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DrawMorseLink

PD Presentation:

X8192 X20,9,21,10 X6,21,1,22 X16,8,17,7 X14,6,15,5 X4,14,5,13 X12,18,13,17 X10,4,11,3 X18,12,19,11 X22,16,7,15 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

q-3/2 - 4q-1/2 + 9q1/2 - 16q3/2 + 20q5/2 - 25q7/2 + 24q9/2 - 21q11/2 + 16q13/2 - 9q15/2 + 4q17/2 - q19/2

A2 (sl(3)) Invariant:

- q-4 + 2q-2 - 2 + 4q4 - 3q6 + 7q8 + q10 + 3q12 + 4q14 - 4q16 + 3q18 - 5q20 - q22 + 2q24 - 2q26 + q28

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-11z3 - a-11z5 - a-10z2 + 5a-10z4 - 4a-10z6 + a-9z - 4a-9z3 + 11a-9z5 - 8a-9z7 + a-8 + 4a-8z2 - 14a-8z4 + 19a-8z6 - 11a-8z8 - a-7z-1 + 2a-7z - 4a-7z3 - 2a-7z5 + 11a-7z7 - 9a-7z9 + 3a-6 + 9a-6z2 - 41a-6z4 + 44a-6z6 - 14a-6z8 - 3a-6z10 - 3a-5z-1 + 5a-5z - 3a-5z3 - 17a-5z5 + 32a-5z7 - 16a-5z9 + 3a-4 + 6a-4z2 - 30a-4z4 + 36a-4z6 - 10a-4z8 - 3a-4z10 - 2a-3z-1 + 6a-3z - 10a-3z3 + 6a-3z5 + 9a-3z7 - 7a-3z9 + a-2z2 - 6a-2z4 + 14a-2z6 - 7a-2z8 + 2a-1z - 6a-1z3 + 9a-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 r = 7 r = 8 j = 20                       1 j = 18                     3   j = 16                   6 1   j = 14                 10 3     j = 12               11 6       j = 10             13 10         j = 8           12 11           j = 6         9 14             j = 4       7 11               j = 2     3 10                 j = 0   1 6                   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, Alternating, 179]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 179]]
Out[4]=	PD[X[8, 1, 9, 2], X[20, 9, 21, 10], X[6, 21, 1, 22], X[16, 8, 17, 7], > X[14, 6, 15, 5], X[4, 14, 5, 13], X[12, 18, 13, 17], X[10, 4, 11, 3], > X[18, 12, 19, 11], X[22, 16, 7, 15], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 8, -6, 5, -3}, > {4, -1, 2, -8, 9, -7, 6, -5, 10, -4, 7, -9, 11, -2, 3, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 4 3/2 5/2 7/2 9/2 q - ------- + 9 Sqrt[q] - 16 q + 20 q - 25 q + 24 q - Sqrt[q] 11/2 13/2 15/2 17/2 19/2 > 21 q + 16 q - 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 2 4 6 8 10 12 14 16 18 -2 - q + -- + 4 q - 3 q + 7 q + q + 3 q + 4 q - 4 q + 3 q - 2 q 20 22 24 26 28 > 5 q - q + 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 179]][a, z]
Out[8]=	3 3 3 3 5 1 3 2 z z 4 z z 2 z 3 z 4 z 2 z z ---- - ---- + ---- - -- - -- + --- - - - ---- + ---- + ---- - ---- - -- + 7 5 3 7 5 3 a 7 5 3 a 7 a z a z a z a a a a a a a 5 5 5 7 7 3 z 3 z z z z > ---- + ---- - -- + -- + -- 5 3 a 5 3 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 179]][a, z]
Out[9]=	2 -8 3 3 1 3 2 z 2 z 5 z 6 z 2 z 2 z a + -- + -- - ---- - ---- - ---- + -- + --- + --- + --- + --- - z - --- + 6 4 7 5 3 9 7 5 3 a 10 a a a z a z a z a a a a a 2 2 2 2 3 3 3 3 3 3 4 z 9 z 6 z z z 4 z 4 z 3 z 10 z 6 z 4 > ---- + ---- + ---- + -- + --- - ---- - ---- - ---- - ----- - ---- + 2 z + 8 6 4 2 11 9 7 5 3 a a a a a a a a a a 4 4 4 4 4 5 5 5 5 5 5 z 14 z 41 z 30 z 6 z z 11 z 2 z 17 z 6 z > ---- - ----- - ----- - ----- - ---- - --- + ----- - ---- - ----- + ---- + 10 8 6 4 2 11 9 7 5 3 a a a a a a a a a a 5 6 6 6 6 6 7 7 7 9 z 6 4 z 19 z 44 z 36 z 14 z 8 z 11 z 32 z > ---- - z - ---- + ----- + ----- + ----- + ----- - ---- + ----- + ----- + a 10 8 6 4 2 9 7 5 a a a a a a a a 7 7 8 8 8 8 9 9 9 10 9 z 4 z 11 z 14 z 10 z 7 z 9 z 16 z 7 z 3 z > ---- - ---- - ----- - ----- - ----- - ---- - ---- - ----- - ---- - ----- - 3 a 8 6 4 2 7 5 3 6 a a a a a a a a a 10 3 z > ----- 4 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 -2 3 6 3 q 4 6 6 2 10 q + 7 q + ----- + t + ----- + - + ---- + 11 q t + 9 q t + 14 q t + 4 3 2 2 t t q t q t 8 2 8 3 10 3 10 4 12 4 12 5 > 12 q t + 11 q t + 13 q t + 10 q t + 11 q t + 6 q t + 14 5 14 6 16 6 16 7 18 7 20 8 > 10 q t + 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a179

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