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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X10,5,11,6 X12,3,13,4 X16,11,5,12 X2,9,3,10 X8,13,9,14

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 4q-15/2 + 6q-13/2 - 7q-11/2 + 5q-9/2 - 6q-7/2 + 3q-5/2 - q-3/2

A2 (sl(3)) Invariant:

q-30 - q-28 - 3q-26 - q-22 + q-20 + 3q-18 + 2q-16 + 4q-14 + q-12 + 2q-10 + q-8 - 2q-6 + q-4

HOMFLY-PT Polynomial:

- a3z3 - 2a5z-1 - 5a5z - 3a5z3 + 3a7z-1 + 2a7z - a7z3 - a9z-1 + a9z

Kauffman Polynomial:

- a3z3 - 3a4z4 + 2a5z-1 - 5a5z + 8a5z3 - 6a5z5 - 3a6 + a6z2 + 5a6z4 - 5a6z6 + 3a7z-1 - 6a7z + 12a7z3 - 4a7z5 - 2a7z7 - 3a8 - 4a8z2 + 16a8z4 - 8a8z6 + a9z-1 - 2a9z + 5a9z3 + a9z5 - 2a9z7 - a10 - 5a10z2 + 8a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                 1 j = -4               3 1 j = -6             3     j = -8           2 3     j = -10         5 3       j = -12       2 3         j = -14     2 4           j = -16   1 2             j = -18   2               j = -20 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[8, Alternating, 7]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[8, Alternating, 7]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[10, 5, 11, 6], > X[12, 3, 13, 4], X[16, 11, 5, 12], X[2, 9, 3, 10], X[8, 13, 9, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, 5, -3}, {4, -1, 2, -8, 7, -4, 6, -5, 8, -2, 3, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 4 6 7 5 6 3 -(3/2) -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - q 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 3 -22 -20 3 2 4 -12 2 -8 2 -4 q - q - --- - q + q + --- + --- + --- + q + --- + q - -- + q 26 18 16 14 10 6 q q q q q q
In[8]:=	HOMFLYPT[Link[8, Alternating, 7]][a, z]
Out[8]=	5 7 9 -2 a 3 a a 5 7 9 3 3 5 3 7 3 ----- + ---- - -- - 5 a z + 2 a z + a z - a z - 3 a z - a z z z z
In[9]:=	Kauffman[Link[8, Alternating, 7]][a, z]
Out[9]=	5 7 9 6 8 10 2 a 3 a a 5 7 9 11 -3 a - 3 a - a + ---- + ---- + -- - 5 a z - 6 a z - 2 a z - a z + z z z 6 2 8 2 10 2 3 3 5 3 7 3 9 3 > a z - 4 a z - 5 a z - a z + 8 a z + 12 a z + 5 a z + 11 3 4 4 6 4 8 4 10 4 5 5 7 5 > 2 a z - 3 a z + 5 a z + 16 a z + 8 a z - 6 a z - 4 a z + 9 5 11 5 6 6 8 6 10 6 7 7 9 7 > a z - a z - 5 a z - 8 a z - 3 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 -2 1 2 1 2 2 4 2 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q t q t q t q t q t q t q t 3 5 3 2 3 3 3 > ------ + ------ + ------ + ----- + ----- + ----- + ---- 12 4 10 4 10 3 8 3 8 2 6 2 4 q t q t q t q t q t q t q t

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