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L11a130 L11a132

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DrawMorseLink

PD Presentation:

X6172 X14,4,15,3 X20,8,21,7 X22,10,5,9 X8,22,9,21 X18,12,19,11 X16,14,17,13 X12,18,13,17 X10,20,11,19 X2536 X4,16,1,15

Gauss Code:

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

Jones Polynomial:

- q-1/2 + q1/2 - 4q3/2 + 5q5/2 - 8q7/2 + 10q9/2 - 10q11/2 + 9q13/2 - 7q15/2 + 5q17/2 - 3q19/2 + q21/2

A2 (sl(3)) Invariant:

q-2 + 1 + 2q2 + 4q4 + q6 + 3q8 + q10 - 2q12 - 2q16 - q22 + 2q24 - q26 + q30 - q32

HOMFLY-PT Polynomial:

a-9z + a-9z3 - a-7z - 2a-7z3 - a-7z5 + a-5z-1 + a-5z - a-5z3 - a-5z5 - 3a-3z-1 - 4a-3z - 3a-3z3 - a-3z5 + 2a-1z-1 + 3a-1z + a-1z3

Kauffman Polynomial:

- a-12z2 + 3a-12z4 - a-12z6 - 6a-11z3 + 10a-11z5 - 3a-11z7 + 3a-10z2 - 12a-10z4 + 14a-10z6 - 4a-10z8 - 2a-9z + 7a-9z3 - 11a-9z5 + 10a-9z7 - 3a-9z9 + a-8z2 - 7a-8z4 + 6a-8z6 - a-8z10 - 2a-7z + 13a-7z3 - 24a-7z5 + 15a-7z7 - 4a-7z9 - a-6 - a-6z2 + 6a-6z4 - 8a-6z6 + 3a-6z8 - a-6z10 + a-5z-1 - 2a-5z + 3a-5z3 - 3a-5z5 + a-5z7 - a-5z9 - 3a-4 + 5a-4z2 - a-4z4 - a-4z8 + 3a-3z-1 - 7a-3z + 7a-3z3 - a-3z5 - a-3z7 - 3a-2 + 3a-2z2 + a-2z4 - a-2z6 + 2a-1z-1 - 5a-1z + 4a-1z3 - a-1z5

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 22                       1 j = 20                     2   j = 18                   3 1   j = 16                 4 2     j = 14               5 3       j = 12             5 4         j = 10           5 5           j = 8         3 5             j = 6       2 5               j = 4     2 3                 j = 2   1 4                   j = 0                         j = -2 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, 131]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 131]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[20, 8, 21, 7], X[22, 10, 5, 9], > X[8, 22, 9, 21], X[18, 12, 19, 11], X[16, 14, 17, 13], X[12, 18, 13, 17], > X[10, 20, 11, 19], X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -5, 4, -9, 6, -8, 7, -2, 11, -7, 8, -6, > 9, -3, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + Sqrt[q] - 4 q + 5 q - 8 q + 10 q - 10 q + Sqrt[q] 13/2 15/2 17/2 19/2 21/2 > 9 q - 7 q + 5 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 2 4 6 8 10 12 16 22 24 26 1 + q + 2 q + 4 q + q + 3 q + q - 2 q - 2 q - q + 2 q - q + 30 32 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 131]][a, z]
Out[8]=	3 3 3 3 3 1 3 2 z z z 4 z 3 z z 2 z z 3 z z ---- - ---- + --- + -- - -- + -- - --- + --- + -- - ---- - -- - ---- + -- - 5 3 a z 9 7 5 3 a 9 7 5 3 a a z a z a a a a a a a a 5 5 5 z z z > -- - -- - -- 7 5 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 131]][a, z]
Out[9]=	2 2 -6 3 3 1 3 2 2 z 2 z 2 z 7 z 5 z z 3 z -a - -- - -- + ---- + ---- + --- - --- - --- - --- - --- - --- - --- + ---- + 4 2 5 3 a z 9 7 5 3 a 12 10 a a a z a z a a a a a a 2 2 2 2 3 3 3 3 3 3 4 z z 5 z 3 z 6 z 7 z 13 z 3 z 7 z 4 z 3 z > -- - -- + ---- + ---- - ---- + ---- + ----- + ---- + ---- + ---- + ---- - 8 6 4 2 11 9 7 5 3 a 12 a a a a a a a a a a 4 4 4 4 4 5 5 5 5 5 5 12 z 7 z 6 z z z 10 z 11 z 24 z 3 z z z > ----- - ---- + ---- - -- + -- + ----- - ----- - ----- - ---- - -- - -- - 10 8 6 4 2 11 9 7 5 3 a a a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 8 z 14 z 6 z 8 z z 3 z 10 z 15 z z z 4 z > --- + ----- + ---- - ---- - -- - ---- + ----- + ----- + -- - -- - ---- + 12 10 8 6 2 11 9 7 5 3 10 a a a a a a a a a a a 8 8 9 9 9 10 10 3 z z 3 z 4 z z z z > ---- - -- - ---- - ---- - -- - --- - --- 6 4 9 7 5 8 6 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 q 4 6 6 2 8 2 8 3 4 q + 2 q + ----- + -- + 3 q t + 2 q t + 5 q t + 3 q t + 5 q t + 2 2 t q t 10 3 10 4 12 4 12 5 14 5 14 6 > 5 q t + 5 q t + 5 q t + 4 q t + 5 q t + 3 q t + 16 6 16 7 18 7 18 8 20 8 22 9 > 4 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 L11a131

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