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DrawMorseLink

PD Presentation:

X8192 X14,4,15,3 X10,20,11,19 X18,10,19,9 X20,12,13,11 X12,14,7,13 X16,6,17,5 X2738 X4,16,5,15 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

q - q2 + 3q3 - 3q4 + 5q5 - 4q6 + 5q7 - 4q8 + 3q9 - 2q10 + q11

A2 (sl(3)) Invariant:

q4 + q6 + 2q8 + 2q10 + 2q12 + 4q14 + 3q16 + 5q18 + 2q20 + 2q22 + q24 + q28 + q32

HOMFLY-PT Polynomial:

a-8z-2 + 3a-8 + 7a-8z2 + 5a-8z4 + a-8z6 - 2a-6z-2 - 9a-6 - 18a-6z2 - 17a-6z4 - 7a-6z6 - a-6z8 + a-4z-2 + 6a-4 + 11a-4z2 + 6a-4z4 + a-4z6

Kauffman Polynomial:

a-14z2 + 2a-13z3 - 2a-12z2 + 3a-12z4 - 6a-11z3 + 4a-11z5 - 2a-10 + 9a-10z2 - 14a-10z4 + 5a-10z6 + 6a-9z3 - 12a-9z5 + 4a-9z7 - a-8z-2 + 3a-8 - 3a-8z2 + 9a-8z4 - 11a-8z6 + 3a-8z8 + 2a-7z-1 - 9a-7z + 20a-7z3 - 11a-7z5 - a-7z7 + a-7z9 - 2a-6z-2 + 11a-6 - 32a-6z2 + 43a-6z4 - 23a-6z6 + 4a-6z8 + 2a-5z-1 - 9a-5z + 6a-5z3 + 5a-5z5 - 5a-5z7 + a-5z9 - a-4z-2 + 7a-4 - 17a-4z2 + 17a-4z4 - 7a-4z6 + a-4z8

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 j = 23                     1 j = 21                   2 1 j = 19                 1     j = 17               3 2     j = 15             2 1       j = 13           2 3         j = 11         3 2           j = 9       2 4             j = 7     1 1               j = 5   1 3                 j = 3                       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[10, Alternating, 161]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 161]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 4, 15, 3], X[10, 20, 11, 19], X[18, 10, 19, 9], > X[20, 12, 13, 11], X[12, 14, 7, 13], X[16, 6, 17, 5], X[2, 7, 3, 8], > X[4, 16, 5, 15], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9, 7, -10}, {8, -1, 4, -3, 5, -6}, > {6, -2, 9, -7, 10, -4, 3, -5}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 5 6 7 8 9 10 11 q - q + 3 q - 3 q + 5 q - 4 q + 5 q - 4 q + 3 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	4 6 8 10 12 14 16 18 20 22 24 q + q + 2 q + 2 q + 2 q + 4 q + 3 q + 5 q + 2 q + 2 q + q + 28 32 > q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 161]][a, z]
Out[8]=	2 2 2 4 4 3 9 6 1 2 1 7 z 18 z 11 z 5 z 17 z -- - -- + -- + ----- - ----- + ----- + ---- - ----- + ----- + ---- - ----- + 8 6 4 8 2 6 2 4 2 8 6 4 8 6 a a a a z a z a z a a a a a 4 6 6 6 8 6 z z 7 z z z > ---- + -- - ---- + -- - -- 4 8 6 4 6 a a a a a
In[9]:=	Kauffman[Link[10, Alternating, 161]][a, z]
Out[9]=	2 -2 3 11 7 1 2 1 2 2 9 z 9 z z --- + -- + -- + -- - ----- - ----- - ----- + ---- + ---- - --- - --- + --- - 10 8 6 4 8 2 6 2 4 2 7 5 7 5 14 a a a a a z a z a z a z a z a a a 2 2 2 2 2 3 3 3 3 3 2 z 9 z 3 z 32 z 17 z 2 z 6 z 6 z 20 z 6 z > ---- + ---- - ---- - ----- - ----- + ---- - ---- + ---- + ----- + ---- + 12 10 8 6 4 13 11 9 7 5 a a a a a a a a a a 4 4 4 4 4 5 5 5 5 6 3 z 14 z 9 z 43 z 17 z 4 z 12 z 11 z 5 z 5 z > ---- - ----- + ---- + ----- + ----- + ---- - ----- - ----- + ---- + ---- - 12 10 8 6 4 11 9 7 5 10 a a a a a a a a a a 6 6 6 7 7 7 8 8 8 9 9 11 z 23 z 7 z 4 z z 5 z 3 z 4 z z z z > ----- - ----- - ---- + ---- - -- - ---- + ---- + ---- + -- + -- + -- 8 6 4 9 7 5 8 6 4 7 5 a a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	5 5 7 q q 7 9 9 2 11 2 11 3 3 q + q + -- + -- + q t + 2 q t + 4 q t + 3 q t + 2 q t + 2 t t 13 3 13 4 15 4 15 5 17 5 17 6 19 6 > 2 q t + 3 q t + 2 q t + q t + 3 q t + 2 q t + q t + 21 7 21 8 23 8 > 2 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a161

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