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DrawMorseLink

PD Presentation:

X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X18,10,19,9 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 X10,20,5,19

Gauss Code:

{{1, -4, 3, -10}, {2, -1, 5, -3, 6, -11}, {-8, -2, 4, -5, 10, 9, -7, -6, 11, 8, -9, 7}}

Jones Polynomial:

- q-1 + 3 - 3q + 6q2 - 4q3 + 6q4 - 4q5 + 3q6 - 2q7

A2 (sl(3)) Invariant:

- q-2 + 1 + q2 + 3q4 + 5q6 + 7q8 + 9q10 + 5q12 + 6q14 - q18 - 3q20 - 3q22 - q24 - 2q26 + q28

HOMFLY-PT Polynomial:

- a-8z-2 + a-8z2 + 4a-6z-2 + 2a-6 - 5a-6z2 - 5a-6z4 - a-6z6 - 5a-4z-2 - 5a-4 + 6a-4z2 + 11a-4z4 + 6a-4z6 + a-4z8 + 2a-2z-2 + 3a-2 - 2a-2z2 - 4a-2z4 - a-2z6

Kauffman Polynomial:

- a-9z-1 + a-9z + a-8z-2 - 2a-8z2 + a-8z4 - 5a-7z-1 + 5a-7z + 6a-7z3 - 8a-7z5 + 2a-7z7 + 4a-6z-2 - 2a-6 - 14a-6z2 + 29a-6z4 - 20a-6z6 + 4a-6z8 - 9a-5z-1 + 9a-5z + 7a-5z3 - 3a-5z5 - 6a-5z7 + 2a-5z9 + 5a-4z-2 - 4a-4 - 21a-4z2 + 49a-4z4 - 35a-4z6 + 7a-4z8 - 5a-3z-1 + 5a-3z + 4a-3z3 + a-3z5 - 7a-3z7 + 2a-3z9 + 2a-2z-2 - 3a-2 - 9a-2z2 + 21a-2z4 - 15a-2z6 + 3a-2z8 + 3a-1z3 - 4a-1z5 + a-1z7

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 15                 2 j = 13               2 1 j = 11             3 2   j = 9           3 2 1   j = 7         3 5       j = 5       3 2 1       j = 3     2 5           j = 1   1 1             j = -1   2               j = -3 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[11, NonAlternating, 284]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 284]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 6, 13, 5], X[8, 4, 9, 3], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[18, 10, 19, 9], X[17, 11, 18, 22], X[11, 21, 12, 20], > X[21, 17, 22, 16], X[4, 15, 1, 16], X[10, 20, 5, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -10}, {2, -1, 5, -3, 6, -11}, > {-8, -2, 4, -5, 10, 9, -7, -6, 11, 8, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	1 2 3 4 5 6 7 3 - - - 3 q + 6 q - 4 q + 6 q - 4 q + 3 q - 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 2 4 6 8 10 12 14 18 20 1 - q + q + 3 q + 5 q + 7 q + 9 q + 5 q + 6 q - q - 3 q - 22 24 26 28 > 3 q - q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 284]][a, z]
Out[8]=	2 2 2 2 4 2 5 3 1 4 5 2 z 5 z 6 z 2 z 5 z -- - -- + -- - ----- + ----- - ----- + ----- + -- - ---- + ---- - ---- - ---- + 6 4 2 8 2 6 2 4 2 2 2 8 6 4 2 6 a a a a z a z a z a z a a a a a 4 4 6 6 6 8 11 z 4 z z 6 z z z > ----- - ---- - -- + ---- - -- + -- 4 2 6 4 2 4 a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 284]][a, z]
Out[9]=	-2 4 3 1 4 5 2 1 5 9 5 z -- - -- - -- + ----- + ----- + ----- + ----- - ---- - ---- - ---- - ---- + -- + 6 4 2 8 2 6 2 4 2 2 2 9 7 5 3 9 a a a a z a z a z a z a z a z a z a z a 2 2 2 2 3 3 3 3 5 z 9 z 5 z 2 z 14 z 21 z 9 z 6 z 7 z 4 z 3 z > --- + --- + --- - ---- - ----- - ----- - ---- + ---- + ---- + ---- + ---- + 7 5 3 8 6 4 2 7 5 3 a a a a a a a a a a a 4 4 4 4 5 5 5 5 6 6 z 29 z 49 z 21 z 8 z 3 z z 4 z 20 z 35 z > -- + ----- + ----- + ----- - ---- - ---- + -- - ---- - ----- - ----- - 8 6 4 2 7 5 3 a 6 4 a a a a a a a a a 6 7 7 7 7 8 8 8 9 9 15 z 2 z 6 z 7 z z 4 z 7 z 3 z 2 z 2 z > ----- + ---- - ---- - ---- + -- + ---- + ---- + ---- + ---- + ---- 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 2 q q 2 q 5 7 5 2 5 q + 3 q + ----- + ---- + -- + - + ---- + 2 q t + 3 q t + q t + 3 3 2 2 t t q t q t t 7 2 9 2 9 3 11 3 9 4 11 4 13 4 > 5 q t + 3 q t + 2 q t + 3 q t + q t + 2 q t + 2 q t + 13 5 15 5 > q t + 2 q t

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