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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,17,12,16 X7,15,8,14 X15,9,16,8 X13,21,14,20 X17,5,18,22 X21,19,22,18 X19,13,20,12 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-5/2 + q-3/2 - 3q-1/2 + 2q1/2 - 3q3/2 + 2q5/2 - q7/2 + q9/2 - q13/2 + q15/2

A2 (sl(3)) Invariant:

q-8 + q-6 + 2q-4 + 3q-2 + 3 + 3q2 + q4 + q6 - 2q10 - q12 - 2q14 - q24

HOMFLY-PT Polynomial:

a-7z + a-5z-1 + a-5z - a-3z-1 - 3a-3z - 4a-3z3 - a-3z5 - 2a-1z-1 - 4a-1z - 4a-1z3 - a-1z5 + 2az-1 + 3az + az3

Kauffman Polynomial:

a-8 - 5a-8z2 + 5a-8z4 - a-8z6 + 2a-7z - 8a-7z3 + 6a-7z5 - a-7z7 - 2a-6z2 + 2a-6z4 + a-5z-1 + a-5z - 9a-5z3 + 7a-5z5 - a-5z7 - 3a-4 + 17a-4z2 - 24a-4z4 + 13a-4z6 - 2a-4z8 + a-3z-1 - 4a-3z + 5a-3z3 - 6a-3z5 + 5a-3z7 - a-3z9 + 9a-2z2 - 22a-2z4 + 16a-2z6 - 3a-2z8 - 2a-1z-1 + 5a-1z - 5a-1z3 - a-1z5 + 4a-1z7 - a-1z9 + 3 - 5z2 - z4 + 4z6 - z8 - 2az-1 + 8az - 11az3 + 6az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 16                       1 j = 14                         j = 12                 1 1 1   j = 10               2 1       j = 8             1 2 1       j = 6           3 3 1         j = 4         1 1 1           j = 2       2 4 1             j = 0     1 1 1               j = -2     2                   j = -4 1 1                     j = -6 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, 54]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 54]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 17, 12, 16], X[7, 15, 8, 14], > X[15, 9, 16, 8], X[13, 21, 14, 20], X[17, 5, 18, 22], X[21, 19, 22, 18], > X[19, 13, 20, 12], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9, -6, 4, -5, 3, -7, 8, > -9, 6, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) -(3/2) 3 3/2 5/2 7/2 9/2 -q + q - ------- + 2 Sqrt[q] - 3 q + 2 q - q + q - Sqrt[q] 13/2 15/2 > q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 2 3 2 4 6 10 12 14 24 3 + q + q + -- + -- + 3 q + q + q - 2 q - q - 2 q - q 4 2 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 54]][a, z]
Out[8]=	3 3 1 1 2 2 a z z 3 z 4 z 4 z 4 z 3 ---- - ---- - --- + --- + -- + -- - --- - --- + 3 a z - ---- - ---- + a z - 5 3 a z z 7 5 3 a 3 a a z a z a a a a 5 5 z z > -- - -- 3 a a
In[9]:=	Kauffman[Link[11, NonAlternating, 54]][a, z]
Out[9]=	-8 3 1 1 2 2 a 2 z z 4 z 5 z 2 3 + a - -- + ---- + ---- - --- - --- + --- + -- - --- + --- + 8 a z - 5 z - 4 5 3 a z z 7 5 3 a a a z a z a a a 2 2 2 2 3 3 3 3 5 z 2 z 17 z 9 z 8 z 9 z 5 z 5 z 3 4 > ---- - ---- + ----- + ---- - ---- - ---- + ---- - ---- - 11 a z - z + 8 6 4 2 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 5 5 z 2 z 24 z 22 z 6 z 7 z 6 z z 5 6 > ---- + ---- - ----- - ----- + ---- + ---- - ---- - -- + 6 a z + 4 z - 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 7 7 7 7 8 8 9 z 13 z 16 z z z 5 z 4 z 7 8 2 z 3 z z > -- + ----- + ----- - -- - -- + ---- + ---- - a z - z - ---- - ---- - -- - 8 4 2 7 5 3 a 4 2 3 a a a a a a a a a 9 z > -- a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 2 1 2 q 2 4 1 + 4 q + q + ----- + ----- + ----- + t + ----- + - + ---- + q t + q t + 6 4 4 4 4 3 2 2 t t q t q t q t q t 6 4 2 6 2 8 2 6 3 8 3 10 3 8 4 > 3 q t + q t + 3 q t + q t + q t + 2 q t + 2 q t + q t + 10 4 12 4 12 5 12 6 16 7 > q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n54

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