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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q1/2 + 2q3/2 - 5q5/2 + 7q7/2 - 11q9/2 + 13q11/2 - 13q13/2 + 12q15/2 - 10q17/2 + 6q19/2 - 3q21/2 + q23/2

A2 (sl(3)) Invariant:

q2 + q6 + 2q8 + 4q12 - q14 + q16 - 2q20 + 3q22 - q24 + 2q26 - q30 + q32 - q34

HOMFLY-PT Polynomial:

2a-9z + 3a-9z3 + a-9z5 - 3a-7z - 5a-7z3 - 4a-7z5 - a-7z7 - a-5z-1 - a-5z - 4a-5z3 - 4a-5z5 - a-5z7 + a-3z-1 + 4a-3z + 4a-3z3 + a-3z5

Kauffman Polynomial:

a-14z2 - a-14z4 - a-13z + 3a-13z3 - 3a-13z5 - 2a-12z2 + 5a-12z4 - 5a-12z6 - a-11z - 3a-11z3 + 7a-11z5 - 6a-11z7 - a-10z4 + 5a-10z6 - 5a-10z8 - a-9z + a-9z3 + 3a-9z5 + a-9z7 - 3a-9z9 + 5a-8z2 - 8a-8z4 + 12a-8z6 - 4a-8z8 - a-8z10 - 2a-7z + 6a-7z3 - 10a-7z5 + 13a-7z7 - 5a-7z9 + 2a-6z2 - 9a-6z4 + 10a-6z6 - a-6z8 - a-6z10 - a-5z-1 + 4a-5z - 9a-5z3 + 2a-5z5 + 5a-5z7 - 2a-5z9 + a-4 - 8a-4z4 + 8a-4z6 - 2a-4z8 - a-3z-1 + 5a-3z - 8a-3z3 + 5a-3z5 - a-3z7

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 24                       1 j = 22                     2   j = 20                   4 1   j = 18                 6 2     j = 16               6 4       j = 14             7 6         j = 12           7 7           j = 10         4 6             j = 8       3 7               j = 6     2 4                 j = 4   1 4                   j = 2   1                     j = 0 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, 299]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 299]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[22, 12, 9, 11], X[16, 6, 17, 5], > X[20, 16, 21, 15], X[14, 22, 15, 21], X[18, 8, 19, 7], X[2, 9, 3, 10], > X[4, 14, 5, 13], X[6, 18, 7, 17], X[8, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9, 4, -10, 7, -11}, > {8, -1, 3, -2, 9, -6, 5, -4, 10, -7, 11, -5, 6, -3}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 -Sqrt[q] + 2 q - 5 q + 7 q - 11 q + 13 q - 13 q + 15/2 17/2 19/2 21/2 23/2 > 12 q - 10 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 6 8 12 14 16 20 22 24 26 30 32 q + q + 2 q + 4 q - q + q - 2 q + 3 q - q + 2 q - q + q - 34 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 299]][a, z]
Out[8]=	3 3 3 3 5 5 1 1 2 z 3 z z 4 z 3 z 5 z 4 z 4 z z 4 z -(----) + ---- + --- - --- - -- + --- + ---- - ---- - ---- + ---- + -- - ---- - 5 3 9 7 5 3 9 7 5 3 9 7 a z a z a a a a a a a a a a 5 5 7 7 4 z z z z > ---- + -- - -- - -- 5 3 7 5 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 299]][a, z]
Out[9]=	2 2 2 -4 1 1 z z z 2 z 4 z 5 z z 2 z 5 z a - ---- - ---- - --- - --- - -- - --- + --- + --- + --- - ---- + ---- + 5 3 13 11 9 7 5 3 14 12 8 a z a z a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 4 2 z 3 z 3 z z 6 z 9 z 8 z z 5 z z 8 z > ---- + ---- - ---- + -- + ---- - ---- - ---- - --- + ---- - --- - ---- - 6 13 11 9 7 5 3 14 12 10 8 a a a a a a a a a a a 4 4 5 5 5 5 5 5 6 6 9 z 8 z 3 z 7 z 3 z 10 z 2 z 5 z 5 z 5 z > ---- - ---- - ---- + ---- + ---- - ----- + ---- + ---- - ---- + ---- + 6 4 13 11 9 7 5 3 12 10 a a a a a a a a a a 6 6 6 7 7 7 7 7 8 8 8 12 z 10 z 8 z 6 z z 13 z 5 z z 5 z 4 z z > ----- + ----- + ---- - ---- + -- + ----- + ---- - -- - ---- - ---- - -- - 8 6 4 11 9 7 5 3 10 8 6 a a a a a a a a a a a 8 9 9 9 10 10 2 z 3 z 5 z 2 z z z > ---- - ---- - ---- - ---- - --- - --- 4 9 7 5 8 6 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 4 6 -2 q q 6 8 8 2 10 2 10 3 4 q + 2 q + t + -- + -- + 4 q t + 3 q t + 7 q t + 4 q t + 6 q t + t t 12 3 12 4 14 4 14 5 16 5 16 6 > 7 q t + 7 q t + 7 q t + 6 q t + 6 q t + 4 q t + 18 6 18 7 20 7 20 8 22 8 24 9 > 6 q t + 2 q t + 4 q t + q t + 2 q t + q t

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