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L11n125 L11n127

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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X22,10,7,9 X2738 X15,5,16,4 X5,13,6,12 X11,16,12,17 X6,18,1,17 X14,20,15,19 X20,14,21,13 X18,21,19,22

Gauss Code:

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

Jones Polynomial:

- 2q-3/2 + 4q-1/2 - 7q1/2 + 9q3/2 - 10q5/2 + 9q7/2 - 8q9/2 + 5q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2a-8z2 + 3a-8z4 - a-8z6 + 3a-7z - 9a-7z3 + 10a-7z5 - 3a-7z7 - a-6z2 - a-6z4 + 7a-6z6 - 3a-6z8 - a-5z-1 + 9a-5z - 23a-5z3 + 23a-5z5 - 5a-5z7 - a-5z9 + 2a-4z2 - 10a-4z4 + 15a-4z6 - 6a-4z8 - 2a-3z-1 + 13a-3z - 26a-3z3 + 20a-3z5 - 5a-3z7 - a-3z9 - a-2 + 3a-2z2 - 8a-2z4 + 6a-2z6 - 3a-2z8 - 2a-1z-1 + 11a-1z - 15a-1z3 + 7a-1z5 - 3a-1z7 + 2z2 - 2z4 - z6 - az-1 + 4az - 3az3

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 16                   1 j = 14                 2   j = 12               3 1   j = 10             5 2     j = 8           4 3       j = 6         6 5         j = 4       4 5           j = 2     3 5             j = 0   2 5               j = -2   2                 j = -4 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, 126]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 126]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[2, 7, 3, 8], > X[15, 5, 16, 4], X[5, 13, 6, 12], X[11, 16, 12, 17], X[6, 18, 1, 17], > X[14, 20, 15, 19], X[20, 14, 21, 13], X[18, 21, 19, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, 5, -6, -8}, > {4, -1, 3, -2, -7, 6, 10, -9, -5, 7, 8, -11, 9, -10, 11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-2 4 3/2 5/2 7/2 9/2 11/2 ---- + ------- - 7 Sqrt[q] + 9 q - 10 q + 9 q - 8 q + 5 q - 3/2 Sqrt[q] q 13/2 15/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 -4 -2 2 6 8 10 12 14 16 18 20 3 + -- + q - q - q + q - q + 2 q - q + 2 q + 2 q - q + 2 q - 6 q 24 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 126]][a, z]
Out[8]=	3 3 3 1 2 2 a z 2 z 4 z 4 z 2 z 2 z 3 z -(----) + ---- - --- + - + -- - --- + --- - --- + 2 a z - ---- + ---- - ---- + 5 3 a z z 7 5 3 a 5 3 a a z a z a a a a a 5 z > -- 3 a
In[9]:=	Kauffman[Link[11, NonAlternating, 126]][a, z]
Out[9]=	2 -2 1 2 2 a 3 z 9 z 13 z 11 z 2 2 z -a - ---- - ---- - --- - - + --- + --- + ---- + ---- + 4 a z + 2 z - ---- - 5 3 a z z 7 5 3 a 8 a z a z a a a a 2 2 2 3 3 3 3 4 z 2 z 3 z 9 z 23 z 26 z 15 z 3 4 3 z > -- + ---- + ---- - ---- - ----- - ----- - ----- - 3 a z - 2 z + ---- - 6 4 2 7 5 3 a 8 a a a a a a a 4 4 4 5 5 5 5 6 6 6 z 10 z 8 z 10 z 23 z 20 z 7 z 6 z 7 z 15 z > -- - ----- - ---- + ----- + ----- + ----- + ---- - z - -- + ---- + ----- + 6 4 2 7 5 3 a 8 6 4 a a a a a a a a a 6 7 7 7 7 8 8 8 9 9 6 z 3 z 5 z 5 z 3 z 3 z 6 z 3 z z z > ---- - ---- - ---- - ---- - ---- - ---- - ---- - ---- - -- - -- 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 2 2 2 4 4 2 6 2 6 3 5 + 3 q + ----- + - + ---- + 5 q t + 4 q t + 5 q t + 6 q t + 5 q t + 4 2 t 2 q t q t 8 3 8 4 10 4 10 5 12 5 12 6 14 6 > 4 q t + 3 q t + 5 q t + 2 q t + 3 q t + q t + 2 q t + 16 7 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n126

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