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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X8,12,9,11 X18,8,11,7 X16,13,17,14 X14,6,15,5 X10,16,5,15 X2,9,3,10 X4,18,1,17

Gauss Code:

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

Jones Polynomial:

- q-2 + 4q-1 - 6 + 9q - 10q2 + 11q3 - 8q4 + 7q5 - 3q6 + q7

A2 (sl(3)) Invariant:

- q-6 + 2q-4 + 1 + 3q2 - 2q4 + 3q6 + q8 + 5q10 + 6q12 + 3q14 + 5q16 + q22

HOMFLY-PT Polynomial:

a-6z-2 + a-6 + a-6z2 - 2a-4z-2 - 3a-4 - 4a-4z2 - 2a-4z4 + a-2z-2 + a-2 + 3a-2z2 + 3a-2z4 + a-2z6 + 1 - z2 - z4

Kauffman Polynomial:

- a-8z2 + a-8z4 - 2a-7z3 + 3a-7z5 + a-6z-2 - 5a-6 + 10a-6z2 - 10a-6z4 + 6a-6z6 - 2a-5z-1 + 6a-5z - 4a-5z3 - 3a-5z5 + 5a-5z7 + 2a-4z-2 - 8a-4 + 17a-4z2 - 21a-4z4 + 8a-4z6 + 2a-4z8 - 2a-3z-1 + 6a-3z + 2a-3z3 - 16a-3z5 + 10a-3z7 + a-2z-2 - 3a-2 + 9a-2z2 - 18a-2z4 + 6a-2z6 + 2a-2z8 + 3a-1z3 - 9a-1z5 + 5a-1z7 + 1 + 3z2 - 8z4 + 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 j = 15                   1 j = 13                 3 1 j = 11               4     j = 9             4 3     j = 7           7 4       j = 5         4 5         j = 3       5 6           j = 1     3 6             j = -1   1 3               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[9, Alternating, 51]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 51]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[8, 12, 9, 11], X[18, 8, 11, 7], > X[16, 13, 17, 14], X[14, 6, 15, 5], X[10, 16, 5, 15], X[2, 9, 3, 10], > X[4, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {6, -1, 4, -3, 8, -7}, {3, -2, 5, -6, 7, -5, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-2 4 2 3 4 5 6 7 -6 - q + - + 9 q - 10 q + 11 q - 8 q + 7 q - 3 q + q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 2 4 6 8 10 12 14 16 22 1 - q + -- + 3 q - 2 q + 3 q + q + 5 q + 6 q + 3 q + 5 q + q 4 q
In[8]:=	HOMFLYPT[Link[9, Alternating, 51]][a, z]
Out[8]=	2 2 2 -6 3 -2 1 2 1 2 z 4 z 3 z 4 1 + a - -- + a + ----- - ----- + ----- - z + -- - ---- + ---- - z - 4 6 2 4 2 2 2 6 4 2 a a z a z a z a a a 4 4 6 2 z 3 z z > ---- + ---- + -- 4 2 2 a a a
In[9]:=	Kauffman[Link[9, Alternating, 51]][a, z]
Out[9]=	5 8 3 1 2 1 2 2 6 z 6 z 2 1 - -- - -- - -- + ----- + ----- + ----- - ---- - ---- + --- + --- + 3 z - 6 4 2 6 2 4 2 2 2 5 3 5 3 a a a a z a z a z a z a z a a 2 2 2 2 3 3 3 3 4 z 10 z 17 z 9 z 2 z 4 z 2 z 3 z 3 4 z > -- + ----- + ----- + ---- - ---- - ---- + ---- + ---- - a z - 8 z + -- - 8 6 4 2 7 5 3 a 8 a a a a a a a a 4 4 4 5 5 5 5 6 10 z 21 z 18 z 3 z 3 z 16 z 9 z 5 6 6 z > ----- - ----- - ----- + ---- - ---- - ----- - ---- + a z + 4 z + ---- + 6 4 2 7 5 3 a 6 a a a a a a a 6 6 7 7 7 8 8 8 z 6 z 5 z 10 z 5 z 2 z 2 z > ---- + ---- + ---- + ----- + ---- + ---- + ---- 4 2 5 3 a 4 2 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 3 1 3 3 q 3 5 5 2 6 q + 5 q + ----- + ----- + ---- + --- + --- + 6 q t + 4 q t + 5 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 13 5 13 6 > 7 q t + 4 q t + 4 q t + 3 q t + 4 q t + 3 q t + q t + 15 6 > q t

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