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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X16,8,17,7 X20,10,21,9 X22,18,5,17 X18,22,19,21 X8,20,9,19 X14,12,15,11 X10,16,11,15 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-1/2 + 2q1/2 - 6q3/2 + 9q5/2 - 14q7/2 + 16q9/2 - 17q11/2 + 15q13/2 - 11q15/2 + 8q17/2 - 4q19/2 + q21/2

A2 (sl(3)) Invariant:

q-2 + q2 + 4q4 - q6 + 4q8 + 3q10 - q12 + 3q14 - 2q16 + q18 - q20 - 3q22 + 2q24 - 3q26 + 2q30 - q32

HOMFLY-PT Polynomial:

a-9z3 + a-7z-1 + 2a-7z - a-7z5 - 2a-5z-1 - 4a-5z - 4a-5z3 - 2a-5z5 - a-3z3 - a-3z5 + a-1z-1 + 2a-1z + a-1z3

Kauffman Polynomial:

- a-12z2 + 2a-12z4 - a-12z6 - 5a-11z3 + 10a-11z5 - 4a-11z7 - 8a-10z4 + 15a-10z6 - 6a-10z8 - 3a-9z3 + 6a-9z5 + 4a-9z7 - 4a-9z9 + 2a-8 - 6a-8z2 - 5a-8z4 + 19a-8z6 - 8a-8z8 - a-8z10 - a-7z-1 + 4a-7z - 10a-7z3 + 6a-7z5 + 6a-7z7 - 6a-7z9 + 5a-6 - 13a-6z2 + 7a-6z4 + 5a-6z6 - 5a-6z8 - a-6z10 - 2a-5z-1 + 7a-5z - 15a-5z3 + 14a-5z5 - 5a-5z7 - 2a-5z9 + 3a-4 - 6a-4z2 + 5a-4z4 - 3a-4z8 + 3a-3z5 - 3a-3z7 - a-2 + 3a-2z4 - 2a-2z6 + a-1z-1 - 3a-1z + 3a-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                   5 1   j = 16                 6 3     j = 14               9 5       j = 12             8 6         j = 10           8 9           j = 8         6 8             j = 6       3 8               j = 4     3 6                 j = 2   1 5                   j = 0   1                     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, 83]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 83]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[16, 8, 17, 7], X[20, 10, 21, 9], > X[22, 18, 5, 17], X[18, 22, 19, 21], X[8, 20, 9, 19], X[14, 12, 15, 11], > X[10, 16, 11, 15], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -7, 4, -9, 8, -2, 11, -8, 9, -3, 5, -6, > 7, -4, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + 2 Sqrt[q] - 6 q + 9 q - 14 q + 16 q - 17 q + Sqrt[q] 13/2 15/2 17/2 19/2 21/2 > 15 q - 11 q + 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 2 4 6 8 10 12 14 16 18 20 22 q + q + 4 q - q + 4 q + 3 q - q + 3 q - 2 q + q - q - 3 q + 24 26 30 32 > 2 q - 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 83]][a, z]
Out[8]=	3 3 3 3 5 5 5 1 2 1 2 z 4 z 2 z z 4 z z z z 2 z z ---- - ---- + --- + --- - --- + --- + -- - ---- - -- + -- - -- - ---- - -- 7 5 a z 7 5 a 9 5 3 a 7 5 3 a z a z a a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 83]][a, z]
Out[9]=	2 2 2 2 5 3 -2 1 2 1 4 z 7 z 3 z z 6 z 13 z -- + -- + -- - a - ---- - ---- + --- + --- + --- - --- - --- - ---- - ----- - 8 6 4 7 5 a z 7 5 a 12 8 6 a a a a z a z a a a a a 2 3 3 3 3 3 4 4 4 4 6 z 5 z 3 z 10 z 15 z 3 z 2 z 8 z 5 z 7 z > ---- - ---- - ---- - ----- - ----- + ---- + ---- - ---- - ---- + ---- + 4 11 9 7 5 a 12 10 8 6 a a a a a a a a a 4 4 5 5 5 5 5 5 6 6 5 z 3 z 10 z 6 z 6 z 14 z 3 z z z 15 z > ---- + ---- + ----- + ---- + ---- + ----- + ---- - -- - --- + ----- + 4 2 11 9 7 5 3 a 12 10 a a a a a a a a a 6 6 6 7 7 7 7 7 8 8 19 z 5 z 2 z 4 z 4 z 6 z 5 z 3 z 6 z 8 z > ----- + ---- - ---- - ---- + ---- + ---- - ---- - ---- - ---- - ---- - 8 6 2 11 9 7 5 3 10 8 a a a a a a a a a a 8 8 9 9 9 10 10 5 z 3 z 4 z 6 z 2 z z z > ---- - ---- - ---- - ---- - ---- - --- - --- 6 4 9 7 5 8 6 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 q 4 6 6 2 8 2 8 3 5 q + 3 q + ----- + - + -- + 6 q t + 3 q t + 8 q t + 6 q t + 8 q t + 2 2 t t q t 10 3 10 4 12 4 12 5 14 5 14 6 > 8 q t + 9 q t + 8 q t + 6 q t + 9 q t + 5 q t + 16 6 16 7 18 7 18 8 20 8 22 9 > 6 q t + 3 q t + 5 q t + q t + 3 q t + q t

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