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DrawMorseLink

PD Presentation:

X8192 X20,9,21,10 X5,15,6,14 X18,12,19,11 X10,4,11,3 X7,13,8,12 X13,17,14,16 X17,7,18,22 X15,1,16,6 X4,21,5,22 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

q-3/2 - 4q-1/2 + 7q1/2 - 11q3/2 + 12q5/2 - 14q7/2 + 12q9/2 - 9q11/2 + 6q13/2 - 2q15/2

A2 (sl(3)) Invariant:

- q-4 + 2q-2 - 1 + 2q2 + 3q4 - q6 + 5q8 - q10 + 3q12 - 2q16 + q18 - 3q20 + q22 + q24

HOMFLY-PT Polynomial:

- a-7z + a-5z - a-5z3 - a-5z5 - a-3z-1 + a-3z + 5a-3z3 + 4a-3z5 + a-3z7 + a-1z-1 - 2a-1z3 - a-1z5

Kauffman Polynomial:

a-9z - 3a-9z3 + 5a-8z2 - 6a-8z4 - a-8z6 + a-7z - 4a-7z3 + 3a-7z5 - 4a-7z7 + 6a-6z2 - 10a-6z4 + 8a-6z6 - 5a-6z8 + a-5z - 12a-5z3 + 19a-5z5 - 6a-5z7 - 2a-5z9 - 8a-4z4 + 21a-4z6 - 10a-4z8 + a-3z-1 + 2a-3z - 19a-3z3 + 27a-3z5 - 6a-3z7 - 2a-3z9 - a-2 - 2a-2z2 - 2a-2z4 + 11a-2z6 - 5a-2z8 + a-1z-1 + a-1z - 8a-1z3 + 11a-1z5 - 4a-1z7 - z2 + 2z4 - z6

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 16                   2 j = 14                 4   j = 12               5 2   j = 10             7 4     j = 8           7 5       j = 6         6 8         j = 4       5 6           j = 2     3 7             j = 0   1 4               j = -2   3                 j = -4 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, NonAlternating, 177]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 177]]
Out[4]=	PD[X[8, 1, 9, 2], X[20, 9, 21, 10], X[5, 15, 6, 14], X[18, 12, 19, 11], > X[10, 4, 11, 3], X[7, 13, 8, 12], X[13, 17, 14, 16], X[17, 7, 18, 22], > X[15, 1, 16, 6], X[4, 21, 5, 22], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -10, -3, 9}, > {-6, -1, 2, -5, 4, 6, -7, 3, -9, 7, -8, -4, 11, -2, 10, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 4 3/2 5/2 7/2 9/2 q - ------- + 7 Sqrt[q] - 11 q + 12 q - 14 q + 12 q - Sqrt[q] 11/2 13/2 15/2 > 9 q + 6 q - 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 2 2 4 6 8 10 12 16 18 20 -1 - q + -- + 2 q + 3 q - q + 5 q - q + 3 q - 2 q + q - 3 q + 2 q 22 24 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 177]][a, z]
Out[8]=	3 3 3 5 5 5 7 1 1 z z z z 5 z 2 z z 4 z z z -(----) + --- - -- + -- + -- - -- + ---- - ---- - -- + ---- - -- + -- 3 a z 7 5 3 5 3 a 5 3 a 3 a z a a a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 177]][a, z]
Out[9]=	2 2 2 3 -2 1 1 z z z 2 z z 2 5 z 6 z 2 z 3 z -a + ---- + --- + -- + -- + -- + --- + - - z + ---- + ---- - ---- - ---- - 3 a z 9 7 5 3 a 8 6 2 9 a z a a a a a a a a 3 3 3 3 4 4 4 4 5 4 z 12 z 19 z 8 z 4 6 z 10 z 8 z 2 z 3 z > ---- - ----- - ----- - ---- + 2 z - ---- - ----- - ---- - ---- + ---- + 7 5 3 a 8 6 4 2 7 a a a a a a a a 5 5 5 6 6 6 6 7 7 19 z 27 z 11 z 6 z 8 z 21 z 11 z 4 z 6 z > ----- + ----- + ----- - z - -- + ---- + ----- + ----- - ---- - ---- - 5 3 a 8 6 4 2 7 5 a a a a a a a a 7 7 8 8 8 9 9 6 z 4 z 5 z 10 z 5 z 2 z 2 z > ---- - ---- - ---- - ----- - ---- - ---- - ---- 3 a 6 4 2 5 3 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 -2 3 4 3 q 4 6 6 2 7 q + 5 q + ----- + t + ----- + - + ---- + 6 q t + 6 q t + 8 q t + 4 3 2 2 t t q t q t 8 2 8 3 10 3 10 4 12 4 12 5 14 5 > 7 q t + 5 q t + 7 q t + 4 q t + 5 q t + 2 q t + 4 q t + 16 6 > 2 q t

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