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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X15,22,16,19 X7,20,8,21 X19,8,20,9 X13,18,14,5 X11,14,12,15 X17,12,18,13 X21,16,22,17 X2536 X4,9,1,10

Gauss Code:

{{1, -10, 2, -11}, {-5, 4, -9, 3}, {10, -1, -4, 5, 11, -2, -7, 8, -6, 7, -3, 9, -8, 6}}

Jones Polynomial:

- q-13 + 2q-12 - 4q-11 + 6q-10 - 5q-9 + 6q-8 - 4q-7 + 4q-6 - q-5 + q-3

A2 (sl(3)) Invariant:

- q-42 - 3q-40 - 2q-38 - q-36 - q-34 + 5q-32 + 5q-30 + 6q-28 + 6q-26 + 3q-24 + 4q-22 + 2q-18 + 2q-16 + q-12 + q-10

HOMFLY-PT Polynomial:

3a6 + 8a6z2 + 6a6z4 + a6z6 + 2a8z-2 + 2a8 + a8z2 + a8z4 - 5a10z-2 - 9a10 - 5a10z2 + 4a12z-2 + 4a12 - a14z-2

Kauffman Polynomial:

- 3a6 + 8a6z2 - 6a6z4 + a6z6 + a7z + a7z3 - a7z5 - 2a8z-2 + 5a8 - 3a8z2 + 3a8z4 - a8z6 + 5a9z-1 - 20a9z + 30a9z3 - 17a9z5 + 3a9z7 - 5a10z-2 + 21a10 - 37a10z2 + 38a10z4 - 21a10z6 + 4a10z8 + 9a11z-1 - 39a11z + 57a11z3 - 32a11z5 + 3a11z7 + a11z9 - 4a12z-2 + 18a12 - 34a12z2 + 41a12z4 - 28a12z6 + 6a12z8 + 5a13z-1 - 23a13z + 36a13z3 - 21a13z5 + a13z7 + a13z9 - a14z-2 + 4a14 - 8a14z2 + 12a14z4 - 9a14z6 + 2a14z8 + a15z-1 - 5a15z + 8a15z3 - 5a15z5 + a15z7

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -5                       1 j = -7                       1 j = -9                 2 1     j = -11               3         j = -13             4 5 1       j = -15           3 1           j = -17         2 4 1           j = -19       4 3               j = -21     1 3                 j = -23   1 3                   j = -25   1                     j = -27 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 398]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 398]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[15, 22, 16, 19], X[7, 20, 8, 21], > X[19, 8, 20, 9], X[13, 18, 14, 5], X[11, 14, 12, 15], X[17, 12, 18, 13], > X[21, 16, 22, 17], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {-5, 4, -9, 3}, > {10, -1, -4, 5, 11, -2, -7, 8, -6, 7, -3, 9, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-13 2 4 6 5 6 4 4 -5 -3 -q + --- - --- + --- - -- + -- - -- + -- - q + q 12 11 10 9 8 7 6 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 3 2 -36 -34 5 5 6 6 3 4 2 -q - --- - --- - q - q + --- + --- + --- + --- + --- + --- + --- + 40 38 32 30 28 26 24 22 18 q q q q q q q q q 2 -12 -10 > --- + q + q 16 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 398]][a, z]
Out[8]=	8 10 12 14 6 8 10 12 2 a 5 a 4 a a 6 2 8 2 3 a + 2 a - 9 a + 4 a + ---- - ----- + ----- - --- + 8 a z + a z - 2 2 2 2 z z z z 10 2 6 4 8 4 6 6 > 5 a z + 6 a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 398]][a, z]
Out[9]=	8 10 12 14 9 6 8 10 12 14 2 a 5 a 4 a a 5 a -3 a + 5 a + 21 a + 18 a + 4 a - ---- - ----- - ----- - --- + ---- + 2 2 2 2 z z z z z 11 13 15 9 a 5 a a 7 9 11 13 15 > ----- + ----- + --- + a z - 20 a z - 39 a z - 23 a z - 5 a z + z z z 6 2 8 2 10 2 12 2 14 2 7 3 9 3 > 8 a z - 3 a z - 37 a z - 34 a z - 8 a z + a z + 30 a z + 11 3 13 3 15 3 6 4 8 4 10 4 > 57 a z + 36 a z + 8 a z - 6 a z + 3 a z + 38 a z + 12 4 14 4 7 5 9 5 11 5 13 5 > 41 a z + 12 a z - a z - 17 a z - 32 a z - 21 a z - 15 5 6 6 8 6 10 6 12 6 14 6 9 7 > 5 a z + a z - a z - 21 a z - 28 a z - 9 a z + 3 a z + 11 7 13 7 15 7 10 8 12 8 14 8 11 9 > 3 a z + a z + a z + 4 a z + 6 a z + 2 a z + a z + 13 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 -5 1 1 1 3 1 3 4 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 27 11 25 10 23 10 23 9 21 9 21 8 19 8 q t q t q t q t q t q t q t 3 2 4 3 1 1 4 5 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 19 7 17 7 17 6 15 6 17 5 15 5 13 5 13 4 q t q t q t q t q t q t q t q t 3 1 2 1 > ------ + ------ + ----- + ----- 11 4 13 3 9 3 9 2 q t q t q t q t

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