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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X22,10,7,9 X16,6,17,5 X18,12,19,11 X20,16,21,15 X12,18,13,17 X14,22,15,21 X2738 X4,14,5,13 X6,20,1,19

Gauss Code:

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

Jones Polynomial:

- q-1/2 + 4q1/2 - 10q3/2 + 15q5/2 - 22q7/2 + 24q9/2 - 24q11/2 + 21q13/2 - 15q15/2 + 9q17/2 - 4q19/2 + q21/2

A2 (sl(3)) Invariant:

q-2 - 2 + 5q4 - 3q6 + 5q8 + 3q10 - q12 + 5q14 - 3q16 + 3q18 - 2q20 - 3q22 + 4q24 - 4q26 + 2q30 - q32

HOMFLY-PT Polynomial:

a-9z3 + a-7z-1 + 2a-7z + a-7z3 - a-7z5 - 3a-5z-1 - 6a-5z - 6a-5z3 - 3a-5z5 + 2a-3z-1 + 5a-3z + 2a-3z3 - a-3z5 + a-1z3

Kauffman Polynomial:

- a-12z2 + 2a-12z4 - a-12z6 + a-11z - 6a-11z3 + 9a-11z5 - 4a-11z7 + 3a-10z2 - 10a-10z4 + 15a-10z6 - 7a-10z8 + 3a-9z - 11a-9z3 + 12a-9z5 + 4a-9z7 - 6a-9z9 + a-8 + 6a-8z2 - 29a-8z4 + 41a-8z6 - 15a-8z8 - 2a-8z10 - a-7z-1 + 8a-7z - 25a-7z3 + 18a-7z5 + 13a-7z7 - 13a-7z9 + 3a-6 + 2a-6z2 - 31a-6z4 + 46a-6z6 - 19a-6z8 - 2a-6z10 - 3a-5z-1 + 13a-5z - 33a-5z3 + 31a-5z5 - 4a-5z7 - 7a-5z9 + 3a-4 - 10a-4z4 + 17a-4z6 - 11a-4z8 - 2a-3z-1 + 7a-3z - 12a-3z3 + 15a-3z5 - 9a-3z7 + 4a-2z4 - 4a-2z6 + a-1z3 - a-1z5

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 22                       1 j = 20                     3   j = 18                   6 1   j = 16                 9 3     j = 14               12 6       j = 12             12 9         j = 10           12 12           j = 8         10 12             j = 6       6 13               j = 4     4 9                 j = 2   1 7                   j = 0   3                     j = -2 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 197]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 197]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[16, 6, 17, 5], > X[18, 12, 19, 11], X[20, 16, 21, 15], X[12, 18, 13, 17], X[14, 22, 15, 21], > X[2, 7, 3, 8], X[4, 14, 5, 13], X[6, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 4, -11}, > {9, -1, 3, -2, 5, -7, 10, -8, 6, -4, 7, -5, 11, -6, 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + 4 Sqrt[q] - 10 q + 15 q - 22 q + 24 q - 24 q + Sqrt[q] 13/2 15/2 17/2 19/2 21/2 > 21 q - 15 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 4 6 8 10 12 14 16 18 20 -2 + q + 5 q - 3 q + 5 q + 3 q - q + 5 q - 3 q + 3 q - 2 q - 22 24 26 30 32 > 3 q + 4 q - 4 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 197]][a, z]
Out[8]=	3 3 3 3 3 5 5 1 3 2 2 z 6 z 5 z z z 6 z 2 z z z 3 z ---- - ---- + ---- + --- - --- + --- + -- + -- - ---- + ---- + -- - -- - ---- - 7 5 3 7 5 3 9 7 5 3 a 7 5 a z a z a z a a a a a a a a a 5 z > -- 3 a
In[9]:=	Kauffman[Link[11, Alternating, 197]][a, z]
Out[9]=	2 -8 3 3 1 3 2 z 3 z 8 z 13 z 7 z z a + -- + -- - ---- - ---- - ---- + --- + --- + --- + ---- + --- - --- + 6 4 7 5 3 11 9 7 5 3 12 a a a z a z a z a a a a a a 2 2 2 3 3 3 3 3 3 4 3 z 6 z 2 z 6 z 11 z 25 z 33 z 12 z z 2 z > ---- + ---- + ---- - ---- - ----- - ----- - ----- - ----- + -- + ---- - 10 8 6 11 9 7 5 3 a 12 a a a a a a a a a 4 4 4 4 4 5 5 5 5 10 z 29 z 31 z 10 z 4 z 9 z 12 z 18 z 31 z > ----- - ----- - ----- - ----- + ---- + ---- + ----- + ----- + ----- + 10 8 6 4 2 11 9 7 5 a a a a a a a a a 5 5 6 6 6 6 6 6 7 7 15 z z z 15 z 41 z 46 z 17 z 4 z 4 z 4 z > ----- - -- - --- + ----- + ----- + ----- + ----- - ---- - ---- + ---- + 3 a 12 10 8 6 4 2 11 9 a a a a a a a a a 7 7 7 8 8 8 8 9 9 9 13 z 4 z 9 z 7 z 15 z 19 z 11 z 6 z 13 z 7 z > ----- - ---- - ---- - ---- - ----- - ----- - ----- - ---- - ----- - ---- - 7 5 3 10 8 6 4 9 7 5 a a a a a a a a a a 10 10 2 z 2 z > ----- - ----- 8 6 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 3 q 4 6 6 2 8 2 7 q + 4 q + ----- + - + -- + 9 q t + 6 q t + 13 q t + 10 q t + 2 2 t t q t 8 3 10 3 10 4 12 4 12 5 14 5 > 12 q t + 12 q t + 12 q t + 12 q t + 9 q t + 12 q t + 14 6 16 6 16 7 18 7 18 8 20 8 22 9 > 6 q t + 9 q t + 3 q t + 6 q t + q t + 3 q t + q t

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