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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X8493 X22,14,5,13 X14,22,15,21 X18,12,19,11 X20,10,21,9 X10,20,11,19 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-5/2 - 4q-3/2 + 8q-1/2 - 14q1/2 + 17q3/2 - 20q5/2 + 18q7/2 - 16q9/2 + 12q11/2 - 6q13/2 + 3q15/2 - q17/2

A2 (sl(3)) Invariant:

- q-8 + 2q-6 - 3q-2 + 5 + 3q4 + 5q6 + 4q10 - 3q12 - 5q18 + 2q20 - q24 + q26

HOMFLY-PT Polynomial:

- a-7z - a-7z3 + a-5z-1 + 2a-5z + a-5z3 + a-5z5 - 3a-3z-1 - 2a-3z + 2a-3z3 + 2a-3z5 + 2a-1z-1 + a-1z + a-1z5 - az3

Kauffman Polynomial:

- 4a-9z3 + 4a-9z5 - a-9z7 + 8a-8z2 - 16a-8z4 + 12a-8z6 - 3a-8z8 - 2a-7z + 6a-7z3 - 13a-7z5 + 13a-7z7 - 4a-7z9 - a-6 + 17a-6z2 - 34a-6z4 + 26a-6z6 - 3a-6z8 - 2a-6z10 + a-5z-1 - 4a-5z + 7a-5z3 - 20a-5z5 + 29a-5z7 - 11a-5z9 - 3a-4 + 13a-4z2 - 36a-4z4 + 42a-4z6 - 12a-4z8 - 2a-4z10 + 3a-3z-1 - 3a-3z - 13a-3z3 + 18a-3z5 + 3a-3z7 - 7a-3z9 - 3a-2 + 4a-2z2 - 11a-2z4 + 20a-2z6 - 12a-2z8 + 2a-1z-1 - a-1z - 8a-1z3 + 17a-1z5 - 12a-1z7 + 6z4 - 8z6 + 2az3 - 4az5 - a2z4

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 18                       1 j = 16                     2   j = 14                   4 1   j = 12                 8 2     j = 10               8 4       j = 8             10 8         j = 6           10 8           j = 4         7 10             j = 2       7 10               j = 0     3 9                 j = -2   1 5                   j = -4   3                     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, 9]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 9]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[12, 6, 13, 5], > X[8, 4, 9, 3], X[22, 14, 5, 13], X[14, 22, 15, 21], X[18, 12, 19, 11], > X[20, 10, 21, 9], X[10, 20, 11, 19], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {4, -1, 2, -5, 9, -10, 8, -4, 6, -7, 11, -2, 3, -8, > 10, -9, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 4 8 3/2 5/2 7/2 9/2 q - ---- + ------- - 14 Sqrt[q] + 17 q - 20 q + 18 q - 16 q + 3/2 Sqrt[q] q 11/2 13/2 15/2 17/2 > 12 q - 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 2 3 4 6 10 12 18 20 24 26 5 - q + -- - -- + 3 q + 5 q + 4 q - 3 q - 5 q + 2 q - q + q 6 2 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 9]][a, z]
Out[8]=	3 3 3 5 5 5 1 3 2 z 2 z 2 z z z z 2 z 3 z 2 z z ---- - ---- + --- - -- + --- - --- + - - -- + -- + ---- - a z + -- + ---- + -- 5 3 a z 7 5 3 a 7 5 3 5 3 a a z a z a a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 9]][a, z]
Out[9]=	2 2 -6 3 3 1 3 2 2 z 4 z 3 z z 8 z 17 z -a - -- - -- + ---- + ---- + --- - --- - --- - --- - - + ---- + ----- + 4 2 5 3 a z 7 5 3 a 8 6 a a a z a z a a a a a 2 2 3 3 3 3 3 4 13 z 4 z 4 z 6 z 7 z 13 z 8 z 3 4 16 z > ----- + ---- - ---- + ---- + ---- - ----- - ---- + 2 a z + 6 z - ----- - 4 2 9 7 5 3 a 8 a a a a a a a 4 4 4 5 5 5 5 5 34 z 36 z 11 z 2 4 4 z 13 z 20 z 18 z 17 z > ----- - ----- - ----- - a z + ---- - ----- - ----- + ----- + ----- - 6 4 2 9 7 5 3 a a a a a a a a 6 6 6 6 7 7 7 7 5 6 12 z 26 z 42 z 20 z z 13 z 29 z 3 z > 4 a z - 8 z + ----- + ----- + ----- + ----- - -- + ----- + ----- + ---- - 8 6 4 2 9 7 5 3 a a a a a a a a 7 8 8 8 8 9 9 9 10 10 12 z 3 z 3 z 12 z 12 z 4 z 11 z 7 z 2 z 2 z > ----- - ---- - ---- - ----- - ----- - ---- - ----- - ---- - ----- - ----- a 8 6 4 2 7 5 3 6 4 a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 3 5 2 4 4 2 9 + 7 q + ----- + ----- + ----- + - + ---- + 10 q t + 7 q t + 10 q t + 6 3 4 2 2 2 t 2 q t q t q t q t 6 2 6 3 8 3 8 4 10 4 10 5 12 5 > 10 q t + 8 q t + 10 q t + 8 q t + 8 q t + 4 q t + 8 q t + 12 6 14 6 14 7 16 7 18 8 > 2 q t + 4 q t + q t + 2 q t + q t

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