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L11n128 L11n130

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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X22,10,7,9 X2738 X4,15,5,16 X5,13,6,12 X16,12,17,11 X17,6,18,1 X19,14,20,15 X13,20,14,21 X21,19,22,18

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:

- q-9/2 + 2q-7/2 - 5q-5/2 + 6q-3/2 - 7q-1/2 + 7q1/2 - 6q3/2 + 4q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-16 + 2q-14 + q-10 + 2q-8 - q-6 + q-4 - q-2 + q2 + 3q6 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - a-1z-1 - 4a-1z - 3a-1z3 - a-1z5 + 2az-1 + 5az + 3az3 - 2a3z-1 - 3a3z + a5z-1

Kauffman Polynomial:

- a-4z2 + 3a-4z4 - a-4z6 + 2a-3z - 9a-3z3 + 11a-3z5 - 3a-3z7 + 2a-2z2 - 7a-2z4 + 10a-2z6 - 3a-2z8 - a-1z-1 + 8a-1z - 22a-1z3 + 18a-1z5 - 2a-1z7 - a-1z9 + 9z2 - 23z4 + 19z6 - 5z8 - 2az-1 + 12az - 21az3 + 10az5 - az9 - a2 + 8a2z2 - 15a2z4 + 8a2z6 - 2a2z8 - 2a3z-1 + 8a3z - 9a3z3 + 3a3z5 - a3z7 + 2a4z2 - 2a4z4 - a5z-1 + 2a5z - a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                   1 j = 8                 2   j = 6               2 1   j = 4             4 2     j = 2           3 2       j = 0         4 4         j = -2       3 4           j = -4     2 3             j = -6   1 4               j = -8   1                 j = -10 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, 129]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 129]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[2, 7, 3, 8], > X[4, 15, 5, 16], X[5, 13, 6, 12], X[16, 12, 17, 11], X[17, 6, 18, 1], > X[19, 14, 20, 15], X[13, 20, 14, 21], X[21, 19, 22, 18]]
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]=	-(9/2) 2 5 6 7 3/2 5/2 -q + ---- - ---- + ---- - ------- + 7 Sqrt[q] - 6 q + 4 q - 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 -10 2 -6 -4 -2 2 6 12 14 q + --- + q + -- - q + q - q + q + 3 q + q - q 14 8 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 129]][a, z]
Out[8]=	3 5 3 3 5 1 2 a 2 a a z 4 z 3 z 3 z 3 z -(---) + --- - ---- + -- + -- - --- + 5 a z - 3 a z + -- - ---- + 3 a z - -- a z z z z 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 129]][a, z]
Out[9]=	3 5 2 1 2 a 2 a a 2 z 8 z 3 5 2 -a - --- - --- - ---- - -- + --- + --- + 12 a z + 8 a z + 2 a z + 9 z - a z z z z 3 a a 2 2 3 3 z 2 z 2 2 4 2 9 z 22 z 3 3 3 5 3 > -- + ---- + 8 a z + 2 a z - ---- - ----- - 21 a z - 9 a z - a z - 4 2 3 a a a a 4 4 5 5 4 3 z 7 z 2 4 4 4 11 z 18 z 5 > 23 z + ---- - ---- - 15 a z - 2 a z + ----- + ----- + 10 a z + 4 2 3 a a a a 6 6 7 7 3 5 6 z 10 z 2 6 3 z 2 z 3 7 8 > 3 a z + 19 z - -- + ----- + 8 a z - ---- - ---- - a z - 5 z - 4 2 3 a a a a 8 9 3 z 2 8 z 9 > ---- - 2 a z - -- - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 1 4 2 3 3 2 4 + -- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + 4 t + 3 q t + 2 10 4 8 3 6 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t 2 2 4 2 4 3 6 3 6 4 8 4 10 5 > 2 q t + 4 q t + 2 q t + 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n129

L11n128 L11n130