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L11n430 L11n432

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DrawMorseLink

PD Presentation:

X8192 X20,10,21,9 X5,15,6,14 X12,14,7,13 X16,8,17,7 X22,18,13,17 X10,4,11,3 X18,11,19,12 X15,1,16,6 X4,20,5,19 X2,21,3,22

Gauss Code:

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

Jones Polynomial:

- 1 + 5q - 7q2 + 11q3 - 11q4 + 12q5 - 9q6 + 7q7 - 4q8 + q9

A2 (sl(3)) Invariant:

- 1 + 3q2 + 3q6 + 6q8 + 2q10 + 6q12 + 2q14 + 4q16 + 3q18 - q20 + 2q22 - 2q24 - q26 + q28

HOMFLY-PT Polynomial:

a-8z2 + a-6z-2 - 3a-6z2 - 2a-6z4 - 2a-4z-2 - 2a-4 + 3a-4z2 + 3a-4z4 + a-4z6 + a-2z-2 + 2a-2 - a-2z4

Kauffman Polynomial:

- 2a-10z4 + a-10z6 + 5a-9z3 - 11a-9z5 + 4a-9z7 - 4a-8z2 + 15a-8z4 - 19a-8z6 + 6a-8z8 + 8a-7z3 - 10a-7z5 - 3a-7z7 + 3a-7z9 + a-6z-2 - 2a-6 - 11a-6z2 + 37a-6z4 - 36a-6z6 + 11a-6z8 - 2a-5z-1 + 2a-5z + 2a-5z3 + a-5z5 - 5a-5z7 + 3a-5z9 + 2a-4z-2 - 3a-4 - 10a-4z2 + 25a-4z4 - 16a-4z6 + 5a-4z8 - 2a-3z-1 + 2a-3z + 2a-3z7 + a-2z-2 - 2a-2 - 3a-2z2 + 5a-2z4 + a-1z3

Khovanov Homology:

trqj r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                   1 j = 17                 3   j = 15               4 1   j = 13             5 3     j = 11           7 4       j = 9         5 6         j = 7       6 6           j = 5     3 7             j = 3   2 4               j = 1   4                 j = -1 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 431]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 431]]
Out[4]=	PD[X[8, 1, 9, 2], X[20, 10, 21, 9], X[5, 15, 6, 14], X[12, 14, 7, 13], > X[16, 8, 17, 7], X[22, 18, 13, 17], X[10, 4, 11, 3], X[18, 11, 19, 12], > X[15, 1, 16, 6], X[4, 20, 5, 19], X[2, 21, 3, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 7, -10, -3, 9}, {5, -1, 2, -7, 8, -4}, > {4, 3, -9, -5, 6, -8, 10, -2, 11, -6}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 5 6 7 8 9 -1 + 5 q - 7 q + 11 q - 11 q + 12 q - 9 q + 7 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 6 8 10 12 14 16 18 20 22 -1 + 3 q + 3 q + 6 q + 2 q + 6 q + 2 q + 4 q + 3 q - q + 2 q - 24 26 28 > 2 q - q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 431]][a, z]
Out[8]=	2 2 2 4 4 4 6 -2 2 1 2 1 z 3 z 3 z 2 z 3 z z z -- + -- + ----- - ----- + ----- + -- - ---- + ---- - ---- + ---- - -- + -- 4 2 6 2 4 2 2 2 8 6 4 6 4 2 4 a a a z a z a z a a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 431]][a, z]
Out[9]=	2 2 -2 3 2 1 2 1 2 2 2 z 2 z 4 z 11 z -- - -- - -- + ----- + ----- + ----- - ---- - ---- + --- + --- - ---- - ----- - 6 4 2 6 2 4 2 2 2 5 3 5 3 8 6 a a a a z a z a z a z a z a a a a 2 2 3 3 3 3 4 4 4 4 10 z 3 z 5 z 8 z 2 z z 2 z 15 z 37 z 25 z > ----- - ---- + ---- + ---- + ---- + -- - ---- + ----- + ----- + ----- + 4 2 9 7 5 a 10 8 6 4 a a a a a a a a a 4 5 5 5 6 6 6 6 7 7 5 z 11 z 10 z z z 19 z 36 z 16 z 4 z 3 z > ---- - ----- - ----- + -- + --- - ----- - ----- - ----- + ---- - ---- - 2 9 7 5 10 8 6 4 9 7 a a a a a a a a a a 7 7 8 8 8 9 9 5 z 2 z 6 z 11 z 5 z 3 z 3 z > ---- + ---- + ---- + ----- + ---- + ---- + ---- 5 3 8 6 4 7 5 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 3 5 5 2 7 2 7 3 9 3 4 q + 2 q + --- + 4 q t + 3 q t + 7 q t + 6 q t + 6 q t + 5 q t + q t 9 4 11 4 11 5 13 5 13 6 15 6 15 7 > 6 q t + 7 q t + 4 q t + 5 q t + 3 q t + 4 q t + q t + 17 7 19 8 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n431

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