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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X9,19,10,18 X14,6,15,5 X16,12,17,11 X3,10,4,11 X12,7,13,8 X20,15,7,16 X6,14,1,13 X4,19,5,20 X17,2,18,3

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 5q-9/2 + 6q-7/2 - 8q-5/2 + 7q-3/2 - 6q-1/2 + 4q1/2 - 2q3/2

A2 (sl(3)) Invariant:

q-20 - q-18 + q-16 + 2q-14 - q-12 + 2q-10 + q-6 + q-4 - q-2 + 2 - q2 + q4 + 2q6

HOMFLY-PT Polynomial:

- a-1z-1 - 2a-1z + 2az-1 + 5az + 3az3 - 2a3z-1 - 5a3z - 3a3z3 - a3z5 + a5z-1 + a5z + a5z3

Kauffman Polynomial:

- a-1z-1 + 4a-1z - 3a-1z3 + 3z2 - 2z4 - z6 - 2az-1 + 12az - 18az3 + 9az5 - 3az7 - a2 + 4a2z2 - 7a2z4 + 4a2z6 - 2a2z8 - 2a3z-1 + 12a3z - 26a3z3 + 21a3z5 - 7a3z7 - a4z2 + 2a4z4 + 2a4z6 - 2a4z8 - a5z-1 + 4a5z - 9a5z3 + 11a5z5 - 4a5z7 - 2a6z2 + 7a6z4 - 3a6z6 + 2a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 4                 2 j = 2               2   j = 0             4 2   j = -2           4 3     j = -4         4 3       j = -6       3 5         j = -8     2 3           j = -10   1 3             j = -12   2               j = -14 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, 52]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 52]]
Out[4]=	PD[X[8, 1, 9, 2], X[9, 19, 10, 18], X[14, 6, 15, 5], X[16, 12, 17, 11], > X[3, 10, 4, 11], X[12, 7, 13, 8], X[20, 15, 7, 16], X[6, 14, 1, 13], > X[4, 19, 5, 20], X[17, 2, 18, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 10, -5, -9, 3, -8}, > {6, -1, -2, 5, 4, -6, 8, -3, 7, -4, -10, 2, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 5 6 8 7 6 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 4 Sqrt[q] - 2 q 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 2 -12 2 -6 -4 -2 2 4 6 2 + q - q + q + --- - q + --- + q + q - q - q + q + 2 q 14 10 q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 52]][a, z]
Out[8]=	3 5 1 2 a 2 a a 2 z 3 5 3 3 3 -(---) + --- - ---- + -- - --- + 5 a z - 5 a z + a z + 3 a z - 3 a z + a z z z z a 5 3 3 5 > a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 52]][a, z]
Out[9]=	3 5 2 1 2 a 2 a a 4 z 3 5 2 -a - --- - --- - ---- - -- + --- + 12 a z + 12 a z + 4 a z + 3 z + a z z z z a 3 2 2 4 2 6 2 3 z 3 3 3 5 3 7 3 > 4 a z - a z - 2 a z - ---- - 18 a z - 26 a z - 9 a z + 2 a z - a 4 2 4 4 4 6 4 5 3 5 5 5 7 5 > 2 z - 7 a z + 2 a z + 7 a z + 9 a z + 21 a z + 11 a z - a z - 6 2 6 4 6 6 6 7 3 7 5 7 2 8 > z + 4 a z + 2 a z - 3 a z - 3 a z - 7 a z - 4 a z - 2 a z - 4 8 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 2 1 3 2 3 3 5 4 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 4 3 4 2 4 2 > ----- + ---- + ---- + 2 t + 2 q t + 2 q t 4 2 4 2 q t q t q t

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