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L11n242 L11n244

Knotscape

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DrawMorseLink

PD Presentation:

X12,1,13,2 X16,7,17,8 X5,1,6,10 X3746 X9,5,10,4 X20,14,21,13 X22,17,11,18 X18,21,19,22 X14,20,15,19 X2,11,3,12 X8,15,9,16

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 5q-7/2 + 6q-5/2 - 7q-3/2 + 7q-1/2 - 7q1/2 + 5q3/2 - 3q5/2 + q7/2

A2 (sl(3)) Invariant:

q-18 + 2q-16 + q-14 + 3q-12 - q-8 - 2q-4 + 2q-2 + 1 + 2q2 + 2q4 - q6 + q8 - q10 - q12

HOMFLY-PT Polynomial:

a-3z-1 + a-3z - 4a-1z-1 - 7a-1z - 3a-1z3 + 6az-1 + 11az + 8az3 + 2az5 - 5a3z-1 - 8a3z - 3a3z3 + 2a5z-1 + a5z

Kauffman Polynomial:

a-4 - a-4z2 - a-3z-1 + 3a-3z - 3a-3z3 + 3a-2 - 8a-2z2 + 5a-2z4 - 2a-2z6 - 4a-1z-1 + 18a-1z - 30a-1z3 + 20a-1z5 - 5a-1z7 + 3 - 6z2 - 5z4 + 12z6 - 4z8 - 6az-1 + 30az - 54az3 + 38az5 - 6az7 - az9 + a2 + 2a2z2 - 18a2z4 + 22a2z6 - 6a2z8 - 5a3z-1 + 22a3z - 36a3z3 + 23a3z5 - 2a3z7 - a3z9 + a4 + a4z2 - 8a4z4 + 8a4z6 - 2a4z8 - 2a5z-1 + 7a5z - 9a5z3 + 5a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 8                   1 j = 6                 2   j = 4               3 1   j = 2             4 2     j = 0           4 4       j = -2         3 4 1       j = -4       3 4           j = -6     2 3             j = -8   1 4               j = -10   1                 j = -12 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, 243]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 243]]
Out[4]=	PD[X[12, 1, 13, 2], X[16, 7, 17, 8], X[5, 1, 6, 10], X[3, 7, 4, 6], > X[9, 5, 10, 4], X[20, 14, 21, 13], X[22, 17, 11, 18], X[18, 21, 19, 22], > X[14, 20, 15, 19], X[2, 11, 3, 12], X[8, 15, 9, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -4, 5, -3, 4, 2, -11, -5, 3}, > {10, -1, 6, -9, 11, -2, 7, -8, 9, -6, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 5 6 7 7 3/2 5/2 -q + ---- - ---- + ---- - ---- + ------- - 7 Sqrt[q] + 5 q - 3 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 7/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 -14 3 -8 2 2 2 4 6 8 10 12 1 + q + --- + q + --- - q - -- + -- + 2 q + 2 q - q + q - q - q 16 12 4 2 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 243]][a, z]
Out[8]=	3 5 3 1 4 6 a 5 a 2 a z 7 z 3 5 3 z ---- - --- + --- - ---- + ---- + -- - --- + 11 a z - 8 a z + a z - ---- + 3 a z z z z 3 a a a z a 3 3 3 5 > 8 a z - 3 a z + 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 243]][a, z]
Out[9]=	3 5 -4 3 2 4 1 4 6 a 5 a 2 a 3 z 18 z 3 + a + -- + a + a - ---- - --- - --- - ---- - ---- + --- + ---- + 30 a z + 2 3 a z z z z 3 a a a z a 2 2 3 3 3 5 2 z 8 z 2 2 4 2 3 z 30 z > 22 a z + 7 a z - 6 z - -- - ---- + 2 a z + a z - ---- - ----- - 4 2 3 a a a a 4 5 3 3 3 5 3 4 5 z 2 4 4 4 20 z > 54 a z - 36 a z - 9 a z - 5 z + ---- - 18 a z - 8 a z + ----- + 2 a a 6 7 5 3 5 5 5 6 2 z 2 6 4 6 5 z > 38 a z + 23 a z + 5 a z + 12 z - ---- + 22 a z + 8 a z - ---- - 2 a a 7 3 7 5 7 8 2 8 4 8 9 3 9 > 6 a z - 2 a z - a z - 4 z - 6 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-2 2 1 1 1 4 2 3 3 4 + q + 4 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 q t q t q t q t q t q t q t 4 3 4 4 2 4 4 2 6 2 8 3 > ----- + ----- + - + ---- + 2 q t + 3 q t + q t + 2 q t + q t 4 2 2 2 t 2 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n243

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