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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X18,14,9,13 X16,12,17,11 X12,18,13,17 X8,16,5,15 X14,8,15,7 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-3 - q-2 + 5q-1 - 5 + 8q - 7q2 + 7q3 - 6q4 + 3q5 - q6

A2 (sl(3)) Invariant:

q-10 + 2q-8 + 3q-6 + 7q-4 + 6q-2 + 7 + 6q2 + q4 + 2q6 - 3q8 - q10 - 2q12 - 2q14 + q16 - q18

HOMFLY-PT Polynomial:

- a-4z-2 - 2a-4 - 2a-4z2 - a-4z4 + 4a-2z-2 + 8a-2 + 7a-2z2 + 4a-2z4 + a-2z6 - 5z-2 - 9 - 7z2 - 2z4 + 2a2z-2 + 3a2 + a2z2

Kauffman Polynomial:

a-7z3 + 3a-6z4 - a-5z-1 + 3a-5z - 6a-5z3 + 6a-5z5 + a-4z-2 - 2a-4 + 6a-4z2 - 11a-4z4 + 7a-4z6 - 5a-3z-1 + 13a-3z - 12a-3z3 - a-3z5 + 4a-3z7 + 4a-2z-2 - 10a-2 + 16a-2z2 - 21a-2z4 + 7a-2z6 + a-2z8 - 9a-1z-1 + 21a-1z - 11a-1z3 - 8a-1z5 + 5a-1z7 + 5z-2 - 14 + 19z2 - 12z4 + z6 + z8 - 5az-1 + 11az - 6az3 - az5 + az7 + 2a2z-2 - 7a2 + 9a2z2 - 5a2z4 + a2z6

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                   1 j = 11                 2   j = 9               4 1   j = 7             3 2     j = 5           4 4       j = 3         4 3         j = 1       4 7           j = -1     1 1             j = -3     4               j = -5 1 1                 j = -7 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[9, Alternating, 44]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 44]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 14, 9, 13], X[16, 12, 17, 11], > X[12, 18, 13, 17], X[8, 16, 5, 15], X[14, 8, 15, 7], X[2, 5, 3, 6], > X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, 7, -6}, {9, -2, 4, -5, 3, -7, 6, -4, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-3 -2 5 2 3 4 5 6 -5 + q - q + - + 8 q - 7 q + 7 q - 6 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 3 7 6 2 4 6 8 10 12 14 7 + q + -- + -- + -- + -- + 6 q + q + 2 q - 3 q - q - 2 q - 2 q + 8 6 4 2 q q q q 16 18 > q - q
In[8]:=	HOMFLYPT[Link[9, Alternating, 44]][a, z]
Out[8]=	2 2 2 2 8 2 5 1 4 2 a 2 2 z 7 z 2 2 -9 - -- + -- + 3 a - -- - ----- + ----- + ---- - 7 z - ---- + ---- + a z - 4 2 2 4 2 2 2 2 4 2 a a z a z a z z a a 4 4 6 4 z 4 z z > 2 z - -- + ---- + -- 4 2 2 a a a
In[9]:=	Kauffman[Link[9, Alternating, 44]][a, z]
Out[9]=	2 2 10 2 5 1 4 2 a 1 5 9 5 a -14 - -- - -- - 7 a + -- + ----- + ----- + ---- - ---- - ---- - --- - --- + 4 2 2 4 2 2 2 2 5 3 a z z a a z a z a z z a z a z 2 2 3 3 3 z 13 z 21 z 2 6 z 16 z 2 2 z 6 z > --- + ---- + ---- + 11 a z + 19 z + ---- + ----- + 9 a z + -- - ---- - 5 3 a 4 2 7 5 a a a a a a 3 3 4 4 4 5 12 z 11 z 3 4 3 z 11 z 21 z 2 4 6 z > ----- - ----- - 6 a z - 12 z + ---- - ----- - ----- - 5 a z + ---- - 3 a 6 4 2 5 a a a a a 5 5 6 6 7 7 8 z 8 z 5 6 7 z 7 z 2 6 4 z 5 z 7 8 z > -- - ---- - a z + z + ---- + ---- + a z + ---- + ---- + a z + z + -- 3 a 4 2 3 a 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 4 1 1 4 q 3 7 q + 4 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 3 q t + 7 4 5 4 5 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 13 5 > 4 q t + 4 q t + 3 q t + 2 q t + 4 q t + q t + 2 q t + q t

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