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DrawMorseLink

PD Presentation:

X6172 X20,7,21,8 X4,21,1,22 X11,14,12,15 X8493 X5,13,6,12 X13,5,14,22 X15,19,16,18 X9,17,10,16 X17,11,18,10 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

- q-3/2 + 3q-1/2 - 7q1/2 + 8q3/2 - 10q5/2 + 9q7/2 - 8q9/2 + 6q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

q-6 + q-4 - q-2 + 3 + q4 + 3q6 + 3q10 - q12 + q14 + q16 - 2q18 + q20 - q22 - q24

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-8 - 3a-8z2 + 3a-8z4 - a-8z6 - a-7z-1 + 3a-7z - 7a-7z3 + 9a-7z5 - 3a-7z7 + 2a-6 - 7a-6z2 + 6a-6z4 + 5a-6z6 - 3a-6z8 - 3a-5z-1 + 14a-5z - 28a-5z3 + 27a-5z5 - 6a-5z7 - a-5z9 + a-4z2 - 9a-4z4 + 15a-4z6 - 6a-4z8 - 3a-3z-1 + 20a-3z - 36a-3z3 + 24a-3z5 - 5a-3z7 - a-3z9 - 2a-2 + 7a-2z2 - 15a-2z4 + 9a-2z6 - 3a-2z8 - 2a-1z-1 + 11a-1z - 16a-1z3 + 6a-1z5 - 2a-1z7 + 2z2 - 3z4 - az-1 + 2az - az3

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 16                   1 j = 14                 2   j = 12               4 1   j = 10             4 2     j = 8           5 4       j = 6         5 4         j = 4       3 5           j = 2     4 5             j = 0   1 5               j = -2   2                 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, 37]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 37]]
Out[4]=	PD[X[6, 1, 7, 2], X[20, 7, 21, 8], X[4, 21, 1, 22], X[11, 14, 12, 15], > X[8, 4, 9, 3], X[5, 13, 6, 12], X[13, 5, 14, 22], X[15, 19, 16, 18], > X[9, 17, 10, 16], X[17, 11, 18, 10], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-6, -1, 2, -5, -9, 10, -4, 6, -7, 4, -8, 9, -10, 8, > 11, -2, 3, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 3 3/2 5/2 7/2 9/2 11/2 -q + ------- - 7 Sqrt[q] + 8 q - 10 q + 9 q - 8 q + 6 q - Sqrt[q] 13/2 15/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -4 -2 4 6 10 12 14 16 18 20 22 3 + q + q - q + q + 3 q + 3 q - q + q + q - 2 q + q - q - 24 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 37]][a, z]
Out[8]=	3 3 1 3 3 2 a z 6 z 9 z 5 z 3 z 7 z ---- - ---- + ---- - --- + - + -- - --- + --- - --- + a z - ---- + ---- - 7 5 3 a z z 7 5 3 a 5 3 a z a z a z a a a a a 3 5 3 z 2 z > ---- + ---- a 3 a
In[9]:=	Kauffman[Link[11, NonAlternating, 37]][a, z]
Out[9]=	-8 2 2 1 3 3 2 a 3 z 14 z 20 z 11 z a + -- - -- - ---- - ---- - ---- - --- - - + --- + ---- + ---- + ---- + 6 2 7 5 3 a z z 7 5 3 a a a a z a z a z a a a 2 2 2 2 3 3 3 3 2 3 z 7 z z 7 z 7 z 28 z 36 z 16 z > 2 a z + 2 z - ---- - ---- + -- + ---- - ---- - ----- - ----- - ----- - 8 6 4 2 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 5 3 4 3 z 6 z 9 z 15 z 9 z 27 z 24 z 6 z > a z - 3 z + ---- + ---- - ---- - ----- + ---- + ----- + ----- + ---- - 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 6 7 7 7 7 8 8 8 z 5 z 15 z 9 z 3 z 6 z 5 z 2 z 3 z 6 z 3 z > -- + ---- + ----- + ---- - ---- - ---- - ---- - ---- - ---- - ---- - ---- - 8 6 4 2 7 5 3 a 6 4 2 a a a a a a a a a a 9 9 z z > -- - -- 5 3 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 2 4 4 2 6 2 6 3 5 + 4 q + ----- + - + ---- + 5 q t + 3 q t + 5 q t + 5 q t + 4 q t + 4 2 t 2 q t q t 8 3 8 4 10 4 10 5 12 5 12 6 14 6 > 5 q t + 4 q t + 4 q t + 2 q t + 4 q t + q t + 2 q t + 16 7 > q t

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