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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 3q-3/2 - 6q-1/2 + 9q1/2 - 12q3/2 + 14q5/2 - 15q7/2 + 12q9/2 - 10q11/2 + 6q13/2 - 3q15/2 + q17/2

A2 (sl(3)) Invariant:

q-8 - q-4 + q-2 - 2 + q2 + q4 - q6 + 4q8 - q10 + 3q12 + q14 + 3q18 - q20 + q22 - q26

HOMFLY-PT Polynomial:

2a-7z + a-7z3 - a-5z-1 - 4a-5z - 6a-5z3 - 2a-5z5 + a-3z-1 + 5a-3z + 6a-3z3 + 4a-3z5 + a-3z7 - 3a-1z - 6a-1z3 - 2a-1z5 + 2az + az3

Kauffman Polynomial:

a-10z2 - a-10z4 - a-9z + 3a-9z3 - 3a-9z5 - 2a-8z2 + 5a-8z4 - 5a-8z6 + 2a-7z - 5a-7z3 + 7a-7z5 - 6a-7z7 - 3a-6z2 + 4a-6z4 + 3a-6z6 - 5a-6z8 - a-5z-1 + 9a-5z - 18a-5z3 + 18a-5z5 - 3a-5z7 - 3a-5z9 + a-4 - 12a-4z4 + 21a-4z6 - 7a-4z8 - a-4z10 - a-3z-1 + 6a-3z - 13a-3z3 + 4a-3z5 + 12a-3z7 - 6a-3z9 + 5a-2z2 - 24a-2z4 + 25a-2z6 - 5a-2z8 - a-2z10 + 2a-1z - 8a-1z3 + 8a-1z7 - 3a-1z9 + 5z2 - 14z4 + 12z6 - 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                   4 1   j = 12                 6 2     j = 10               7 5       j = 8             8 5         j = 6           6 7           j = 4         6 8             j = 2       4 7               j = 0     2 5                 j = -2   1 4                   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, 264]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 264]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[22, 12, 9, 11], X[14, 6, 15, 5], > X[2, 9, 3, 10], X[4, 14, 5, 13], X[20, 17, 21, 18], X[8, 16, 1, 15], > X[6, 20, 7, 19], X[18, 8, 19, 7], X[16, 21, 17, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, 2, -6, 4, -9, 10, -8}, > {5, -1, 3, -2, 6, -4, 8, -11, 7, -10, 9, -7, 11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 3 6 3/2 5/2 7/2 9/2 -q + ---- - ------- + 9 Sqrt[q] - 12 q + 14 q - 15 q + 12 q - 3/2 Sqrt[q] q 11/2 13/2 15/2 17/2 > 10 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -4 -2 2 4 6 8 10 12 14 18 20 -2 + q - q + q + q + q - q + 4 q - q + 3 q + q + 3 q - q + 22 26 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 264]][a, z]
Out[8]=	3 3 3 3 1 1 2 z 4 z 5 z 3 z z 6 z 6 z 6 z -(----) + ---- + --- - --- + --- - --- + 2 a z + -- - ---- + ---- - ---- + 5 3 7 5 3 a 7 5 3 a a z a z a a a a a a 5 5 5 7 3 2 z 4 z 2 z z > a z - ---- + ---- - ---- + -- 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 264]][a, z]
Out[9]=	2 2 -4 1 1 z 2 z 9 z 6 z 2 z 2 z 2 z a - ---- - ---- - -- + --- + --- + --- + --- + 2 a z + 5 z + --- - ---- - 5 3 9 7 5 3 a 10 8 a z a z a a a a a a 2 2 3 3 3 3 3 4 3 z 5 z 3 z 5 z 18 z 13 z 8 z 3 4 z > ---- + ---- + ---- - ---- - ----- - ----- - ---- - 5 a z - 14 z - --- + 6 2 9 7 5 3 a 10 a a a a a a a 4 4 4 4 5 5 5 5 5 z 4 z 12 z 24 z 3 z 7 z 18 z 4 z 5 6 > ---- + ---- - ----- - ----- - ---- + ---- + ----- + ---- + 4 a z + 12 z - 8 6 4 2 9 7 5 3 a a a a a a a a 6 6 6 6 7 7 7 7 5 z 3 z 21 z 25 z 6 z 3 z 12 z 8 z 7 8 > ---- + ---- + ----- + ----- - ---- - ---- + ----- + ---- - a z - 3 z - 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 9 10 10 5 z 7 z 5 z 3 z 6 z 3 z z z > ---- - ---- - ---- - ---- - ---- - ---- - --- - --- 6 4 2 5 3 a 4 2 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 1 2 4 5 4 q 4 6 7 q + 6 q + ----- + ----- + ----- + -- + ----- + - + ---- + 8 q t + 6 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 > 7 q t + 8 q t + 5 q t + 7 q t + 5 q t + 6 q t + 2 q t + 14 5 14 6 16 6 18 7 > 4 q t + q t + 2 q t + q t

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