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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 5q-3/2 + 7q-1/2 - 10q1/2 + 10q3/2 - 10q5/2 + 8q7/2 - 5q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant:

q-12 + q-10 + 3q-6 + q-4 + 3 - q2 + 2q4 + q10 - 3q12 + q14 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z - a-5z3 + 2a-3z + 2a-3z3 + a-3z5 + a-1z3 + a-1z5 - az-1 - 3az - 2az3 + a3z-1 + a3z

Kauffman Polynomial:

2a-7z3 - a-7z5 - 3a-6z2 + 7a-6z4 - 3a-6z6 + 2a-5z - 6a-5z3 + 9a-5z5 - 4a-5z7 - 2a-4z2 + 4a-4z6 - 3a-4z8 + 4a-3z - 13a-3z3 + 12a-3z5 - 4a-3z7 - a-3z9 + 4a-2z2 - 11a-2z4 + 10a-2z6 - 5a-2z8 - a-1z3 + 3a-1z5 - 2a-1z7 - a-1z9 + 2z2 + z6 - 2z8 + az-1 - 5az + 7az3 - 2az7 - a2 - a2z2 + 4a2z4 - 2a2z6 + a3z-1 - 3a3z + 3a3z3 - a3z5

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 j = 14                     1 j = 12                   2   j = 10                 3 1   j = 8               5 2     j = 6             5 3       j = 4           5 5         j = 2         5 5           j = 0       3 6             j = -2     2 4               j = -4   1 4                 j = -6   1                   j = -8 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[10, Alternating, 64]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 64]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[20, 10, 7, 9], X[2, 7, 3, 8], > X[16, 14, 17, 13], X[14, 5, 15, 6], X[4, 15, 5, 16], X[18, 12, 19, 11], > X[12, 18, 13, 17], X[6, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -7, 6, -10}, > {4, -1, 3, -2, 8, -9, 5, -6, 7, -5, 9, -8, 10, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 2 5 7 3/2 5/2 7/2 -q + ---- - ---- + ------- - 10 Sqrt[q] + 10 q - 10 q + 8 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 > 5 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 3 -4 2 4 10 12 14 18 20 3 + q + q + -- + q - q + 2 q + q - 3 q + q - q + q 6 q
In[8]:=	HOMFLYPT[Link[10, Alternating, 64]][a, z]
Out[8]=	3 3 3 3 5 5 a a z 2 z 3 z 2 z z 3 z z -(-) + -- - -- + --- - 3 a z + a z - -- + ---- + -- - 2 a z + -- + -- z z 5 3 5 3 a 3 a a a a a a
In[9]:=	Kauffman[Link[10, Alternating, 64]][a, z]
Out[9]=	3 2 2 2 2 a a 2 z 4 z 3 2 3 z 2 z 4 z 2 2 -a + - + -- + --- + --- - 5 a z - 3 a z + 2 z - ---- - ---- + ---- - a z + z z 5 3 6 4 2 a a a a a 3 3 3 3 4 4 5 2 z 6 z 13 z z 3 3 3 7 z 11 z 2 4 z > ---- - ---- - ----- - -- + 7 a z + 3 a z + ---- - ----- + 4 a z - -- + 7 5 3 a 6 2 7 a a a a a a 5 5 5 6 6 6 7 9 z 12 z 3 z 3 5 6 3 z 4 z 10 z 2 6 4 z > ---- + ----- + ---- - a z + z - ---- + ---- + ----- - 2 a z - ---- - 5 3 a 6 4 2 5 a a a a a a 7 7 8 8 9 9 4 z 2 z 7 8 3 z 5 z z z > ---- - ---- - 2 a z - 2 z - ---- - ---- - -- - -- 3 a 4 2 3 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 4 2 3 4 2 4 6 + 5 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 5 q t + 5 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 10 5 > 5 q t + 5 q t + 3 q t + 5 q t + 2 q t + 3 q t + q t + 12 5 14 6 > 2 q t + q t

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