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DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,17,12,16 X15,9,16,18 X17,13,18,12 X2536 X9,1,10,4

Gauss Code:

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

Jones Polynomial:

3q - 3q2 + 6q3 - 4q4 + 5q5 - 4q6 + 2q7 - q8

A2 (sl(3)) Invariant:

3q2 + 2q4 + 5q6 + 7q8 + 6q10 + 7q12 + 3q14 + 2q16 - q18 - 3q20 - q22 - 2q24 - q26

HOMFLY-PT Polynomial:

- a-8z-2 - a-8 + 4a-6z-2 + 6a-6 + 3a-6z2 - 5a-4z-2 - 10a-4 - 7a-4z2 - 2a-4z4 + 2a-2z-2 + 5a-2 + 3a-2z2

Kauffman Polynomial:

- a-9z-1 + 3a-9z - 3a-9z3 + a-9z5 + a-8z-2 - 2a-8 + 3a-8z2 - 5a-8z4 + 2a-8z6 - 5a-7z-1 + 13a-7z - 15a-7z3 + 3a-7z5 + a-7z7 + 4a-6z-2 - 10a-6 + 15a-6z2 - 16a-6z4 + 6a-6z6 - 9a-5z-1 + 21a-5z - 18a-5z3 + 5a-5z5 + a-5z7 + 5a-4z-2 - 14a-4 + 18a-4z2 - 11a-4z4 + 4a-4z6 - 5a-3z-1 + 11a-3z - 6a-3z3 + 3a-3z5 + 2a-2z-2 - 7a-2 + 6a-2z2

Khovanov Homology:

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17               1 j = 15             1   j = 13           3 1   j = 11         2 1     j = 9       3 4       j = 7     3 1         j = 5     3           j = 3 3 3             j = 1 3

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, NonAlternating, 20]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 20]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[7, 15, 8, 14], X[13, 5, 14, 8], > X[11, 17, 12, 16], X[15, 9, 16, 18], X[17, 13, 18, 12], X[2, 5, 3, 6], > X[9, 1, 10, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, -2, 9}, {8, -1, -3, 4}, {-9, 2, -5, 7, -4, 3, -6, 5, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 5 6 7 8 3 q - 3 q + 6 q - 4 q + 5 q - 4 q + 2 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 6 8 10 12 14 16 18 20 22 3 q + 2 q + 5 q + 7 q + 6 q + 7 q + 3 q + 2 q - q - 3 q - q - 24 26 > 2 q - q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 20]][a, z]
Out[8]=	2 2 2 4 -8 6 10 5 1 4 5 2 3 z 7 z 3 z 2 z -a + -- - -- + -- - ----- + ----- - ----- + ----- + ---- - ---- + ---- - ---- 6 4 2 8 2 6 2 4 2 2 2 6 4 2 4 a a a a z a z a z a z a a a a
In[9]:=	Kauffman[Link[9, NonAlternating, 20]][a, z]
Out[9]=	-2 10 14 7 1 4 5 2 1 5 9 5 -- - -- - -- - -- + ----- + ----- + ----- + ----- - ---- - ---- - ---- - ---- + 8 6 4 2 8 2 6 2 4 2 2 2 9 7 5 3 a a a a a z a z a z a z a z a z a z a z 2 2 2 2 3 3 3 z 13 z 21 z 11 z 3 z 15 z 18 z 6 z 3 z 15 z > --- + ---- + ---- + ---- + ---- + ----- + ----- + ---- - ---- - ----- - 9 7 5 3 8 6 4 2 9 7 a a a a a a a a a a 3 3 4 4 4 5 5 5 5 6 18 z 6 z 5 z 16 z 11 z z 3 z 5 z 3 z 2 z > ----- - ---- - ---- - ----- - ----- + -- + ---- + ---- + ---- + ---- + 5 3 8 6 4 9 7 5 3 8 a a a a a a a a a a 6 6 7 7 6 z 4 z z z > ---- + ---- + -- + -- 6 4 7 5 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 2 7 2 7 3 9 3 9 4 3 q + 3 q + 3 q t + 3 q t + 3 q t + q t + 3 q t + 4 q t + 11 4 11 5 13 5 13 6 15 6 17 7 > 2 q t + q t + 3 q t + q t + q t + q t

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