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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-3/2 + q-1/2 - 3q1/2 + 3q3/2 - 4q5/2 + 4q7/2 - 4q9/2 + 4q11/2 - 3q13/2 + 2q15/2 - 2q17/2 + q19/2

A2 (sl(3)) Invariant:

q-6 + q-4 + q-2 + 3 + q2 + q4 + q6 - q18 + q22 + q26 - q30

HOMFLY-PT Polynomial:

a-9z - a-7z - a-7z3 - a-5z3 - a-3z3 - a-1z-1 - a-1z - a-1z3 + az-1 + az

Kauffman Polynomial:

4a-10z2 - 10a-10z4 + 6a-10z6 - a-10z8 - 2a-9z + 17a-9z3 - 26a-9z5 + 13a-9z7 - 2a-9z9 + a-8z4 - 6a-8z6 + 5a-8z8 - a-8z10 - 2a-7z + 17a-7z3 - 29a-7z5 + 17a-7z7 - 3a-7z9 - 4a-6z2 + 10a-6z4 - 9a-6z6 + 5a-6z8 - a-6z10 - a-5z5 + 3a-5z7 - a-5z9 + 2a-4z6 - a-4z8 + a-3z5 - a-3z7 - a-2z6 - a-1z-1 + 2a-1z - a-1z3 - a-1z5 + 1 - z4 - az-1 + 2az - az3

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 = 20                       1 j = 18                     1   j = 16                   1 1   j = 14                 2 1     j = 12               2 1       j = 10             2 2         j = 8           2 2           j = 6         2 2             j = 4       1 2               j = 2     2 2                 j = 0   1 3                   j = -2                         j = -4 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, 132]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 132]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[22, 8, 5, 7], X[20, 10, 21, 9], > X[18, 12, 19, 11], X[16, 14, 17, 13], X[12, 18, 13, 17], X[10, 20, 11, 19], > X[8, 22, 9, 21], X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 4, -8, 5, -7, 6, -2, 11, -6, 7, -5, > 8, -4, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 1 3/2 5/2 7/2 9/2 11/2 -q + ------- - 3 Sqrt[q] + 3 q - 4 q + 4 q - 4 q + 4 q - Sqrt[q] 13/2 15/2 17/2 19/2 > 3 q + 2 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -4 -2 2 4 6 18 22 26 30 3 + q + q + q + q + q + q - q + q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 132]][a, z]
Out[8]=	3 3 3 3 1 a z z z z z z z -(---) + - + -- - -- - - + a z - -- - -- - -- - -- a z z 9 7 a 7 5 3 a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 132]][a, z]
Out[9]=	2 2 3 3 3 1 a 2 z 2 z 2 z 4 z 4 z 17 z 17 z z 1 - --- - - - --- - --- + --- + 2 a z + ---- - ---- + ----- + ----- - -- - a z z 9 7 a 10 6 9 7 a a a a a a a 4 4 4 5 5 5 5 5 6 3 4 10 z z 10 z 26 z 29 z z z z 6 z > a z - z - ----- + -- + ----- - ----- - ----- - -- + -- - -- + ---- - 10 8 6 9 7 5 3 a 10 a a a a a a a a 6 6 6 6 7 7 7 7 8 8 8 6 z 9 z 2 z z 13 z 17 z 3 z z z 5 z 5 z > ---- - ---- + ---- - -- + ----- + ----- + ---- - -- - --- + ---- + ---- - 8 6 4 2 9 7 5 3 10 8 6 a a a a a a a a a a a 8 9 9 9 10 10 z 2 z 3 z z z z > -- - ---- - ---- - -- - --- - --- 4 9 7 5 8 6 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 4 4 2 6 2 6 3 8 3 3 + 2 q + ----- + - + 2 q t + q t + 2 q t + 2 q t + 2 q t + 2 q t + 4 2 t q t 8 4 10 4 10 5 12 5 12 6 14 6 14 7 > 2 q t + 2 q t + 2 q t + 2 q t + q t + 2 q t + q t + 16 7 16 8 18 8 20 9 > q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a132

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