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DrawMorseLink

PD Presentation:

X8192 X18,12,19,11 X10,4,11,3 X2,20,3,19 X16,8,17,7 X20,9,21,10 X12,18,7,17 X22,16,13,15 X14,6,15,5 X4,14,5,13 X6,21,1,22

Gauss Code:

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

Jones Polynomial:

- q-2 + 4q-1 - 8 + 14q - 18q2 + 22q3 - 20q4 + 19q5 - 13q6 + 8q7 - 4q8 + q9

A2 (sl(3)) Invariant:

- q-6 + 2q-4 - q-2 - 1 + 5q2 - 3q4 + 4q6 + 4q8 + 3q10 + 8q12 + q14 + 6q16 + q18 - 2q20 + 3q22 - 2q24 - q26 + q28

HOMFLY-PT Polynomial:

a-8z2 + a-6z-2 - 2a-6z2 - 2a-6z4 - 2a-4z-2 - 2a-4 + a-4z4 + a-4z6 + a-2z-2 + 2a-2 + 3a-2z2 + 2a-2z4 + a-2z6 - z2 - z4

Kauffman Polynomial:

- 2a-10z4 + a-10z6 + 5a-9z3 - 10a-9z5 + 4a-9z7 - 6a-8z2 + 16a-8z4 - 19a-8z6 + 7a-8z8 + 5a-7z3 - a-7z5 - 10a-7z7 + 6a-7z9 + a-6z-2 - 2a-6 - 15a-6z2 + 44a-6z4 - 41a-6z6 + 11a-6z8 + 2a-6z10 - 2a-5z-1 + 2a-5z + 14a-5z5 - 24a-5z7 + 12a-5z9 + 2a-4z-2 - 3a-4 - 10a-4z2 + 34a-4z4 - 35a-4z6 + 12a-4z8 + 2a-4z10 - 2a-3z-1 + 2a-3z + 4a-3z3 - 6a-3z5 - 3a-3z7 + 6a-3z9 + a-2z-2 - 2a-2 + a-2z2 + 2a-2z4 - 10a-2z6 + 8a-2z8 + 3a-1z3 - 10a-1z5 + 7a-1z7 + 2z2 - 6z4 + 4z6 - az3 + az5

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 = 19                       1 j = 17                     3   j = 15                   5 1   j = 13                 8 3     j = 11               11 5       j = 9             10 9         j = 7           12 10           j = 5         8 12             j = 3       6 10               j = 1     3 9                 j = -1   1 5                   j = -3   3                     j = -5 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 507]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 507]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 12, 19, 11], X[10, 4, 11, 3], X[2, 20, 3, 19], > X[16, 8, 17, 7], X[20, 9, 21, 10], X[12, 18, 7, 17], X[22, 16, 13, 15], > X[14, 6, 15, 5], X[4, 14, 5, 13], X[6, 21, 1, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -10, 9, -11}, {5, -1, 6, -3, 2, -7}, > {10, -9, 8, -5, 7, -2, 4, -6, 11, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-2 4 2 3 4 5 6 7 8 9 -8 - q + - + 14 q - 18 q + 22 q - 20 q + 19 q - 13 q + 8 q - 4 q + q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 -2 2 4 6 8 10 12 14 16 -1 - q + -- - q + 5 q - 3 q + 4 q + 4 q + 3 q + 8 q + q + 6 q + 4 q 18 20 22 24 26 28 > q - 2 q + 3 q - 2 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 507]][a, z]
Out[8]=	2 2 2 4 4 -2 2 1 2 1 2 z 2 z 3 z 4 2 z z -- + -- + ----- - ----- + ----- - z + -- - ---- + ---- - z - ---- + -- + 4 2 6 2 4 2 2 2 8 6 2 6 4 a a a z a z a z a a a a a 4 6 6 2 z z z > ---- + -- + -- 2 4 2 a a a
In[9]:=	Kauffman[Link[11, Alternating, 507]][a, z]
Out[9]=	2 -2 3 2 1 2 1 2 2 2 z 2 z 2 6 z -- - -- - -- + ----- + ----- + ----- - ---- - ---- + --- + --- + 2 z - ---- - 6 4 2 6 2 4 2 2 2 5 3 5 3 8 a a a a z a z a z a z a z a a a 2 2 2 3 3 3 3 4 15 z 10 z z 5 z 5 z 4 z 3 z 3 4 2 z > ----- - ----- + -- + ---- + ---- + ---- + ---- - a z - 6 z - ---- + 6 4 2 9 7 3 a 10 a a a a a a a 4 4 4 4 5 5 5 5 5 16 z 44 z 34 z 2 z 10 z z 14 z 6 z 10 z 5 > ----- + ----- + ----- + ---- - ----- - -- + ----- - ---- - ----- + a z + 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 6 7 7 7 7 6 z 19 z 41 z 35 z 10 z 4 z 10 z 24 z 3 z > 4 z + --- - ----- - ----- - ----- - ----- + ---- - ----- - ----- - ---- + 10 8 6 4 2 9 7 5 3 a a a a a a a a a 7 8 8 8 8 9 9 9 10 10 7 z 7 z 11 z 12 z 8 z 6 z 12 z 6 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]=	3 1 3 1 5 3 q 3 5 5 2 9 q + 6 q + ----- + ----- + ---- + --- + --- + 10 q t + 8 q t + 12 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 > 12 q t + 10 q t + 10 q t + 9 q t + 11 q t + 5 q t + 13 5 13 6 15 6 15 7 17 7 19 8 > 8 q t + 3 q t + 5 q t + q t + 3 q t + q t

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