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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,19,12,18 X17,9,18,8 X7,17,8,16 X13,15,14,20 X15,5,16,14 X19,13,20,12 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-2 + 1 + q - q2 + 2q3 - 2q4 + 2q5 - q6 + q7

A2 (sl(3)) Invariant:

q-6 + 2q-4 + 3q-2 + 3 + 4q2 + 2q4 + 2q6 + 2q8 + q10 + 2q12 + q14 + 2q16 + q18 + q22

HOMFLY-PT Polynomial:

a-6 + a-6z2 + a-4z-2 + 3a-4 + 2a-4z2 - 2a-2z-2 - 10a-2 - 14a-2z2 - 7a-2z4 - a-2z6 + z-2 + 6 + 5z2 + z4

Kauffman Polynomial:

a-8 - 3a-8z2 + a-8z4 - 2a-7z3 + a-7z5 - a-6 + 3a-6z2 - 3a-6z4 + a-6z6 - a-5z3 + a-5z5 - a-4z-2 + 4a-4 - 2a-4z2 + a-4z4 + 2a-3z-1 - 13a-3z + 19a-3z3 - 8a-3z5 + a-3z7 - 2a-2z-2 + 14a-2 - 30a-2z2 + 26a-2z4 - 9a-2z6 + a-2z8 + 2a-1z-1 - 13a-1z + 18a-1z3 - 8a-1z5 + a-1z7 - z-2 + 9 - 22z2 + 21z4 - 8z6 + z8

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 = 15                     1 j = 13                   1 1 j = 11                 1     j = 9               1 1     j = 7           1 2 1       j = 5           1 2         j = 3       1 2 1           j = 1         2             j = -1     1                 j = -3 1                     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[10, NonAlternating, 87]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 87]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 19, 12, 18], X[17, 9, 18, 8], > X[7, 17, 8, 16], X[13, 15, 14, 20], X[15, 5, 16, 14], X[19, 13, 20, 12], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {-7, 5, -4, 3, -8, 6}, > {9, -1, -5, 4, 10, -2, -3, 8, -6, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-2 2 3 4 5 6 7 1 + q + q - q + 2 q - 2 q + 2 q - q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 3 2 4 6 8 10 12 14 16 3 + q + -- + -- + 4 q + 2 q + 2 q + 2 q + q + 2 q + q + 2 q + 4 2 q q 18 22 > q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 87]][a, z]
Out[8]=	2 2 2 -6 3 10 -2 1 2 2 z 2 z 14 z 4 6 + a + -- - -- + z + ----- - ----- + 5 z + -- + ---- - ----- + z - 4 2 4 2 2 2 6 4 2 a a a z a z a a a 4 6 7 z z > ---- - -- 2 2 a a
In[9]:=	Kauffman[Link[10, NonAlternating, 87]][a, z]
Out[9]=	-8 -6 4 14 -2 1 2 2 2 13 z 13 z 9 + a - a + -- + -- - z - ----- - ----- + ---- + --- - ---- - ---- - 4 2 4 2 2 2 3 a z 3 a a a a z a z a z a 2 2 2 2 3 3 3 3 2 3 z 3 z 2 z 30 z 2 z z 19 z 18 z 4 > 22 z - ---- + ---- - ---- - ----- - ---- - -- + ----- + ----- + 21 z + 8 6 4 2 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 5 6 6 7 z 3 z z 26 z z z 8 z 8 z 6 z 9 z z > -- - ---- + -- + ----- + -- + -- - ---- - ---- - 8 z + -- - ---- + -- + 8 6 4 2 7 5 3 a 6 2 3 a a a a a a a a a a 7 8 z 8 z > -- + z + -- a 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 1 1 q 3 5 7 5 2 2 q + 2 q + ----- + ----- + ---- + -- + q t + q t + q t + 2 q t + 5 4 3 4 2 t q t q t q t 7 2 7 3 9 3 9 4 11 4 13 5 13 6 15 6 > 2 q t + q t + q t + q t + q t + q t + q t + q t

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