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DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X14,6,15,5 X16,13,17,14 X20,18,7,17 X18,12,19,11 X12,20,13,19 X4,16,5,15 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 5q-5/2 - 9q-3/2 + 10q-1/2 - 12q1/2 + 11q3/2 - 9q5/2 + 6q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant:

- q-14 + q-12 - q-8 + 4q-6 + q-4 + 3q-2 + 4 + 2q4 - 3q6 - 2q12 + 2q14 - q18

HOMFLY-PT Polynomial:

a-5z + a-3z-1 - a-3z - 2a-3z3 - 3a-1z-1 - 3a-1z + a-1z5 + 2az-1 + 3az + 2az3 + az5 - a3z - a3z3

Kauffman Polynomial:

a-6z2 - a-6z4 - a-5z + 3a-5z3 - 3a-5z5 + a-4 - 4a-4z2 + 6a-4z4 - 5a-4z6 - a-3z-1 + a-3z - a-3z3 + 4a-3z5 - 5a-3z7 + 3a-2 - 12a-2z2 + 15a-2z4 - 4a-2z6 - 3a-2z8 - 3a-1z-1 + 10a-1z - 17a-1z3 + 20a-1z5 - 8a-1z7 - a-1z9 + 3 - 9z2 + 6z4 + 8z6 - 6z8 - 2az-1 + 11az - 23az3 + 23az5 - 6az7 - az9 - 4a2z2 + a2z4 + 6a2z6 - 3a2z8 + 3a3z - 10a3z3 + 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                 4 1   j = 6               5 2     j = 4             6 4       j = 2           6 5         j = 0         5 7           j = -2       4 5             j = -4     2 6               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, 59]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 59]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[14, 6, 15, 5], X[16, 13, 17, 14], > X[20, 18, 7, 17], X[18, 12, 19, 11], X[12, 20, 13, 19], X[4, 16, 5, 15], > X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -8, 3, -10}, > {9, -1, 10, -2, 6, -7, 4, -3, 8, -4, 5, -6, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 5 9 10 3/2 5/2 q - ---- + ---- - ---- + ------- - 12 Sqrt[q] + 11 q - 9 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 > 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 -8 4 -4 3 4 6 12 14 18 4 - q + q - q + -- + q + -- + 2 q - 3 q - 2 q + 2 q - q 6 2 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 59]][a, z]
Out[8]=	3 5 1 3 2 a z z 3 z 3 2 z 3 3 3 z ---- - --- + --- + -- - -- - --- + 3 a z - a z - ---- + 2 a z - a z + -- + 3 a z z 5 3 a 3 a a z a a a 5 > a z
In[9]:=	Kauffman[Link[10, Alternating, 59]][a, z]
Out[9]=	-4 3 1 3 2 a z z 10 z 3 2 3 + a + -- - ---- - --- - --- - -- + -- + ---- + 11 a z + 3 a z - 9 z + 2 3 a z z 5 3 a a a z a a 2 2 2 3 3 3 z 4 z 12 z 2 2 4 2 3 z z 17 z 3 > -- - ---- - ----- - 4 a z - 2 a z + ---- - -- - ----- - 23 a z - 6 4 2 5 3 a a a a a a 4 4 4 5 5 3 3 4 z 6 z 15 z 2 4 4 4 3 z 4 z > 10 a z + 6 z - -- + ---- + ----- + a z + 3 a z - ---- + ---- + 6 4 2 5 3 a a a a a 5 6 6 7 20 z 5 3 5 6 5 z 4 z 2 6 4 6 5 z > ----- + 23 a z + 10 a z + 8 z - ---- - ---- + 6 a z - a z - ---- - a 4 2 3 a a a 7 8 9 8 z 7 3 7 8 3 z 2 8 z 9 > ---- - 6 a z - 3 a z - 6 z - ---- - 3 a z - -- - a z a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 3 2 6 4 5 5 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 + 6 q t + 4 q t + 5 q t + 2 q t + 4 q t + q t + 10 4 12 5 > 2 q t + q t

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