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L11n190 L11n192

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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X16,6,17,5 X15,21,16,20 X21,19,22,18 X19,15,20,14 X2738 X4,11,5,12 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 5q-1/2 + 5q1/2 - 7q3/2 + 7q5/2 - 6q7/2 + 5q9/2 - 3q11/2 + q13/2

A2 (sl(3)) Invariant:

q-8 + q-6 + q-4 + 3q-2 + 2 + 3q2 + q4 + q8 - 3q10 - 2q14 + q18 + q22 - q24

HOMFLY-PT Polynomial:

a-7z - 3a-5z - 4a-5z3 - a-5z5 + a-3z-1 + 6a-3z + 8a-3z3 + 5a-3z5 + a-3z7 - 3a-1z-1 - 8a-1z - 8a-1z3 - 2a-1z5 + 2az-1 + 3az + az3

Kauffman Polynomial:

- a-8z2 + a-7z - 3a-7z3 + a-6z2 - a-6z4 - a-6z6 + 3a-5z - 8a-5z3 + 8a-5z5 - 3a-5z7 + a-4 - 4a-4z4 + 8a-4z6 - 3a-4z8 - a-3z-1 + 8a-3z - 23a-3z3 + 21a-3z5 - 3a-3z7 - a-3z9 + 3a-2 - 4a-2z2 - 10a-2z4 + 17a-2z6 - 5a-2z8 - 3a-1z-1 + 13a-1z - 27a-1z3 + 18a-1z5 - a-1z7 - a-1z9 + 3 - 2z2 - 7z4 + 8z6 - 2z8 - 2az-1 + 7az - 9az3 + 5az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 14                   1 j = 12                 2   j = 10               3 1   j = 8             3 2     j = 6           4 3       j = 4         3 3         j = 2       3 5           j = 0     2 2             j = -2   1 4               j = -4   1                 j = -6 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, 191]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 191]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[22, 10, 7, 9], X[10, 14, 11, 13], > X[16, 6, 17, 5], X[15, 21, 16, 20], X[21, 19, 22, 18], X[19, 15, 20, 14], > X[2, 7, 3, 8], X[4, 11, 5, 12], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 5, -11}, > {9, -1, 3, -4, 10, -2, 4, 8, -6, -5, 11, 7, -8, 6, -7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 5 3/2 5/2 7/2 9/2 -q + ---- - ------- + 5 Sqrt[q] - 7 q + 7 q - 6 q + 5 q - 3/2 Sqrt[q] q 11/2 13/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -4 3 2 4 8 10 14 18 22 24 2 + q + q + q + -- + 3 q + q + q - 3 q - 2 q + q + q - q 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 191]][a, z]
Out[8]=	3 3 3 1 3 2 a z 3 z 6 z 8 z 4 z 8 z 8 z 3 ---- - --- + --- + -- - --- + --- - --- + 3 a z - ---- + ---- - ---- + a z - 3 a z z 7 5 3 a 5 3 a a z a a a a a 5 5 5 7 z 5 z 2 z z > -- + ---- - ---- + -- 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 191]][a, z]
Out[9]=	2 -4 3 1 3 2 a z 3 z 8 z 13 z 2 z 3 + a + -- - ---- - --- - --- + -- + --- + --- + ---- + 7 a z - 2 z - -- + 2 3 a z z 7 5 3 a 8 a a z a a a a 2 2 3 3 3 3 4 4 z 4 z 3 z 8 z 23 z 27 z 3 4 z 4 z > -- - ---- - ---- - ---- - ----- - ----- - 9 a z - 7 z - -- - ---- - 6 2 7 5 3 a 6 4 a a a a a a a 4 5 5 5 6 6 6 7 10 z 8 z 21 z 18 z 5 6 z 8 z 17 z 3 z > ----- + ---- + ----- + ----- + 5 a z + 8 z - -- + ---- + ----- - ---- - 2 5 3 a 6 4 2 5 a a a a a a a 7 7 8 8 9 9 3 z z 7 8 3 z 5 z z z > ---- - -- - a z - 2 z - ---- - ---- - -- - -- 3 a 4 2 3 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 2 4 2 3 q 4 6 5 q + 3 q + ----- + ----- + ----- + -- + ----- + - + ---- + 3 q t + 4 q t + 6 4 4 3 2 3 2 2 2 t t q t q t q t t q t 6 2 8 2 8 3 10 3 10 4 12 4 14 5 > 3 q t + 3 q t + 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n191

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