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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X16,5,17,6 X20,13,9,14 X14,19,15,20 X18,7,19,8 X8,9,1,10 X4,15,5,16 X6,17,7,18

Gauss Code:

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

Jones Polynomial:

- q-27/2 + 2q-25/2 - 4q-23/2 + 6q-21/2 - 6q-19/2 + 6q-17/2 - 6q-15/2 + 4q-13/2 - 3q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant:

q-40 + q-36 - q-32 - 2q-28 + 2q-26 + q-24 + 2q-22 + 2q-20 + 2q-16 + q-12

HOMFLY-PT Polynomial:

- a7z-1 - 6a7z - 11a7z3 - 6a7z5 - a7z7 + a9z-1 - 2a9z - 7a9z3 - 5a9z5 - a9z7 + 4a11z + 4a11z3 + a11z5

Kauffman Polynomial:

- a7z-1 + 6a7z - 11a7z3 + 6a7z5 - a7z7 + a8 - 2a8z2 - 3a8z4 + 4a8z6 - a8z8 - a9z-1 + 3a9z3 - 5a9z5 + 4a9z7 - a9z9 + 6a10z2 - 12a10z4 + 11a10z6 - 3a10z8 - 5a11z + 11a11z3 - 4a11z5 + 2a11z7 - a11z9 + a12z2 + 3a12z6 - 2a12z8 - a13z + 4a13z5 - 3a13z7 - 6a14z2 + 7a14z4 - 4a14z6 - a15z + 2a15z3 - 3a15z5 + a16z2 - 2a16z4 + a17z - a17z3

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6                     1 j = -8                   1 1 j = -10                 2     j = -12               2 1     j = -14             4 2       j = -16           2 2         j = -18         4 4           j = -20       2 2             j = -22     2 4               j = -24   1 3                 j = -26   1                   j = -28 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, Alternating, 102]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 102]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[16, 5, 17, 6], > X[20, 13, 9, 14], X[14, 19, 15, 20], X[18, 7, 19, 8], X[8, 9, 1, 10], > X[4, 15, 5, 16], X[6, 17, 7, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -9, 4, -10, 7, -8}, > {8, -1, 2, -3, 5, -6, 9, -4, 10, -7, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 2 4 6 6 6 6 4 3 -q + ----- - ----- + ----- - ----- + ----- - ----- + ----- - ----- + 25/2 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q q -(9/2) -(7/2) > q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 -36 -32 2 2 -24 2 2 2 -12 q + q - q - --- + --- + q + --- + --- + --- + q 28 26 22 20 16 q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 102]][a, z]
Out[8]=	7 9 a a 7 9 11 7 3 9 3 11 3 -(--) + -- - 6 a z - 2 a z + 4 a z - 11 a z - 7 a z + 4 a z - z z 7 5 9 5 11 5 7 7 9 7 > 6 a z - 5 a z + a z - a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 102]][a, z]
Out[9]=	7 9 8 a a 7 11 13 15 17 8 2 10 2 a - -- - -- + 6 a z - 5 a z - a z - a z + a z - 2 a z + 6 a z + z z 12 2 14 2 16 2 7 3 9 3 11 3 15 3 > a z - 6 a z + a z - 11 a z + 3 a z + 11 a z + 2 a z - 17 3 8 4 10 4 14 4 16 4 7 5 9 5 > a z - 3 a z - 12 a z + 7 a z - 2 a z + 6 a z - 5 a z - 11 5 13 5 15 5 8 6 10 6 12 6 > 4 a z + 4 a z - 3 a z + 4 a z + 11 a z + 3 a z - 14 6 7 7 9 7 11 7 13 7 8 8 10 8 > 4 a z - a z + 4 a z + 2 a z - 3 a z - a z - 3 a z - 12 8 9 9 11 9 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 1 3 2 4 2 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 28 10 26 9 24 9 24 8 22 8 22 7 20 7 q t q t q t q t q t q t q t 2 4 4 2 2 4 2 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 6 18 6 18 5 16 5 16 4 14 4 14 3 12 3 q t q t q t q t q t q t q t q t 1 2 1 > ------ + ------ + ---- 12 2 10 2 8 q t q t q t

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