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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-3 - 3q-2 + 9q-1 - 14 + 22q - 25q2 + 27q3 - 23q4 + 18q5 - 12q6 + 5q7 - q8

A2 (sl(3)) Invariant:

q-10 - q-6 + 5q-4 + 3 + 10q2 + q4 + 9q6 + 2q8 + 2q10 + 3q12 - 6q14 + 3q16 - 4q18 - 3q20 + 3q22 - q24

HOMFLY-PT Polynomial:

- a-6z-2 - a-6 - a-6z4 + 4a-4z-2 + 7a-4 + 3a-4z2 + a-4z4 + a-4z6 - 5a-2z-2 - 10a-2 - 7a-2z2 - a-2z4 + a-2z6 + 2z-2 + 3 - z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

a-9z5 - 3a-8z4 + 5a-8z6 + a-7z-1 - 2a-7z + 5a-7z3 - 16a-7z5 + 12a-7z7 - a-6z-2 + 2a-6 - 4a-6z2 + 11a-6z4 - 23a-6z6 + 15a-6z8 + 5a-5z-1 - 15a-5z + 26a-5z3 - 30a-5z5 + 3a-5z7 + 9a-5z9 - 4a-4z-2 + 14a-4 - 25a-4z2 + 47a-4z4 - 60a-4z6 + 25a-4z8 + 2a-4z10 + 9a-3z-1 - 33a-3z + 52a-3z3 - 31a-3z5 - 11a-3z7 + 14a-3z9 - 5a-2z-2 + 21a-2 - 40a-2z2 + 53a-2z4 - 47a-2z6 + 16a-2z8 + 2a-2z10 + 5a-1z-1 - 20a-1z + 34a-1z3 - 24a-1z5 + a-1z7 + 5a-1z9 - 2z-2 + 9 - 16z2 + 17z4 - 14z6 + 6z8 + 3az3 - 6az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     4   j = 13                   8 1   j = 11                 10 4     j = 9               14 9       j = 7             13 9         j = 5           12 14           j = 3         10 13             j = 1       6 14               j = -1     3 8                 j = -3     6                   j = -5 1 3                     j = -7 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 500]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 500]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[22, 14, 19, 13], X[20, 8, 21, 7], > X[10, 20, 11, 19], X[16, 10, 17, 9], X[14, 18, 15, 17], X[8, 16, 9, 15], > X[18, 22, 5, 21], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {5, -4, 9, -3}, > {10, -1, 4, -8, 6, -5, 11, -2, 3, -7, 8, -6, 7, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-3 3 9 2 3 4 5 6 7 8 -14 + q - -- + - + 22 q - 25 q + 27 q - 23 q + 18 q - 12 q + 5 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -6 5 2 4 6 8 10 12 14 3 + q - q + -- + 10 q + q + 9 q + 2 q + 2 q + 3 q - 6 q + 4 q 16 18 20 22 24 > 3 q - 4 q - 3 q + 3 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 500]][a, z]
Out[8]=	2 2 -6 7 10 2 2 1 4 5 2 3 z 7 z 3 - a + -- - -- + a + -- - ----- + ----- - ----- - z + ---- - ---- + 4 2 2 6 2 4 2 2 2 4 2 a a z a z a z a z a a 4 4 4 6 6 2 2 4 z z z z z > a z - 2 z - -- + -- - -- + -- + -- 6 4 2 4 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 500]][a, z]
Out[9]=	2 14 21 2 2 1 4 5 1 5 9 5 9 + -- + -- + -- - a - -- - ----- - ----- - ----- + ---- + ---- + ---- + --- - 6 4 2 2 6 2 4 2 2 2 7 5 3 a z a a a z a z a z a z a z a z a z 2 2 2 3 2 z 15 z 33 z 20 z 2 4 z 25 z 40 z 2 2 5 z > --- - ---- - ---- - ---- - 16 z - ---- - ----- - ----- + 3 a z + ---- + 7 5 3 a 6 4 2 7 a a a a a a a 3 3 3 4 4 4 4 26 z 52 z 34 z 3 4 3 z 11 z 47 z 53 z > ----- + ----- + ----- + 3 a z + 17 z - ---- + ----- + ----- + ----- - 5 3 a 8 6 4 2 a a a a a a 5 5 5 5 5 6 2 4 z 16 z 30 z 31 z 24 z 5 6 5 z > 3 a z + -- - ----- - ----- - ----- - ----- - 6 a z - 14 z + ---- - 9 7 5 3 a 8 a a a a a 6 6 6 7 7 7 7 23 z 60 z 47 z 2 6 12 z 3 z 11 z z 7 8 > ----- - ----- - ----- + a z + ----- + ---- - ----- + -- + 3 a z + 6 z + 6 4 2 7 5 3 a a a a a a a 8 8 8 9 9 9 10 10 15 z 25 z 16 z 9 z 14 z 5 z 2 z 2 z > ----- + ----- + ----- + ---- + ----- + ---- + ----- + ----- 6 4 2 5 3 a 4 2 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 3 6 3 8 6 q 3 14 q + 10 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 13 q t + 7 4 5 4 5 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 12 q t + 14 q t + 13 q t + 9 q t + 14 q t + 9 q t + 10 q t + 11 5 13 5 13 6 15 6 17 7 > 4 q t + 8 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a500

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