Full Rolfsen Table8ΒAlex@ðD, Alexander@ðD@XD< & �� AllKnots@83, 7<D �� MatrixForm

X - X2+ X3

-1 +

1

X+ X

-1 + 3 X - X2 3 -

1

X- X

X - X2+ X3

- X4+ X5 1 +

1

X2-

1

X- X + X2

2 X2- 3 X3

+ 2 X4-3 +

2

X+ 2 X

-2 X + 5 X2- 2 X3 5 -

2

X- 2 X

-1 + 3 X - 3 X2+ 3 X3

- X4-3 -

1

X2+

3

X+ 3 X - X2

5 +

1

X2-

3

X- 3 X + X2 5 +

1

X2-

3

X- 3 X + X2

X - X2+ X3

- X4+ X5

- X6+ X7

-1 +

1

X3-

1

X2+

1

X+ X - X2

+ X3

3 X3- 5 X4

+ 3 X5-5 +

3

X+ 3 X

2

X5-

3

X4+

3

X3-

3

X2+

2

X3 +

2

X2-

3

X- 3 X + 2 X2

4

X4-

7

X3+

4

X2-7 +

4

X+ 4 X

2 X2- 4 X3

+ 5 X4- 4 X5

+ 2 X6 5 +

2

X2-

4

X- 4 X + 2 X2

-1 + 5 X - 7 X2+ 5 X3

- X4-7 -

1

X2+

5

X+ 5 X - X2

-5 +

1

X+ 9 X - 5 X2

+ X3 9 +

1

X2-

5

X- 5 X + X2

Test@K_D := Factor@ΒAlex@KD � Alexander@KD@XDD

Union@Test �� AllKnots@83, 11<DD

KnotTheory::loading : Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits :

The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

:1,1

X6,

1

X5,

1

X4,

1

X3,

1

X2,

1

X, X, X2, X3, X4, X5, X6>

Avoiding Exponentials

ar@1, 2D **-1

1 + c@1Dar@1, 2D

0

Βalex@K_D := ModuleB8gc, Β<,

gc = GC@KD;

Β = ΒCollectBW@1D + Plus ��

gc �. :Ar@i_, j_, +1D ¦ ar@i, jD, Ar@i_, j_, -1D ¦ -1

1 + c@iDar@i, jD> F;

Do@Β = dm@1, k, 1D@ΒD, 8k, 2, 2 Length@gcD<D;

Expand@ΒD �. c@1D ® Y

F

2 betaAlexander.nb

Βalex@KD

WB-

-1 - 3 Y - 2 Y2+ Y3

+ 3 Y4+ 2 Y5

+ Y6

H1 + YL2F

8Expand@Βalex@ðD �. 8h@1D ® 0, W@a_D ¦ a< �. 8Y ® X - 1<D,

Expand@Βalex@ðD �. 8h@1D ® 0, W@a_D ¦ a<D, ΒAlex@ðD,

Alexander@ðD@XD, Conway@ðD@YD< & �� AllKnots@83, 7<D �� MatrixForm

X - X2+ X3 1 + 2 Y + 2 Y2

+ Y3 X - X2+ X3

-1 + 3 X - X2 1 + Y - Y2-1 + 3 X - X2

X - X2+ X3

- X4+ X5 1 + 3 Y + 6 Y2

+ 7 Y3+ 4 Y4

+ Y5 X - X2+ X3

- X4+ X5

2 X2- 3 X3

+ 2 X4 1 + 3 Y + 5 Y2+ 5 Y3

+ 2 Y4 2 X2- 3 X3

+ 2 X4

-2 X + 5 X2- 2 X3 1 + 2 Y - Y2

- 2 Y3-2 X + 5 X2

- 2 X3

-1 + 3 X - 3 X2+ 3 X3

- X4 1 + 2 Y - Y3- Y4

-1 + 3 X - 3 X2+ 3 X3

- X4

5 +

1

X2-

3

X- 3 X + X2 1

H1+YL2+

2 Y

H1+YL2+

2 Y2

H1+YL2+

Y3

H1+YL2+

Y4

H1+YL25 +

1

X2-

3

X- 3 X + X2

X - X2+ X3

- X4+ X5

- X6+ X7 1 + 4 Y + 12 Y2

+ 22 Y3+ 24 Y4

+ 16 Y5+ 6 Y6

+ Y7 X - X2+ X3

- X4+ X5

- X6+ X7

3 X3- 5 X4

+ 3 X5 1 + 4 Y + 9 Y2+ 13 Y3

+ 10 Y4+ 3 Y5 3 X3

- 5 X4+ 3 X5

2

X5-

3

X4+

3

X3-

3

X2+

2

X

1

H1+YL5+

2 Y

H1+YL5+

6 Y2

H1+YL5+

5 Y3

H1+YL5+

2 Y4

H1+YL5

2

X5-

3

X4+

3

X3-

3

X2+

2

X

4

X4-

7

X3+

4

X2

1

H1+YL4+

Y

H1+YL4+

4 Y2

H1+YL4

4

X4-

7

X3+

4

X2

2 X2- 4 X3

+ 5 X4- 4 X5

+ 2 X6 1 + 4 Y + 10 Y2+ 16 Y3

+ 15 Y4+ 8 Y5

+ 2 Y6 2 X2- 4 X3

+ 5 X4- 4 X5

+ 2 X6

-1 + 5 X - 7 X2+ 5 X3

- X4 1 + 2 Y + 2 Y2+ Y3

- Y4-1 + 5 X - 7 X2

+ 5 X3- X4

-5 +

1

X+ 9 X - 5 X2

+ X3 1

1+Y+

2 Y

1+Y-

Y3

1+Y+

Y4

1+Y-5 +

1

X+ 9 X - 5 X2

+ X3

In[22]:= ΒSimplify = Factor;

gc = GC@KD;

Β = ΒCollectBW@1D +

Plus �� gc �. :Ar@i_, j_, +1D ¦ ar@i, jD, Ar@i_, j_, -1D ¦ -1

1 + c@iDar@i, jD> F;

HTable@8

HΒ = dm@1, k, 1D@ΒDL �. _c ® c - 1 �� ΒCollect �� ΒForm,

Collect@Β �. 8_W ® 0, t@s_D ¦ c@sD<, _h, FactorD<, 8k, 2, 2 Length@gcD<D

L �� ColumnForm

Out[25]= :

W@1D h@1D h@4D h@6D h@8D h@10D h@12D h@14D h@16Dt@1D 0 0 1 0 0 0 0 0

t@3D 0 0 0 -

1

c0 0 0 0

t@5D 0 0 0 0 0 -

1

c0 0

t@7D 0 0 0 0 0 0 1 0

t@9D 0 -

1

c0 0 0 0 0 0

t@11D 0 0 0 0 0 0 0 1

t@13D -

1

c0 0 0 0 0 0 0

t@15D 0 0 0 0 1 0 0 0

, -

c@13D h@1D1+c@13D -

c@9D h@4D1+c@9D + c@1D h@6D -

c@1

betaAlexander.nb 3

Out[25]=

:

W@1D h@1D h@4D h@6D h@8D h@10D h@12D h@14D h@16Dt@1D 0 0 1 -1 0 0 0 0

t@5D 0 0 0 0 0 -

1

c0 0

t@7D 0 0 0 0 0 0 1 0

t@9D 0 -

1

c0 0 0 0 0 0

t@11D 0 0 0 0 0 0 0 1

t@13D -

1

c0 0

-1+c

c0 0 0 0

t@15D 0 0 0 0 1 0 0 0

, -

c@13D h@1D1+c@13D -

c@9D h@4D1+c@9D + c@1D h@6D -

c@1

:

W@1D h@1D h@6D h@8D h@10D h@12D h@14D h@16Dt@1D 0 1 -1 0 0 0 0

t@5D 0 0 0 0 -

1

c0 0

t@7D 0 0 0 0 0 1 0

t@9D -

1

c20 0 0 0 0 0

t@11D 0 0 0 0 0 0 1

t@13D -

1

c0

-1+c

c0 0 0 0

t@15D 0 0 0 1 0 0 0

, -

Hc@9D+c@13D+c@9D c@13DL h@1DH1+c@9DL H1+c@13DL + c@1D h@6D -

c@1D1+c

:

W@1D h@1D h@6D h@8D h@10D h@12D h@14D h@16Dt@1D 0 1 -1 0 -c 0 0

t@7D 0 0 0 0 0 1 0

t@9D -

1

c20 0 0

-1+c

c0 0

t@11D 0 0 0 0 0 0 1

t@13D -

1

c0

-1+c

c0 -1 + c 0 0

t@15D 0 0 0 1 0 0 0

, -

Hc@9D+c@13D+c@9D c@13DL h@1DH1+c@9DL H1+c@13DL + c@1D h@6D -

c@1D1+c

:

W@1D h@1D h@8D h@10D h@12D h@14D h@16Dt@1D 1

c2-1 0 -c 0 0

t@7D 0 0 0 0 1 0

t@9D -

1

c20 0

-1+c

c0 0

t@11D 0 0 0 0 0 1

t@13D -

1

c

-1+c

c0 -1 + c 0 0

t@15D 0 0 1 0 0 0

,Hc@1D-c@9D-c@13D-c@9D c@13DL h@1D

H1+c@9DL H1+c@13DL -

c@1D h@8D1+c@1D + c@15D h@

:

W@1D h@1D h@8D h@10D h@12D h@14D h@16Dt@1D 1

c2-1 0 -c

-1+c+c2

c0

t@9D -

1

c20 0

-1+c

c-

-1+c

c0

t@11D 0 0 0 0 0 1

t@13D -

1

c

-1+c

c0 -1 + c 1 - c 0

t@15D 0 0 1 0 0 0

,Hc@1D-c@9D-c@13D-c@9D c@13DL h@1D

H1+c@9DL H1+c@13DL -

c@1D h@8D1+c@1D + c@15D h@

:

W@1D h@1D h@10D h@12D h@14D h@16Dt@1D -

-1+c

c20 -c

-1+c+c2

c0

t@9D -

1

c20

-1+c

c-

-1+c

c0

t@11D 0 0 0 0 1

t@13D -

1

c20 -1 + c 1 - c 0

t@15D 0 1 0 0 0

, -

Hc@9D+c@13D+c@9D c@13DL h@1DH1+c@9DL H1+c@13DL + c@15D h@10D -

c@1D h@12D1+c@1D + c

4 betaAlexander.nb

Out[25]=

:

W@cD h@1D h@10D h@12D h@14D h@16Dt@1D -

-1+c+c2

c30 -

-1+3 c-2 c2+c3

c2

-1+2 c-c2+c3

c20

t@11D 0 0 0 0 1

t@13D -

1

c30

H-1+cL I1-c+c2Mc2

-

H-1+cL I1-c+c2Mc2

0

t@15D 0 1 0 0 0

, -

Hc@1D+c@13D+c@1D c@13DL h@1DH1+c@1DL H1+c@13DL + c@15D h@10

:

W@cD h@1D h@12D h@14D h@16Dt@1D -

-1+c+c2

c3-

-1+3 c-2 c2+c3

c2

-1+2 c-c2+c3

c20

t@11D 0 0 0 1

t@13D -

1

c3

H-1+cL I1-c+c2Mc2

-

H-1+cL I1-c+c2Mc2

0

t@15D 1

c20 0 0

, -

Hc@1D+c@13D+c@1D c@13D-c@15DL h@1DH1+c@1DL H1+c@13DL -

c@1D h@12D1+c@1D + c

:

W@cD h@1D h@12D h@14D h@16Dt@1D -

-1+c+c2

c3-

-1+3 c-2 c2+c3

c2

-1+2 c-c2+c3

c2

-1+2 c

c2

t@13D -

1

c3

H-1+cL I1-c+c2Mc2

-

H-1+cL I1-c+c2Mc2

-

-1+c

c2

t@15D 1

c20 0

-1+c

c

, -

Hc@1D+c@13D+c@1D c@13D-c@15DL h@1DH1+c@1DL H1+c@13DL -

c@1D h@12D1+c@1D + c

:

W@cD h@1D h@14D h@16Dt@1D -

-2+4 c-c2+c3

c3

-1+2 c-c2+c3

c2

-1+2 c

c2

t@13D -2+2 c-2 c2+c3

c3-

H-1+cL I1-c+c2Mc2

-

-1+c

c2

t@15D 1

c20

-1+c

c

, -

I2 c@1D+c@1D2+c@13D+2 c@1D c@13D+c@1D2 c@13D-c@15DM h@1D

H1+c@1DL2 H1+c@13DL+ c@1D

:

WA-H-2 + cL I-1 + 2 c - c2+ c3ME h@1D h@14D h@16D

t@1D -

2-4 c-c2+c3

-2 c4+c5

H-2+cL c2 I-1+2 c-c2+c3M

-1+5 c-9 c2+7 c3

-4 c4+c5

H-2+cL c I-1+2 c-c2+c3M

1-3 c+3 c2-3 c3

+c4

H-2+cL c I-1+2 c-c2+c3M

t@15D -

1

H-2+cL c I-1+2 c-c2+c3M

H-1+cL2 I1-c+c2MH-2+cL c I-1+2 c-c2

+c3MH-1+cL I1-4 c+4 c2

-3 c3+c4M

H-2+cL c I-1+2 c-c2+c3M

,

:

WA-H-2 + cL I-1 + 2 c - c2+ c3ME h@1D h@16D

t@1D -

1-3 c+5 c2-8 c3

+5 c4-3 c5

+c6

H-2+cL c3 I-1+2 c-c2+c3M

1-3 c+3 c2-3 c3

+c4

H-2+cL c I-1+2 c-c2+c3M

t@15D 1-3 c+3 c2-3 c3

+c4

H-2+cL c3 I-1+2 c-c2+c3M

H-1+cL I1-4 c+4 c2-3 c3

+c4MH-2+cL c I-1+2 c-c2

+c3M

, -

I2 c@1D+c@1D2-c@15D

H1+c@1DL2

:WB-

1-4 c+8 c2-11 c3

+8 c4-4 c5

+c6

c2F h@1D h@16D

t@1D -

1

c1

, -

c@1D h@1D1+c@1D + c@1D h@16D>

:J WB-

1-4 c+8 c2-11 c3

+8 c4-4 c5

+c6

c2F N, 0>

-2 - 4 c - c2 + c3 - 2 c4 + c5

H-2 + cL c2 I-1 + 2 c - c2 + c3M+ -

1

H-2 + cL c I-1 + 2 c - c2 + c3M�� Simplify

-

1 + c

c2

-1 + 5 c - 9 c2 + 7 c3 - 4 c4 + c5

H-2 + cL c I-1 + 2 c - c2 + c3M+

H-1 + cL2 I1 - c + c2M

H-2 + cL c I-1 + 2 c - c2 + c3M�� Simplify

1

1 - 3 c + 3 c2 - 3 c3 + c4

H-2 + cL c I-1 + 2 c - c2 + c3M+

H-1 + cL I1 - 4 c + 4 c2 - 3 c3 + c4M

H-2 + cL c I-1 + 2 c - c2 + c3M�� Simplify

1

betaAlexander.nb 5