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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,4,13,3 X20,12,9,11 X2,9,3,10 X17,5,18,4 X5,19,6,18 X6,14,7,13 X14,8,15,7 X8,16,1,15 X19,17,20,16

Gauss Code:

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

Jones Polynomial:

- q1/2 + q3/2 - 2q5/2 + 2q7/2 - 2q9/2 + q11/2 - 2q13/2 + q15/2

A2 (sl(3)) Invariant:

q2 + q4 + q6 + q8 + 3q16 + 2q18 + 2q20 - q24 - q26 - q28 + q30

HOMFLY-PT Polynomial:

- a-9z + 5a-7z + 5a-7z3 + a-7z5 - a-5z-1 - 8a-5z - 11a-5z3 - 6a-5z5 - a-5z7 + a-3z-1 + 6a-3z + 5a-3z3 + a-3z5

Kauffman Polynomial:

- 2a-8z2 + 4a-8z4 - a-8z6 + 3a-7z - 12a-7z3 + 10a-7z5 - 2a-7z7 - a-6z2 - 2a-6z4 + 4a-6z6 - a-6z8 - a-5z-1 + 10a-5z - 23a-5z3 + 16a-5z5 - 3a-5z7 + a-4 + a-4z2 - 6a-4z4 + 5a-4z6 - a-4z8 - a-3z-1 + 7a-3z - 11a-3z3 + 6a-3z5 - a-3z7

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 16               1 j = 14             1   j = 12           1 2   j = 10         1 1 1   j = 8       1 1       j = 6     1 1         j = 4   1 2           j = 2                 j = 0 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]=	2
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 54]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 54]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[20, 12, 9, 11], X[2, 9, 3, 10], > X[17, 5, 18, 4], X[5, 19, 6, 18], X[6, 14, 7, 13], X[14, 8, 15, 7], > X[8, 16, 1, 15], X[19, 17, 20, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, 5, -6, -7, 8, -9}, > {4, -1, 3, -2, 7, -8, 9, 10, -5, 6, -10, -3}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 15/2 -Sqrt[q] + q - 2 q + 2 q - 2 q + q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 6 8 16 18 20 24 26 28 30 q + q + q + q + 3 q + 2 q + 2 q - q - q - q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 54]][a, z]
Out[8]=	3 3 3 5 5 5 1 1 z 5 z 8 z 6 z 5 z 11 z 5 z z 6 z z -(----) + ---- - -- + --- - --- + --- + ---- - ----- + ---- + -- - ---- + -- - 5 3 9 7 5 3 7 5 3 7 5 3 a z a z a a a a a a a a a a 7 z > -- 5 a
In[9]:=	Kauffman[Link[10, NonAlternating, 54]][a, z]
Out[9]=	2 2 2 3 3 3 -4 1 1 3 z 10 z 7 z 2 z z z 12 z 23 z 11 z a - ---- - ---- + --- + ---- + --- - ---- - -- + -- - ----- - ----- - ----- + 5 3 7 5 3 8 6 4 7 5 3 a z a z a a a a a a a a a 4 4 4 5 5 5 6 6 6 7 4 z 2 z 6 z 10 z 16 z 6 z z 4 z 5 z 2 z > ---- - ---- - ---- + ----- + ----- + ---- - -- + ---- + ---- - ---- - 8 6 4 7 5 3 8 6 4 7 a a a a a a a a a a 7 7 8 8 3 z z z z > ---- - -- - -- - -- 5 3 6 4 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	4 4 6 -2 q 6 8 8 2 10 2 10 3 12 3 2 q + q + t + -- + q t + q t + q t + q t + q t + q t + t 10 4 12 4 14 4 16 5 > q t + 2 q t + q t + q t

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