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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X20,14,21,13 X14,8,15,7 X16,6,17,5 X6,16,7,15 X4,21,5,22 X18,9,19,10 X22,17,9,18 X8,20,1,19

Gauss Code:

{{1, -2, 3, -8, 6, -7, 5, -11}, {9, -1, 2, -3, 4, -5, 7, -6, 10, -9, 11, -4, 8, -10}}

Jones Polynomial:

- q-15/2 + 3q-13/2 - 6q-11/2 + 10q-9/2 - 12q-7/2 + 13q-5/2 - 14q-3/2 + 11q-1/2 - 9q1/2 + 5q3/2 - 3q5/2 + q7/2

A2 (sl(3)) Invariant:

q-22 - q-20 + q-18 - 2q-14 + q-12 - 3q-10 + 2q-8 + q-6 + q-4 + 4q-2 + 3q2 + q4 + q8 - q10

HOMFLY-PT Polynomial:

- a-1z-1 + a-1z + 3a-1z3 + a-1z5 + az-1 - 4az3 - 4az5 - az7 - 3a3z - 5a3z3 - 4a3z5 - a3z7 + 2a5z + 3a5z3 + a5z5

Kauffman Polynomial:

4a-2z2 - 8a-2z4 + 5a-2z6 - a-2z8 - a-1z-1 - 2a-1z + 17a-1z3 - 28a-1z5 + 16a-1z7 - 3a-1z9 + 1 + 11z2 - 23z4 + 7z6 + 5z8 - 2z10 - az-1 + 17az3 - 47az5 + 38az7 - 9az9 + 10a2z2 - 38a2z4 + 34a2z6 - 4a2z8 - 2a2z10 + 6a3z - 19a3z3 + 13a3z5 + 10a3z7 - 6a3z9 - 3a4z2 - 5a4z4 + 22a4z6 - 10a4z8 + 4a5z - 15a5z3 + 26a5z5 - 12a5z7 - 6a6z2 + 15a6z4 - 10a6z6 + 3a7z3 - 6a7z5 - 3a8z4 - a9z3

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 8                       1 j = 6                     2   j = 4                   3 1   j = 2                 6 2     j = 0               5 3       j = -2             9 6         j = -4           6 7           j = -6         6 7             j = -8       4 6               j = -10     2 6                 j = -12   1 4                   j = -14   2                     j = -16 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[11, Alternating, 293]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 293]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[20, 14, 21, 13], > X[14, 8, 15, 7], X[16, 6, 17, 5], X[6, 16, 7, 15], X[4, 21, 5, 22], > X[18, 9, 19, 10], X[22, 17, 9, 18], X[8, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -8, 6, -7, 5, -11}, > {9, -1, 2, -3, 4, -5, 7, -6, 10, -9, 11, -4, 8, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 6 10 12 13 14 11 -q + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 9 Sqrt[q] + 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 5 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 -18 2 -12 3 2 -6 -4 4 2 4 8 q - q + q - --- + q - --- + -- + q + q + -- + 3 q + q + q - 14 10 8 2 q q q q 10 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 293]][a, z]
Out[8]=	3 5 1 a z 3 5 3 z 3 3 3 5 3 z -(---) + - + - - 3 a z + 2 a z + ---- - 4 a z - 5 a z + 3 a z + -- - a z z a a a 5 3 5 5 5 7 3 7 > 4 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 293]][a, z]
Out[9]=	2 1 a 2 z 3 5 2 4 z 2 2 4 2 1 - --- - - - --- + 6 a z + 4 a z + 11 z + ---- + 10 a z - 3 a z - a z z a 2 a 3 6 2 17 z 3 3 3 5 3 7 3 9 3 4 > 6 a z + ----- + 17 a z - 19 a z - 15 a z + 3 a z - a z - 23 z - a 4 5 8 z 2 4 4 4 6 4 8 4 28 z 5 > ---- - 38 a z - 5 a z + 15 a z - 3 a z - ----- - 47 a z + 2 a a 6 3 5 5 5 7 5 6 5 z 2 6 4 6 > 13 a z + 26 a z - 6 a z + 7 z + ---- + 34 a z + 22 a z - 2 a 7 8 6 6 16 z 7 3 7 5 7 8 z 2 8 > 10 a z + ----- + 38 a z + 10 a z - 12 a z + 5 z - -- - 4 a z - a 2 a 9 4 8 3 z 9 3 9 10 2 10 > 10 a z - ---- - 9 a z - 6 a z - 2 z - 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	7 9 1 2 1 4 2 6 4 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 4 2 16 6 14 5 12 5 12 4 10 4 10 3 8 3 8 2 q q q t q t q t q t q t q t q t q t 6 7 6 6 t 2 2 2 2 3 4 3 > ----- + ---- + ---- + 5 t + --- + 3 t + 6 q t + 2 q t + 3 q t + 6 2 6 4 2 q t q t q t q 4 4 6 4 8 5 > q t + 2 q t + q t

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