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L10n41 L10n43

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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X20,10,7,9 X2738 X15,5,16,4 X5,13,6,12 X16,12,17,11 X6,18,1,17 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:

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

A2 (sl(3)) Invariant:

q2 + q4 + q6 + 2q8 + 2q10 + 2q12 + q14 + 2q16 - q22 - q24 - q26 - q28 + q30

HOMFLY-PT Polynomial:

- a-9z + a-7z-1 + 5a-7z + 5a-7z3 + a-7z5 - 3a-5z-1 - 9a-5z - 11a-5z3 - 6a-5z5 - a-5z7 + 2a-3z-1 + 6a-3z + 5a-3z3 + a-3z5

Kauffman Polynomial:

a-9z + a-8 - 5a-8z2 + 5a-8z4 - a-8z6 - a-7z-1 + 8a-7z - 16a-7z3 + 11a-7z5 - 2a-7z7 + 3a-6 - 7a-6z2 + 4a-6z6 - a-6z8 - 3a-5z-1 + 15a-5z - 27a-5z3 + 17a-5z5 - 3a-5z7 + 3a-4 - 2a-4z2 - 5a-4z4 + 5a-4z6 - a-4z8 - 2a-3z-1 + 8a-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 1 j = 12           1 1   j = 10         1 1 1   j = 8       1 2       j = 6     1 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, 42]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 42]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[20, 10, 7, 9], X[2, 7, 3, 8], > X[15, 5, 16, 4], X[5, 13, 6, 12], X[16, 12, 17, 11], X[6, 18, 1, 17], > 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]=	3/2 5/2 7/2 9/2 11/2 13/2 15/2 -Sqrt[q] + q - 2 q + q - 2 q + q - q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 6 8 10 12 14 16 22 24 26 28 30 q + q + q + 2 q + 2 q + 2 q + q + 2 q - q - q - q - q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 42]][a, z]
Out[8]=	3 3 3 5 5 1 3 2 z 5 z 9 z 6 z 5 z 11 z 5 z z 6 z ---- - ---- + ---- - -- + --- - --- + --- + ---- - ----- + ---- + -- - ---- + 7 5 3 9 7 5 3 7 5 3 7 5 a z a z a z a a a a a a a a a 5 7 z z > -- - -- 3 5 a a
In[9]:=	Kauffman[Link[10, NonAlternating, 42]][a, z]
Out[9]=	2 2 -8 3 3 1 3 2 z 8 z 15 z 8 z 5 z 7 z a + -- + -- - ---- - ---- - ---- + -- + --- + ---- + --- - ---- - ---- - 6 4 7 5 3 9 7 5 3 8 6 a a a z a z a z a a a a a a 2 3 3 3 4 4 5 5 5 6 2 z 16 z 27 z 11 z 5 z 5 z 11 z 17 z 6 z z > ---- - ----- - ----- - ----- + ---- - ---- + ----- + ----- + ---- - -- + 4 7 5 3 8 4 7 5 3 8 a a a a a a a a a a 6 6 7 7 7 8 8 4 z 5 z 2 z 3 z z z z > ---- + ---- - ---- - ---- - -- - -- - -- 6 4 7 5 3 6 4 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	4 4 6 -2 q 6 8 6 2 8 2 10 2 10 3 2 q + q + t + -- + q t + q t + q t + 2 q t + q t + q t + t 12 3 10 4 12 4 14 4 14 5 16 5 > q t + q t + q t + q t + q t + q t

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L10n41 L10n43