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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q1/2 + 2q3/2 - 6q5/2 + 8q7/2 - 12q9/2 + 13q11/2 - 14q13/2 + 13q15/2 - 9q17/2 + 6q19/2 - 3q21/2 + q23/2

A2 (sl(3)) Invariant:

q2 + 2q6 + 3q8 + q10 + 5q12 + 3q16 + q18 - 2q20 + q22 - 4q24 - q28 - q30 + q32 - q34

HOMFLY-PT Polynomial:

a-9z-1 + 2a-9z + 3a-9z3 + a-9z5 - a-7z-1 - 3a-7z - 5a-7z3 - 4a-7z5 - a-7z7 - 2a-5z-1 - 4a-5z - 5a-5z3 - 4a-5z5 - a-5z7 + 2a-3z-1 + 5a-3z + 4a-3z3 + a-3z5

Kauffman Polynomial:

a-14z2 - a-14z4 + 3a-13z3 - 3a-13z5 + a-12 - 3a-12z2 + 6a-12z4 - 5a-12z6 + a-11z - 5a-11z3 + 8a-11z5 - 6a-11z7 - a-10z2 - 6a-10z4 + 10a-10z6 - 6a-10z8 + a-9z-1 - 3a-9z3 + a-9z5 + 5a-9z7 - 4a-9z9 - 3a-8 + 10a-8z2 - 23a-8z4 + 25a-8z6 - 7a-8z8 - a-8z10 + a-7z-1 - 2a-7z3 - 6a-7z5 + 15a-7z7 - 6a-7z9 + 4a-6z2 - 15a-6z4 + 17a-6z6 - 3a-6z8 - a-6z10 - 2a-5z-1 + 8a-5z - 16a-5z3 + 9a-5z5 + 3a-5z7 - 2a-5z9 + 3a-4 - 3a-4z2 - 5a-4z4 + 7a-4z6 - 2a-4z8 - 2a-3z-1 + 7a-3z - 9a-3z3 + 5a-3z5 - a-3z7

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 = 24                       1 j = 22                     2   j = 20                   4 1   j = 18                 5 2     j = 16               8 4       j = 14             6 5         j = 12           7 8           j = 10         5 6             j = 8       3 7               j = 6     3 5                 j = 4   1 5                   j = 2   1                     j = 0 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, 56]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 56]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 8, 15, 7], X[16, 10, 17, 9], > X[20, 12, 21, 11], X[22, 18, 5, 17], X[18, 22, 19, 21], X[8, 16, 9, 15], > X[10, 20, 11, 19], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -8, 4, -9, 5, -2, 11, -3, 8, -4, 6, -7, > 9, -5, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 -Sqrt[q] + 2 q - 6 q + 8 q - 12 q + 13 q - 14 q + 15/2 17/2 19/2 21/2 23/2 > 13 q - 9 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 6 8 10 12 16 18 20 22 24 28 q + 2 q + 3 q + q + 5 q + 3 q + q - 2 q + q - 4 q - q - 30 32 34 > q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 56]][a, z]
Out[8]=	3 3 3 3 1 1 2 2 2 z 3 z 4 z 5 z 3 z 5 z 5 z 4 z ---- - ---- - ---- + ---- + --- - --- - --- + --- + ---- - ---- - ---- + ---- + 9 7 5 3 9 7 5 3 9 7 5 3 a z a z a z a z a a a a a a a a 5 5 5 5 7 7 z 4 z 4 z z z z > -- - ---- - ---- + -- - -- - -- 9 7 5 3 7 5 a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 56]][a, z]
Out[9]=	2 2 -12 3 3 1 1 2 2 z 8 z 7 z z 3 z a - -- + -- + ---- + ---- - ---- - ---- + --- + --- + --- + --- - ---- - 8 4 9 7 5 3 11 5 3 14 12 a a a z a z a z a z a a a a a 2 2 2 2 3 3 3 3 3 3 z 10 z 4 z 3 z 3 z 5 z 3 z 2 z 16 z 9 z > --- + ----- + ---- - ---- + ---- - ---- - ---- - ---- - ----- - ---- - 10 8 6 4 13 11 9 7 5 3 a a a a a a a a a a 4 4 4 4 4 4 5 5 5 5 5 z 6 z 6 z 23 z 15 z 5 z 3 z 8 z z 6 z 9 z > --- + ---- - ---- - ----- - ----- - ---- - ---- + ---- + -- - ---- + ---- + 14 12 10 8 6 4 13 11 9 7 5 a a a a a a a a a a a 5 6 6 6 6 6 7 7 7 7 5 z 5 z 10 z 25 z 17 z 7 z 6 z 5 z 15 z 3 z > ---- - ---- + ----- + ----- + ----- + ---- - ---- + ---- + ----- + ---- - 3 12 10 8 6 4 11 9 7 5 a a a a a a a a a a 7 8 8 8 8 9 9 9 10 10 z 6 z 7 z 3 z 2 z 4 z 6 z 2 z z z > -- - ---- - ---- - ---- - ---- - ---- - ---- - ---- - --- - --- 3 10 8 6 4 9 7 5 8 6 a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 4 6 -2 q q 6 8 8 2 10 2 10 3 5 q + 3 q + t + -- + -- + 5 q t + 3 q t + 7 q t + 5 q t + 6 q t + t t 12 3 12 4 14 4 14 5 16 5 16 6 > 7 q t + 8 q t + 6 q t + 5 q t + 8 q t + 4 q t + 18 6 18 7 20 7 20 8 22 8 24 9 > 5 q t + 2 q t + 4 q t + q t + 2 q t + q t

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