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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X9,16,10,17 X17,5,18,20 X13,19,14,18 X19,15,20,14 X15,8,16,9 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

- q-3/2 + 2q-1/2 - 5q1/2 + 5q3/2 - 6q5/2 + 5q7/2 - 4q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant:

q-6 + q-4 + 3 + q2 + 2q4 + 2q6 + q10 - 2q12 - q18 + q20

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

2a-7z3 - a-7z5 - 3a-6z2 + 8a-6z4 - 3a-6z6 + 2a-5z - 5a-5z3 + 8a-5z5 - 3a-5z7 - 5a-4z2 + 7a-4z4 - a-4z6 - a-4z8 + 6a-3z - 15a-3z3 + 12a-3z5 - 4a-3z7 - a-2z2 - 3a-2z4 + 2a-2z6 - a-2z8 - a-1z-1 + 6a-1z - 9a-1z3 + 3a-1z5 - a-1z7 + 1 + z2 - 2z4 - 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 j = 14                 1 j = 12               2   j = 10             2 1   j = 8           3 2     j = 6         3 2       j = 4       2 3         j = 2     3 3           j = 0   1 4             j = -2   1               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[10, NonAlternating, 21]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 21]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[9, 16, 10, 17], > X[17, 5, 18, 20], X[13, 19, 14, 18], X[19, 15, 20, 14], X[15, 8, 16, 9], > X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, 8, -4, -2, 10, -3, -6, 7, -8, 4, -5, 6, > -7, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 2 3/2 5/2 7/2 9/2 11/2 -q + ------- - 5 Sqrt[q] + 5 q - 6 q + 5 q - 4 q + 3 q - Sqrt[q] 13/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -4 2 4 6 10 12 18 20 3 + q + q + q + 2 q + 2 q + q - 2 q - q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 21]][a, z]
Out[8]=	3 3 3 5 1 a z 3 z 3 z z 3 z 2 z z -(---) + - - -- + --- - --- + a z - -- + ---- - ---- + -- a z z 5 3 a 5 3 a 3 a a a a a
In[9]:=	Kauffman[Link[10, NonAlternating, 21]][a, z]
Out[9]=	2 2 2 3 3 1 a 2 z 6 z 6 z 2 3 z 5 z z 2 z 5 z 1 - --- - - + --- + --- + --- + 2 a z + z - ---- - ---- - -- + ---- - ---- - a z z 5 3 a 6 4 2 7 5 a a a a a a a 3 3 4 4 4 5 5 5 15 z 9 z 3 4 8 z 7 z 3 z z 8 z 12 z > ----- - ---- - a z - 2 z + ---- + ---- - ---- - -- + ---- + ----- + 3 a 6 4 2 7 5 3 a a a a a a a 5 6 6 6 7 7 7 8 8 3 z 3 z z 2 z 3 z 4 z z z z > ---- - ---- - -- + ---- - ---- - ---- - -- - -- - -- a 6 4 2 5 3 a 4 2 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 2 4 4 2 6 2 6 3 4 + 3 q + ----- + - + ---- + 3 q t + 2 q t + 3 q t + 3 q t + 2 q t + 4 2 t 2 q t q t 8 3 8 4 10 4 10 5 12 5 14 6 > 3 q t + 2 q t + 2 q t + q t + 2 q t + q t

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