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L11n263 L11n265

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X18,12,19,11 X19,22,20,9 X15,20,16,21 X21,16,22,17 X12,18,13,17 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-8 - 3q-7 + 6q-6 - 5q-5 + 7q-4 - 5q-3 + 5q-2 - 3q-1 + 1

A2 (sl(3)) Invariant:

2q-24 + q-22 + 4q-20 + 4q-18 + 4q-16 + 6q-14 + 2q-12 + 4q-10 - q-2 + 1

HOMFLY-PT Polynomial:

2a2z2 + a2z4 + a4z-2 + 3a4 - 3a4z4 - a4z6 - 2a6z-2 - 3a6 + a6z2 + a6z4 + a8z-2

Kauffman Polynomial:

2a2z2 - 3a2z4 + a2z6 + 6a3z3 - 10a3z5 + 3a3z7 - a4z-2 + 3a4 - 6a4z2 + 8a4z4 - 10a4z6 + 3a4z8 + 2a5z-1 - 3a5z + 7a5z3 - 8a5z5 + a5z9 - 2a6z-2 + 5a6 - 14a6z2 + 22a6z4 - 16a6z6 + 4a6z8 + 2a7z-1 - 3a7z + 3a7z3 + 2a7z5 - 3a7z7 + a7z9 - a8z-2 + 3a8 - 6a8z2 + 11a8z4 - 5a8z6 + a8z8 + 2a9z3

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                 1 j = -1               2   j = -3             3 1   j = -5           3 3     j = -7         4 2       j = -9     1 2 3         j = -11     5 4           j = -13   1 4             j = -15   2               j = -17 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 264]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 264]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], > X[18, 12, 19, 11], X[19, 22, 20, 9], X[15, 20, 16, 21], X[21, 16, 22, 17], > X[12, 18, 13, 17], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -4}, > {11, -2, 5, -9, 4, -3, -7, 8, 9, -5, -6, 7, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-8 3 6 5 7 5 5 3 1 + q - -- + -- - -- + -- - -- + -- - - 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 -22 4 4 4 6 2 4 -2 1 + --- + q + --- + --- + --- + --- + --- + --- - q 24 20 18 16 14 12 10 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 264]][a, z]
Out[8]=	4 6 8 4 6 a 2 a a 2 2 6 2 2 4 4 4 6 4 4 6 3 a - 3 a + -- - ---- + -- + 2 a z + a z + a z - 3 a z + a z - a z 2 2 2 z z z
In[9]:=	Kauffman[Link[11, NonAlternating, 264]][a, z]
Out[9]=	4 6 8 5 7 4 6 8 a 2 a a 2 a 2 a 5 7 2 2 3 a + 5 a + 3 a - -- - ---- - -- + ---- + ---- - 3 a z - 3 a z + 2 a z - 2 2 2 z z z z z 4 2 6 2 8 2 3 3 5 3 7 3 9 3 > 6 a z - 14 a z - 6 a z + 6 a z + 7 a z + 3 a z + 2 a z - 2 4 4 4 6 4 8 4 3 5 5 5 7 5 > 3 a z + 8 a z + 22 a z + 11 a z - 10 a z - 8 a z + 2 a z + 2 6 4 6 6 6 8 6 3 7 7 7 4 8 > a z - 10 a z - 16 a z - 5 a z + 3 a z - 3 a z + 3 a z + 6 8 8 8 5 9 7 9 > 4 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 2 1 4 5 1 4 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ----- + ------ + ----- + 5 3 17 6 15 5 13 5 13 4 11 4 9 4 11 3 9 3 q q q t q t q t q t q t q t q t q t 3 4 2 3 t 2 t 2 > ----- + ----- + ---- + ---- + -- + --- + q t 9 2 7 2 7 5 3 q q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n264

L11n263 L11n265