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L11n246 L11n248

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DrawMorseLink

PD Presentation:

X12,1,13,2 X3849 X14,6,15,5 X7,18,8,19 X9,21,10,20 X10,11,1,12 X6,14,7,13 X17,4,18,5 X15,11,16,22 X19,3,20,2 X21,17,22,16

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-14 + 2q-8 + 2q-6 + q-4 + q-2 - 1 + q2 + q4 + 2q6 + 2q8 - q14

HOMFLY-PT Polynomial:

2a-3z + a-3z3 - a-1z-1 - 6a-1z - 5a-1z3 - a-1z5 + az-1 + 6az + 5az3 + az5 - 2a3z - a3z3

Kauffman Polynomial:

- 2a-4z2 + 4a-4z4 - a-4z6 + 4a-3z - 11a-3z3 + 10a-3z5 - 2a-3z7 - 4a-2z4 + 5a-2z6 - a-2z8 - a-1z-1 + 12a-1z - 29a-1z3 + 19a-1z5 - 3a-1z7 + 1 + 4z2 - 16z4 + 12z6 - 2z8 - az-1 + 12az - 29az3 + 19az5 - 3az7 - 4a2z4 + 5a2z6 - a2z8 + 4a3z - 11a3z3 + 10a3z5 - 2a3z7 - 2a4z2 + 4a4z4 - 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                   1   j = 6               1 1 1   j = 4             1 2 1     j = 2           2 2 1       j = 0         2 4 2         j = -2       1 2 2           j = -4     1 2 1             j = -6   1 1 1               j = -8   1                   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[11, NonAlternating, 247]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 247]]
Out[4]=	PD[X[12, 1, 13, 2], X[3, 8, 4, 9], X[14, 6, 15, 5], X[7, 18, 8, 19], > X[9, 21, 10, 20], X[10, 11, 1, 12], X[6, 14, 7, 13], X[17, 4, 18, 5], > X[15, 11, 16, 22], X[19, 3, 20, 2], X[21, 17, 22, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 10, -2, 8, 3, -7, -4, 2, -5, -6}, > {6, -1, 7, -3, -9, 11, -8, 4, -10, 5, -11, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 2 -(5/2) -(3/2) 3/2 5/2 7/2 9/2 q - ---- + q - q - q + q - 2 q + q 7/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 2 2 -4 -2 2 4 6 8 14 -1 - q + -- + -- + q + q + q + q + 2 q + 2 q - q 8 6 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 247]][a, z]
Out[8]=	3 3 5 1 a 2 z 6 z 3 z 5 z 3 3 3 z 5 -(---) + - + --- - --- + 6 a z - 2 a z + -- - ---- + 5 a z - a z - -- + a z a z z 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 247]][a, z]
Out[9]=	2 3 1 a 4 z 12 z 3 2 2 z 4 2 11 z 1 - --- - - + --- + ---- + 12 a z + 4 a z + 4 z - ---- - 2 a z - ----- - a z z 3 a 4 3 a a a 3 4 4 29 z 3 3 3 4 4 z 4 z 2 4 4 4 > ----- - 29 a z - 11 a z - 16 z + ---- - ---- - 4 a z + 4 a z + a 4 2 a a 5 5 6 6 10 z 19 z 5 3 5 6 z 5 z 2 6 4 6 > ----- + ----- + 19 a z + 10 a z + 12 z - -- + ---- + 5 a z - a z - 3 a 4 2 a a a 7 7 8 2 z 3 z 7 3 7 8 z 2 8 > ---- - ---- - 3 a z - 2 a z - 2 z - -- - a z 3 a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 1 1 1 2 4 + -- + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 2 10 5 8 4 6 4 6 3 4 3 6 2 4 2 q q t q t q t q t q t q t q t 1 2 1 2 2 4 2 2 4 2 6 2 > ----- + - + ---- + ---- + 2 t + 2 q t + q t + q t + 2 q t + q t + 2 2 t 4 2 q t q t q t 4 3 6 3 6 4 8 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 L11n247

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