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L10n58 L10n60

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DrawMorseLink

PD Presentation:

X12,1,13,2 X7,17,8,16 X5,1,6,10 X3746 X9,5,10,4 X20,17,11,18 X18,13,19,14 X14,19,15,20 X2,11,3,12 X15,9,16,8

Gauss Code:

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

Jones Polynomial:

q-9/2 - q-7/2 + q-5/2 - q-3/2 - q-1/2 - q1/2 - q3/2 + q5/2 - q7/2 + q9/2

A2 (sl(3)) Invariant:

- q-14 - q-12 - 2q-10 - q-8 + 2q-4 + 5q-2 + 5 + 5q2 + 2q4 - q8 - 2q10 - q12 - q14

HOMFLY-PT Polynomial:

2a-3z-1 + 3a-3z + a-3z3 - 7a-1z-1 - 11a-1z - 6a-1z3 - a-1z5 + 7az-1 + 11az + 6az3 + az5 - 2a3z-1 - 3a3z - a3z3

Kauffman Polynomial:

2a-4 - 6a-4z2 + 5a-4z4 - a-4z6 - 2a-3z-1 + 4a-3z - 6a-3z3 + 5a-3z5 - a-3z7 + 8a-2 - 18a-2z2 + 12a-2z4 - 2a-2z6 - 7a-1z-1 + 14a-1z - 14a-1z3 + 7a-1z5 - a-1z7 + 13 - 24z2 + 14z4 - 2z6 - 7az-1 + 14az - 14az3 + 7az5 - az7 + 8a2 - 18a2z2 + 12a2z4 - 2a2z6 - 2a3z-1 + 4a3z - 6a3z3 + 5a3z5 - a3z7 + 2a4 - 6a4z2 + 5a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                     1 j = 8                       j = 6                 1 1   j = 4             1 1       j = 2           1   1       j = 0         1 4 1         j = -2       1   1           j = -4       1 1             j = -6   1 1                 j = -8                       j = -10 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, 59]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 59]]
Out[4]=	PD[X[12, 1, 13, 2], X[7, 17, 8, 16], X[5, 1, 6, 10], X[3, 7, 4, 6], > X[9, 5, 10, 4], X[20, 17, 11, 18], X[18, 13, 19, 14], X[14, 19, 15, 20], > X[2, 11, 3, 12], X[15, 9, 16, 8]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, -4, 5, -3, 4, -2, 10, -5, 3}, > {9, -1, 7, -8, -10, 2, 6, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) -(7/2) -(5/2) -(3/2) 1 3/2 5/2 q - q + q - q - ------- - Sqrt[q] - q + q - Sqrt[q] 7/2 9/2 > q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 2 -8 2 5 2 4 8 10 12 14 5 - q - q - --- - q + -- + -- + 5 q + 2 q - q - 2 q - q - q 10 4 2 q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 59]][a, z]
Out[8]=	3 3 3 2 7 7 a 2 a 3 z 11 z 3 z 6 z 3 ---- - --- + --- - ---- + --- - ---- + 11 a z - 3 a z + -- - ---- + 6 a z - 3 a z z z 3 a 3 a a z a a 5 3 3 z 5 > a z - -- + a z a
In[9]:=	Kauffman[Link[10, NonAlternating, 59]][a, z]
Out[9]=	3 2 8 2 4 2 7 7 a 2 a 4 z 14 z 13 + -- + -- + 8 a + 2 a - ---- - --- - --- - ---- + --- + ---- + 14 a z + 4 2 3 a z z z 3 a a a a z a 2 2 3 3 3 2 6 z 18 z 2 2 4 2 6 z 14 z > 4 a z - 24 z - ---- - ----- - 18 a z - 6 a z - ---- - ----- - 4 2 3 a a a a 4 4 5 3 3 3 4 5 z 12 z 2 4 4 4 5 z > 14 a z - 6 a z + 14 z + ---- + ----- + 12 a z + 5 a z + ---- + 4 2 3 a a a 5 6 6 7 7 7 z 5 3 5 6 z 2 z 2 6 4 6 z z > ---- + 7 a z + 5 a z - 2 z - -- - ---- - 2 a z - a z - -- - -- - a 4 2 3 a a a a 7 3 7 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-2 2 1 1 1 1 1 1 1 4 4 + q + q + ------ + ----- + ----- + ----- + ----- + - + ---- + t + q t + 10 5 6 4 6 3 4 2 2 2 t 4 q t q t q t q t q t q t 2 2 4 2 6 3 6 4 10 5 > q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n59

L10n58 L10n60