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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X4,13,1,14 X5,16,6,17 X8493 X17,20,18,15 X19,10,20,11 X9,18,10,19 X15,14,16,5 X2,12,3,11

Gauss Code:

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

Jones Polynomial:

q-8 - 3q-7 + 5q-6 - 5q-5 + 6q-4 - 4q-3 + 5q-2 - 2q-1 + 1

A2 (sl(3)) Invariant:

q-24 - q-22 + q-20 + q-18 + 2q-16 + 5q-14 + 3q-12 + 6q-10 + 3q-8 + 3q-6 + 2q-4 + 1

HOMFLY-PT Polynomial:

a2z-2 + 3a2 + 3a2z2 + a2z4 - 2a4z-2 - 4a4 - 5a4z2 - 4a4z4 - a4z6 + a6z-2 + a6 + 2a6z2 + a6z4

Kauffman Polynomial:

a2z-2 - 4a2 + 6a2z2 - 4a2z4 + a2z6 - 2a3z-1 + 3a3z + 2a3z3 - 6a3z5 + 2a3z7 + 2a4z-2 - 8a4 + 18a4z2 - 17a4z4 + 2a4z6 + a4z8 - 2a5z-1 + a5z + 9a5z3 - 15a5z5 + 5a5z7 + a6z-2 - 7a6 + 20a6z2 - 18a6z4 + 3a6z6 + a6z8 - 3a7z + 10a7z3 - 9a7z5 + 3a7z7 - 3a8 + 9a8z2 - 5a8z4 + 2a8z6 - a9z + 3a9z3 - a10 + a10z2

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                 1 j = -1               1   j = -3             4 1   j = -5           2 3     j = -7         4 2       j = -9     1 2 2         j = -11     4 4           j = -13   1 3             j = -15   2               j = -17 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]=	3
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 83]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 83]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[5, 16, 6, 17], > X[8, 4, 9, 3], X[17, 20, 18, 15], X[19, 10, 20, 11], X[9, 18, 10, 19], > X[15, 14, 16, 5], X[2, 12, 3, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {-9, 4, -6, 8, -7, 6}, > {-4, -1, 2, -5, -8, 7, 10, -2, 3, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-8 3 5 5 6 4 5 2 1 + q - -- + -- - -- + -- - -- + -- - - 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 -18 2 5 3 6 3 3 2 1 + q - q + q + q + --- + --- + --- + --- + -- + -- + -- 16 14 12 10 8 6 4 q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 83]][a, z]
Out[8]=	2 4 6 2 4 6 a 2 a a 2 2 4 2 6 2 2 4 3 a - 4 a + a + -- - ---- + -- + 3 a z - 5 a z + 2 a z + a z - 2 2 2 z z z 4 4 6 4 4 6 > 4 a z + a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 83]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 10 a 2 a a 2 a 2 a 3 -4 a - 8 a - 7 a - 3 a - a + -- + ---- + -- - ---- - ---- + 3 a z + 2 2 2 z z z z z 5 7 9 2 2 4 2 6 2 8 2 10 2 > a z - 3 a z - a z + 6 a z + 18 a z + 20 a z + 9 a z + a z + 3 3 5 3 7 3 9 3 2 4 4 4 6 4 > 2 a z + 9 a z + 10 a z + 3 a z - 4 a z - 17 a z - 18 a z - 8 4 3 5 5 5 7 5 2 6 4 6 6 6 > 5 a z - 6 a z - 15 a z - 9 a z + a z + 2 a z + 3 a z + 8 6 3 7 5 7 7 7 4 8 6 8 > 2 a z + 2 a z + 5 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 2 1 3 4 1 4 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ----- + ------ + ----- + 5 3 17 6 15 5 13 5 13 4 11 4 9 4 11 3 9 3 q q q t q t q t q t q t q t q t q t 2 4 2 2 t t 2 > ----- + ----- + ---- + ---- + -- + - + q t 9 2 7 2 7 5 3 q q t q t q t q t q

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