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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X10,6,11,5 X8493 X18,12,5,11 X12,18,13,17 X16,10,17,9 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

q-5/2 - 4q-3/2 + 6q-1/2 - 9q1/2 + 9q3/2 - 10q5/2 + 8q7/2 - 5q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-8 + 2q-6 + q-4 + 4 + 3q4 + q6 + q10 - 3q12 + q14 - q18 + q20

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

2a-7z3 - a-7z5 - 4a-6z2 + 7a-6z4 - 3a-6z6 + 2a-5z - 7a-5z3 + 9a-5z5 - 4a-5z7 - 7a-4z2 + 10a-4z4 - 2a-4z6 - 2a-4z8 + 4a-3z - 13a-3z3 + 18a-3z5 - 9a-3z7 - 4a-2z2 + 11a-2z4 - 5a-2z6 - 2a-2z8 - a-1z-1 + 2a-1z + 4a-1z5 - 5a-1z7 + 1 - z2 + 7z4 - 6z6 - az-1 + 4az3 - 4az5 - a2z4

Khovanov Homology:

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

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