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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 4q-1/2 + 5q1/2 - 7q3/2 + 7q5/2 - 8q7/2 + 6q9/2 - 4q11/2 + 3q13/2 - q15/2

A2 (sl(3)) Invariant:

q-8 + q-6 + q-2 + 2q2 + 2q4 + q6 + 3q8 - q10 + q12 - q14 - q16 - q20 + q22

HOMFLY-PT Polynomial:

- a-5z - 3a-5z3 - a-5z5 + 5a-3z + 8a-3z3 + 5a-3z5 + a-3z7 - a-1z-1 - 7a-1z - 8a-1z3 - 2a-1z5 + az-1 + 3az + az3

Kauffman Polynomial:

- a-9z3 + 2a-8z2 - 3a-8z4 - a-7z + 3a-7z3 - 4a-7z5 + 3a-6z2 + 2a-6z4 - 4a-6z6 - 2a-5z + 2a-5z3 + 5a-5z5 - 4a-5z7 - a-4z2 + a-4z4 + 5a-4z6 - 3a-4z8 + 4a-3z - 17a-3z3 + 18a-3z5 - 3a-3z7 - a-3z9 + a-2z2 - 15a-2z4 + 18a-2z6 - 5a-2z8 - a-1z-1 + 10a-1z - 23a-1z3 + 14a-1z5 - a-1z9 + 1 + 3z2 - 11z4 + 9z6 - 2z8 - az-1 + 5az - 8az3 + 5az5 - 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 j = 16                     1 j = 14                   2   j = 12                 2 1   j = 10               4 2     j = 8             4 2       j = 6           3 4         j = 4         4 4           j = 2       3 5             j = 0     1 2               j = -2   1 3                 j = -4   1                   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[10, Alternating, 89]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 89]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[20, 12, 9, 11], X[14, 6, 15, 5], > X[2, 9, 3, 10], X[4, 14, 5, 13], X[18, 16, 19, 15], X[16, 7, 17, 8], > X[6, 17, 7, 18], X[8, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, 2, -6, 4, -9, 8, -10}, > {5, -1, 3, -2, 6, -4, 7, -8, 9, -7, 10, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 4 3/2 5/2 7/2 9/2 -q + ---- - ------- + 5 Sqrt[q] - 7 q + 7 q - 8 q + 6 q - 3/2 Sqrt[q] q 11/2 13/2 15/2 > 4 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -2 2 4 6 8 10 12 14 16 20 22 q + q + q + 2 q + 2 q + q + 3 q - q + q - q - q - q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 89]][a, z]
Out[8]=	3 3 3 5 5 1 a z 5 z 7 z 3 z 8 z 8 z 3 z 5 z -(---) + - - -- + --- - --- + 3 a z - ---- + ---- - ---- + a z - -- + ---- - a z z 5 3 a 5 3 a 5 3 a a a a a a 5 7 2 z z > ---- + -- a 3 a
In[9]:=	Kauffman[Link[10, Alternating, 89]][a, z]
Out[9]=	2 2 2 2 1 a z 2 z 4 z 10 z 2 2 z 3 z z z 1 - --- - - - -- - --- + --- + ---- + 5 a z + 3 z + ---- + ---- - -- + -- - a z z 7 5 3 a 8 6 4 2 a a a a a a a 3 3 3 3 3 4 4 4 z 3 z 2 z 17 z 23 z 3 4 3 z 2 z z > -- + ---- + ---- - ----- - ----- - 8 a z - 11 z - ---- + ---- + -- - 9 7 5 3 a 8 6 4 a a a a a a a 4 5 5 5 5 6 6 6 15 z 4 z 5 z 18 z 14 z 5 6 4 z 5 z 18 z > ----- - ---- + ---- + ----- + ----- + 5 a z + 9 z - ---- + ---- + ----- - 2 7 5 3 a 6 4 2 a a a a a a a 7 7 8 8 9 9 4 z 3 z 7 8 3 z 5 z z z > ---- - ---- - a z - 2 z - ---- - ---- - -- - -- 5 3 4 2 3 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 3 2 3 q 4 5 q + 4 q + ----- + ----- + ----- + t + ----- + - + ---- + 4 q t + 6 4 4 3 2 3 2 2 t t q t q t q t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 > 3 q t + 4 q t + 4 q t + 2 q t + 4 q t + 2 q t + 2 q t + 12 5 14 5 16 6 > q t + 2 q t + q t

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