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L11n186 L11n188

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DrawMorseLink

PD Presentation:

X8192 X12,4,13,3 X22,12,7,11 X15,20,16,21 X18,10,19,9 X10,20,11,19 X21,14,22,15 X16,6,17,5 X2738 X4,14,5,13 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

- 2q-1/2 + 4q1/2 - 7q3/2 + 9q5/2 - 11q7/2 + 10q9/2 - 9q11/2 + 6q13/2 - 3q15/2 + q17/2

A2 (sl(3)) Invariant:

2q-2 + q2 + 2q4 - 2q6 + 3q8 + 2q12 + 2q14 - q16 + 2q18 - 2q20 + q24 - q26

HOMFLY-PT Polynomial:

a-7z + a-7z3 + a-5z - a-5z3 - a-5z5 - a-3z-1 - 5a-3z - 6a-3z3 - 2a-3z5 + a-1z-1 + 4a-1z + 2a-1z3

Kauffman Polynomial:

a-10z2 - a-10z4 + 3a-9z3 - 3a-9z5 - 2a-8z2 + 6a-8z4 - 5a-8z6 + 4a-7z - 11a-7z3 + 11a-7z5 - 6a-7z7 - 3a-6z4 + 5a-6z6 - 4a-6z8 + 4a-5z - 15a-5z3 + 17a-5z5 - 6a-5z7 - a-5z9 + 2a-4z2 - 6a-4z4 + 10a-4z6 - 5a-4z8 + a-3z-1 - 5a-3z + 7a-3z3 - a-3z9 - a-2 - a-2z2 + 4a-2z4 - a-2z8 + a-1z-1 - 5a-1z + 8a-1z3 - 3a-1z5

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 18                   1 j = 16                 2   j = 14               4 1   j = 12             5 2     j = 10           5 4       j = 8         6 5         j = 6       4 6           j = 4     3 5             j = 2   2 5               j = 0   2                 j = -2 2

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, 187]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 187]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 4, 13, 3], X[22, 12, 7, 11], X[15, 20, 16, 21], > X[18, 10, 19, 9], X[10, 20, 11, 19], X[21, 14, 22, 15], X[16, 6, 17, 5], > X[2, 7, 3, 8], X[4, 14, 5, 13], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 8, -11}, > {9, -1, 5, -6, 3, -2, 10, 7, -4, -8, 11, -5, 6, 4, -7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-2 3/2 5/2 7/2 9/2 11/2 13/2 ------- + 4 Sqrt[q] - 7 q + 9 q - 11 q + 10 q - 9 q + 6 q - Sqrt[q] 15/2 17/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 2 4 6 8 12 14 16 18 20 24 26 -- + q + 2 q - 2 q + 3 q + 2 q + 2 q - q + 2 q - 2 q + q - q 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 187]][a, z]
Out[8]=	3 3 3 3 5 5 1 1 z z 5 z 4 z z z 6 z 2 z z 2 z -(----) + --- + -- + -- - --- + --- + -- - -- - ---- + ---- - -- - ---- 3 a z 7 5 3 a 7 5 3 a 5 3 a z a a a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 187]][a, z]
Out[9]=	2 2 2 2 3 -2 1 1 4 z 4 z 5 z 5 z z 2 z 2 z z 3 z -a + ---- + --- + --- + --- - --- - --- + --- - ---- + ---- - -- + ---- - 3 a z 7 5 3 a 10 8 4 2 9 a z a a a a a a a a 3 3 3 3 4 4 4 4 4 5 11 z 15 z 7 z 8 z z 6 z 3 z 6 z 4 z 3 z > ----- - ----- + ---- + ---- - --- + ---- - ---- - ---- + ---- - ---- + 7 5 3 a 10 8 6 4 2 9 a a a a a a a a a 5 5 5 6 6 6 7 7 8 8 11 z 17 z 3 z 5 z 5 z 10 z 6 z 6 z 4 z 5 z > ----- + ----- - ---- - ---- + ---- + ----- - ---- - ---- - ---- - ---- - 7 5 a 8 6 4 7 5 6 4 a a a a a a a a a 8 9 9 z z z > -- - -- - -- 2 5 3 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 2 2 2 q 4 6 6 2 8 2 5 q + 3 q + ----- + - + ---- + 5 q t + 4 q t + 6 q t + 6 q t + 2 2 t t q t 8 3 10 3 10 4 12 4 12 5 14 5 14 6 > 5 q t + 5 q t + 4 q t + 5 q t + 2 q t + 4 q t + q t + 16 6 18 7 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n187

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