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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X20,10,7,9 X2738 X4,15,5,16 X5,13,6,12 X16,12,17,11 X17,6,18,1 X19,15,20,14 X13,19,14,18

Gauss Code:

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

Jones Polynomial:

- 2q-3/2 + 3q-1/2 - 6q1/2 + 6q3/2 - 7q5/2 + 6q7/2 - 4q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant:

2q-6 + 2q-4 + q-2 + 4 + q2 + 2q4 + q6 - q8 - 3q12 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z - a-5z3 + a-3z-1 + 4a-3z + 3a-3z3 + a-3z5 - 3a-1z-1 - 6a-1z - 3a-1z3 + 2az-1 + 2az

Kauffman Polynomial:

2a-7z3 - a-7z5 - 4a-6z2 + 8a-6z4 - 3a-6z6 + 2a-5z - 4a-5z3 + 7a-5z5 - 3a-5z7 + a-4 - 9a-4z2 + 12a-4z4 - 3a-4z6 - a-4z8 - a-3z-1 + 7a-3z - 15a-3z3 + 13a-3z5 - 5a-3z7 + 3a-2 - 7a-2z2 + 4a-2z4 - a-2z6 - a-2z8 - 3a-1z-1 + 10a-1z - 12a-1z3 + 5a-1z5 - 2a-1z7 + 3 - 2z2 - z6 - 2az-1 + 5az - 3az3

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                 1 j = 12               2   j = 10             2 1   j = 8           4 2     j = 6         3 2       j = 4       3 4         j = 2     3 3           j = 0   1 4             j = -2 1 2               j = -4 2

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, 43]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 43]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[20, 10, 7, 9], X[2, 7, 3, 8], > X[4, 15, 5, 16], X[5, 13, 6, 12], X[16, 12, 17, 11], X[17, 6, 18, 1], > X[19, 15, 20, 14], X[13, 19, 14, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -5, -6, 8}, > {4, -1, 3, -2, 7, 6, -10, 9, 5, -7, -8, 10, -9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-2 3 3/2 5/2 7/2 9/2 11/2 13/2 ---- + ------- - 6 Sqrt[q] + 6 q - 7 q + 6 q - 4 q + 3 q - q 3/2 Sqrt[q] q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 2 -2 2 4 6 8 12 18 20 4 + -- + -- + q + q + 2 q + q - q - 3 q - q + q 6 4 q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 43]][a, z]
Out[8]=	3 3 3 5 1 3 2 a z 4 z 6 z z 3 z 3 z z ---- - --- + --- - -- + --- - --- + 2 a z - -- + ---- - ---- + -- 3 a z z 5 3 a 5 3 a 3 a z a a a a a
In[9]:=	Kauffman[Link[10, NonAlternating, 43]][a, z]
Out[9]=	2 -4 3 1 3 2 a 2 z 7 z 10 z 2 4 z 3 + a + -- - ---- - --- - --- + --- + --- + ---- + 5 a z - 2 z - ---- - 2 3 a z z 5 3 a 6 a a z a a a 2 2 3 3 3 3 4 4 4 9 z 7 z 2 z 4 z 15 z 12 z 3 8 z 12 z 4 z > ---- - ---- + ---- - ---- - ----- - ----- - 3 a z + ---- + ----- + ---- - 4 2 7 5 3 a 6 4 2 a a a a a a a a 5 5 5 5 6 6 6 7 7 7 z 7 z 13 z 5 z 6 3 z 3 z z 3 z 5 z 2 z > -- + ---- + ----- + ---- - z - ---- - ---- - -- - ---- - ---- - ---- - 7 5 3 a 6 4 2 5 3 a a a a a a a a a 8 8 z z > -- - -- 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 2 2 4 4 2 6 2 4 + 3 q + ----- + ----- + - + ---- + 3 q t + 3 q t + 4 q t + 3 q t + 4 2 2 2 t 2 q t q t q t 6 3 8 3 8 4 10 4 10 5 12 5 14 6 > 2 q t + 4 q t + 2 q t + 2 q t + q t + 2 q t + q t

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