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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-1/2 + 2q1/2 - 5q3/2 + 6q5/2 - 8q7/2 + 8q9/2 - 7q11/2 + 5q13/2 - 3q15/2 + q17/2

A2 (sl(3)) Invariant:

q-2 + q2 + 3q4 + 3q8 + q14 - q16 + 2q18 - q20 + q24 - q26

HOMFLY-PT Polynomial:

a-7z + a-7z3 - a-5z - 2a-5z3 - a-5z5 - a-3z-1 - a-3z - 2a-3z3 - a-3z5 + a-1z-1 + 2a-1z + a-1z3

Kauffman Polynomial:

a-10z2 - a-10z4 - a-9z + 4a-9z3 - 3a-9z5 - 2a-8z2 + 5a-8z4 - 4a-8z6 - a-7z + 2a-7z3 + a-7z5 - 3a-7z7 - 4a-6z2 + 9a-6z4 - 5a-6z6 - a-6z8 - a-5z + 6a-5z5 - 5a-5z7 - 2a-4z2 + 7a-4z4 - 3a-4z6 - a-4z8 + a-3z-1 - 4a-3z + 5a-3z3 + a-3z5 - 2a-3z7 - a-2 - a-2z2 + 4a-2z4 - 2a-2z6 + a-1z-1 - 3a-1z + 3a-1z3 - a-1z5

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 18                   1 j = 16                 2   j = 14               3 1   j = 12             4 2     j = 10           4 3       j = 8         4 4         j = 6       3 5           j = 4     2 3             j = 2   1 4               j = 0   1                 j = -2 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[9, Alternating, 26]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 26]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[18, 10, 7, 9], X[14, 6, 15, 5], > X[16, 14, 17, 13], X[12, 18, 13, 17], X[2, 7, 3, 8], X[4, 12, 5, 11], > X[6, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, 2, -8, 4, -9}, {7, -1, 3, -2, 8, -6, 5, -4, 9, -5, 6, -3}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + 2 Sqrt[q] - 5 q + 6 q - 8 q + 8 q - 7 q + Sqrt[q] 13/2 15/2 17/2 > 5 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 2 4 8 14 16 18 20 24 26 q + q + 3 q + 3 q + q - q + 2 q - q + q - q
In[8]:=	HOMFLYPT[Link[9, Alternating, 26]][a, z]
Out[8]=	3 3 3 3 5 5 1 1 z z z 2 z z 2 z 2 z z z z -(----) + --- + -- - -- - -- + --- + -- - ---- - ---- + -- - -- - -- 3 a z 7 5 3 a 7 5 3 a 5 3 a z a a a a a a a a
In[9]:=	Kauffman[Link[9, Alternating, 26]][a, z]
Out[9]=	2 2 2 2 2 -2 1 1 z z z 4 z 3 z z 2 z 4 z 2 z z -a + ---- + --- - -- - -- - -- - --- - --- + --- - ---- - ---- - ---- - -- + 3 a z 9 7 5 3 a 10 8 6 4 2 a z a a a a a a a a a 3 3 3 3 4 4 4 4 4 5 5 4 z 2 z 5 z 3 z z 5 z 9 z 7 z 4 z 3 z z > ---- + ---- + ---- + ---- - --- + ---- + ---- + ---- + ---- - ---- + -- + 9 7 3 a 10 8 6 4 2 9 7 a a a a a a a a a a 5 5 5 6 6 6 6 7 7 7 8 8 6 z z z 4 z 5 z 3 z 2 z 3 z 5 z 2 z z z > ---- + -- - -- - ---- - ---- - ---- - ---- - ---- - ---- - ---- - -- - -- 5 3 a 8 6 4 2 7 5 3 6 4 a a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 q 4 6 6 2 8 2 8 3 4 q + 2 q + ----- + - + -- + 3 q t + 3 q t + 5 q t + 4 q t + 4 q t + 2 2 t t q t 10 3 10 4 12 4 12 5 14 5 14 6 16 6 > 4 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 2 q t + 18 7 > q t

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