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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 3q-3/2 - 7q-1/2 + 10q1/2 - 14q3/2 + 16q5/2 - 17q7/2 + 14q9/2 - 11q11/2 + 7q13/2 - 3q15/2 + q17/2

A2 (sl(3)) Invariant:

q-6 - q-4 + 3q-2 + 2q2 + 2q4 - 2q6 + 5q8 - 2q10 + 4q12 - q16 + q18 - 3q20 + q22 - q24

HOMFLY-PT Polynomial:

a-5z-1 + 7a-5z + 9a-5z3 + 5a-5z5 + a-5z7 - 3a-3z-1 - 15a-3z - 25a-3z3 - 19a-3z5 - 7a-3z7 - a-3z9 + 2a-1z-1 + 8a-1z + 9a-1z3 + 5a-1z5 + a-1z7

Kauffman Polynomial:

a-10z2 - a-10z4 + 2a-9z3 - 3a-9z5 - 4a-8z2 + 7a-8z4 - 6a-8z6 + 2a-7z - 9a-7z3 + 12a-7z5 - 8a-7z7 - a-6 + 6a-6z2 - 12a-6z4 + 14a-6z6 - 8a-6z8 + a-5z-1 - 6a-5z + 13a-5z3 - 11a-5z5 + 11a-5z7 - 6a-5z9 - 3a-4 + 18a-4z2 - 33a-4z4 + 29a-4z6 - 6a-4z8 - 2a-4z10 + 3a-3z-1 - 17a-3z + 35a-3z3 - 41a-3z5 + 33a-3z7 - 10a-3z9 - 3a-2 + 10a-2z2 - 24a-2z4 + 20a-2z6 - a-2z8 - 2a-2z10 + 2a-1z-1 - 7a-1z + 6a-1z3 - 11a-1z5 + 13a-1z7 - 4a-1z9 + 3z2 - 11z4 + 11z6 - 3z8 + 2az - 5az3 + 4az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 18                       1 j = 16                     2   j = 14                   5 1   j = 12                 6 2     j = 10               8 5       j = 8             9 6         j = 6           7 8           j = 4         7 9             j = 2       5 9               j = 0     2 5                 j = -2   1 5                   j = -4   2                     j = -6 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, 339]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 339]]
Out[4]=	PD[X[10, 1, 11, 2], X[14, 4, 15, 3], X[22, 5, 9, 6], X[6, 9, 7, 10], > X[18, 12, 19, 11], X[20, 14, 21, 13], X[12, 20, 13, 19], X[16, 8, 17, 7], > X[4, 16, 5, 15], X[8, 18, 1, 17], X[2, 21, 3, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -9, 3, -4, 8, -10}, > {4, -1, 5, -7, 6, -2, 9, -8, 10, -5, 7, -6, 11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 3 7 3/2 5/2 7/2 -q + ---- - ------- + 10 Sqrt[q] - 14 q + 16 q - 17 q + 3/2 Sqrt[q] q 9/2 11/2 13/2 15/2 17/2 > 14 q - 11 q + 7 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -4 3 2 4 6 8 10 12 16 18 q - q + -- + 2 q + 2 q - 2 q + 5 q - 2 q + 4 q - q + q - 2 q 20 22 24 > 3 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 339]][a, z]
Out[8]=	3 3 3 5 5 1 3 2 7 z 15 z 8 z 9 z 25 z 9 z 5 z 19 z ---- - ---- + --- + --- - ---- + --- + ---- - ----- + ---- + ---- - ----- + 5 3 a z 5 3 a 5 3 a 5 3 a z a z a a a a a a 5 7 7 7 9 5 z z 7 z z z > ---- + -- - ---- + -- - -- a 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 339]][a, z]
Out[9]=	-6 3 3 1 3 2 2 z 6 z 17 z 7 z 2 -a - -- - -- + ---- + ---- + --- + --- - --- - ---- - --- + 2 a z + 3 z + 4 2 5 3 a z 7 5 3 a a a a z a z a a a 2 2 2 2 2 3 3 3 3 3 z 4 z 6 z 18 z 10 z 2 z 9 z 13 z 35 z 6 z > --- - ---- + ---- + ----- + ----- + ---- - ---- + ----- + ----- + ---- - 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 4 4 4 4 4 5 5 3 4 z 7 z 12 z 33 z 24 z 3 z 12 z > 5 a z - 11 z - --- + ---- - ----- - ----- - ----- - ---- + ----- - 10 8 6 4 2 9 7 a a a a a a a 5 5 5 6 6 6 6 11 z 41 z 11 z 5 6 6 z 14 z 29 z 20 z > ----- - ----- - ----- + 4 a z + 11 z - ---- + ----- + ----- + ----- - 5 3 a 8 6 4 2 a a a a a a 7 7 7 7 8 8 8 9 8 z 11 z 33 z 13 z 7 8 8 z 6 z z 6 z > ---- + ----- + ----- + ----- - a z - 3 z - ---- - ---- - -- - ---- - 7 5 3 a 6 4 2 5 a a a a a a a 9 9 10 10 10 z 4 z 2 z 2 z > ----- - ---- - ----- - ----- 3 a 4 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 1 2 5 5 5 q 4 6 9 q + 7 q + ----- + ----- + ----- + -- + ----- + - + ---- + 9 q t + 7 q t + 6 4 4 3 2 3 2 2 2 t t q t q t q t t q t 6 2 8 2 8 3 10 3 10 4 12 4 12 5 > 8 q t + 9 q t + 6 q t + 8 q t + 5 q t + 6 q t + 2 q t + 14 5 14 6 16 6 18 7 > 5 q t + q t + 2 q t + q t

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