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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 X6,20,7,19 X16,7,17,8 X18,15,19,16 X8,17,1,18

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 5q-5/2 - 8q-3/2 + 9q-1/2 - 11q1/2 + 10q3/2 - 8q5/2 + 5q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant:

- q-14 + q-10 + 3q-6 + q-2 + 2 - q2 + 3q4 - q6 + 2q8 + q10 - q12 + q14 - q16

HOMFLY-PT Polynomial:

2a-3z + 3a-3z3 + a-3z5 - a-1z-1 - 6a-1z - 9a-1z3 - 5a-1z5 - a-1z7 + az-1 + 6az + 7az3 + 2az5 - 2a3z - a3z3

Kauffman Polynomial:

a-6z2 - a-6z4 - a-5z + 4a-5z3 - 3a-5z5 - a-4z2 + 4a-4z4 - 4a-4z6 + 3a-3z5 - 4a-3z7 - 3a-2z2 + 3a-2z4 + a-2z6 - 3a-2z8 - a-1z-1 + 6a-1z - 17a-1z3 + 18a-1z5 - 6a-1z7 - a-1z9 + 1 - z2 - 6z4 + 13z6 - 6z8 - az-1 + 8az - 22az3 + 22az5 - 5az7 - az9 - 2a2z2 - a2z4 + 7a2z6 - 3a2z8 + 3a3z - 9a3z3 + 10a3z5 - 3a3z7 - 2a4z2 + 3a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                     1 j = 10                   2   j = 8                 3 1   j = 6               5 2     j = 4             5 3       j = 2           6 5         j = 0         5 7           j = -2       3 4             j = -4     2 5               j = -6   1 3                 j = -8   2                   j = -10 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, 90]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 90]]
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[6, 20, 7, 19], X[16, 7, 17, 8], > X[18, 15, 19, 16], X[8, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, 2, -6, 4, -7, 8, -10}, > {5, -1, 3, -2, 6, -4, 9, -8, 10, -9, 7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 5 8 9 3/2 5/2 q - ---- + ---- - ---- + ------- - 11 Sqrt[q] + 10 q - 8 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 > 5 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -10 3 -2 2 4 6 8 10 12 14 16 2 - q + q + -- + q - q + 3 q - q + 2 q + q - q + q - q 6 q
In[8]:=	HOMFLYPT[Link[10, Alternating, 90]][a, z]
Out[8]=	3 3 5 1 a 2 z 6 z 3 3 z 9 z 3 3 3 z -(---) + - + --- - --- + 6 a z - 2 a z + ---- - ---- + 7 a z - a z + -- - a z z 3 a 3 a 3 a a a 5 7 5 z 5 z > ---- + 2 a z - -- a a
In[9]:=	Kauffman[Link[10, Alternating, 90]][a, z]
Out[9]=	2 2 2 1 a z 6 z 3 2 z z 3 z 2 2 1 - --- - - - -- + --- + 8 a z + 3 a z - z + -- - -- - ---- - 2 a z - a z z 5 a 6 4 2 a a a a 3 3 4 4 4 4 2 4 z 17 z 3 3 3 4 z 4 z 3 z > 2 a z + ---- - ----- - 22 a z - 9 a z - 6 z - -- + ---- + ---- - 5 a 6 4 2 a a a a 5 5 5 6 2 4 4 4 3 z 3 z 18 z 5 3 5 6 4 z > a z + 3 a z - ---- + ---- + ----- + 22 a z + 10 a z + 13 z - ---- + 5 3 a 4 a a a 6 7 7 8 z 2 6 4 6 4 z 6 z 7 3 7 8 3 z > -- + 7 a z - a z - ---- - ---- - 5 a z - 3 a z - 6 z - ---- - 2 3 a 2 a a a 9 2 8 z 9 > 3 a z - -- - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 3 2 5 3 5 4 7 + 6 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 8 4 > 5 q t + 5 q t + 3 q t + 5 q t + 2 q t + 3 q t + q t + 10 4 12 5 > 2 q t + q t

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