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L11n113 L11n115

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X18,8,19,7 X22,20,5,19 X20,9,21,10 X8,21,9,22 X11,17,12,16 X17,15,18,14 X15,11,16,10 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 5q-1/2 + 7q1/2 - 9q3/2 + 9q5/2 - 9q7/2 + 7q9/2 - 5q11/2 + 2q13/2

A2 (sl(3)) Invariant:

q-8 + q-6 + q-4 + 3q-2 + q2 - q4 - 2q6 + 2q8 - q10 + 4q12 + q14 + q16 + q18 - 2q20 - q24

HOMFLY-PT Polynomial:

a-7z-1 + a-7z - 4a-5z-1 - 6a-5z - 4a-5z3 - a-5z5 + 6a-3z-1 + 12a-3z + 10a-3z3 + 5a-3z5 + a-3z7 - 5a-1z-1 - 10a-1z - 8a-1z3 - 2a-1z5 + 2az-1 + 3az + az3

Kauffman Polynomial:

a-8 - 3a-8z2 - a-7z-1 + 4a-7z - 5a-7z3 - a-7z5 + 3a-6 - 9a-6z2 + 9a-6z4 - 5a-6z6 - 4a-5z-1 + 18a-5z - 31a-5z3 + 25a-5z5 - 8a-5z7 + 3a-4 - 8a-4z2 + 2a-4z4 + 10a-4z6 - 5a-4z8 - 6a-3z-1 + 31a-3z - 58a-3z3 + 47a-3z5 - 10a-3z7 - a-3z9 + a-2 - 16a-2z4 + 23a-2z6 - 7a-2z8 - 5a-1z-1 + 24a-1z - 41a-1z3 + 26a-1z5 - 3a-1z7 - a-1z9 + 1 + 2z2 - 9z4 + 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                   2 j = 12                 3   j = 10               4 2   j = 8             5 3     j = 6           4 4       j = 4         5 5         j = 2       4 6           j = 0     1 3             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, 114]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 114]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[18, 8, 19, 7], X[22, 20, 5, 19], > X[20, 9, 21, 10], X[8, 21, 9, 22], X[11, 17, 12, 16], X[17, 15, 18, 14], > X[15, 11, 16, 10], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -6, 5, 9, -7, -2, 11, 8, -9, 7, -8, -3, > 4, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 5 3/2 5/2 7/2 9/2 -q + ---- - ------- + 7 Sqrt[q] - 9 q + 9 q - 9 q + 7 q - 3/2 Sqrt[q] q 11/2 13/2 > 5 q + 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -4 3 2 4 6 8 10 12 14 16 18 q + q + q + -- + q - q - 2 q + 2 q - q + 4 q + q + q + q - 2 q 20 24 > 2 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 114]][a, z]
Out[8]=	3 1 4 6 5 2 a z 6 z 12 z 10 z 4 z ---- - ---- + ---- - --- + --- + -- - --- + ---- - ---- + 3 a z - ---- + 7 5 3 a z z 7 5 3 a 5 a z a z a z a a a a 3 3 5 5 5 7 10 z 8 z 3 z 5 z 2 z z > ----- - ---- + a z - -- + ---- - ---- + -- 3 a 5 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 114]][a, z]
Out[9]=	-8 3 3 -2 1 4 6 5 2 a 4 z 18 z 31 z 1 + a + -- + -- + a - ---- - ---- - ---- - --- - --- + --- + ---- + ---- + 6 4 7 5 3 a z z 7 5 3 a a a z a z a z a a a 2 2 2 3 3 3 3 24 z 2 3 z 9 z 8 z 5 z 31 z 58 z 41 z > ---- + 7 a z + 2 z - ---- - ---- - ---- - ---- - ----- - ----- - ----- - a 8 6 4 7 5 3 a a a a a a a 4 4 4 5 5 5 5 3 4 9 z 2 z 16 z z 25 z 47 z 26 z 5 > 9 a z - 9 z + ---- + ---- - ----- - -- + ----- + ----- + ----- + 5 a z + 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 7 8 6 5 z 10 z 23 z 8 z 10 z 3 z 7 8 5 z > 8 z - ---- + ----- + ----- - ---- - ----- - ---- - a z - 2 z - ---- - 6 4 2 5 3 a 4 a a a a a a 8 9 9 7 z z z > ---- - -- - -- 2 3 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 4 3 4 q 4 6 q + 5 q + ----- + ----- + ----- + t + ----- + - + ---- + 5 q t + 6 4 4 3 2 3 2 2 t t q t q t q t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 > 4 q t + 4 q t + 5 q t + 3 q t + 4 q t + 2 q t + 3 q t + 14 5 > 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n114

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