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L11n195 L11n197

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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X13,21,14,20 X16,9,17,10 X10,19,11,20 X15,7,16,22 X21,15,22,14 X18,5,19,6 X2738 X4,11,5,12 X6,17,1,18

Gauss Code:

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

Jones Polynomial:

- 2q-13/2 + 4q-11/2 - 8q-9/2 + 10q-7/2 - 12q-5/2 + 12q-3/2 - 10q-1/2 + 7q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

q-22 + 3q-20 + 2q-16 + 3q-14 - 2q-12 + 2q-10 - q-8 + q-4 - 2q-2 + 3 - 2q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 - az - az3 - az5 - a3z - a3z3 - a3z5 - a5z-1 - a5z + a5z3 + a7z-1

Kauffman Polynomial:

- a-2z4 + 3a-1z3 - 4a-1z5 - 2z2 + 8z4 - 7z6 - az + 7az5 - 7az7 - 7a2z2 + 12a2z4 - 2a2z6 - 4a2z8 + a3z - 7a3z3 + 16a3z5 - 8a3z7 - a3z9 - 4a4z2 + 5a4z4 + 5a4z6 - 5a4z8 + a5z-1 - 4a5z + 4a5z3 + 2a5z5 - a5z7 - a5z9 - a6 + a6z2 + 2a6z4 - a6z8 + a7z-1 - 6a7z + 8a7z3 - 3a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                   1 j = 4                 3   j = 2               4 1   j = 0             6 3     j = -2           7 5       j = -4         5 5         j = -6       5 7           j = -8     3 5             j = -10   1 5               j = -12 1 3                 j = -14 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[11, NonAlternating, 196]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 196]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[13, 21, 14, 20], X[16, 9, 17, 10], > X[10, 19, 11, 20], X[15, 7, 16, 22], X[21, 15, 22, 14], X[18, 5, 19, 6], > X[2, 7, 3, 8], X[4, 11, 5, 12], X[6, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 8, -11}, > {9, -1, 4, -5, 10, -2, -3, 7, -6, -4, 11, -8, 5, 3, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-2 4 8 10 12 12 10 3/2 5/2 ----- + ----- - ---- + ---- - ---- + ---- - ------- + 7 Sqrt[q] - 4 q + q 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 3 2 3 2 2 -8 -4 2 2 6 8 3 + q + --- + --- + --- - --- + --- - q + q - -- - 2 q + 2 q - q 20 16 14 12 10 2 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 196]][a, z]
Out[8]=	5 7 3 a a 3 5 z 3 3 3 5 3 5 3 5 -(--) + -- - a z - a z - a z + -- - a z - a z + a z - a z - a z z z a
In[9]:=	Kauffman[Link[11, NonAlternating, 196]][a, z]
Out[9]=	5 7 6 a a 3 5 7 2 2 2 4 2 -a + -- + -- - a z + a z - 4 a z - 6 a z - 2 z - 7 a z - 4 a z + z z 3 4 6 2 3 z 3 3 5 3 7 3 4 z 2 4 > a z + ---- - 7 a z + 4 a z + 8 a z + 8 z - -- + 12 a z + a 2 a 5 4 4 6 4 4 z 5 3 5 5 5 7 5 6 > 5 a z + 2 a z - ---- + 7 a z + 16 a z + 2 a z - 3 a z - 7 z - a 2 6 4 6 7 3 7 5 7 2 8 4 8 6 8 > 2 a z + 5 a z - 7 a z - 8 a z - a z - 4 a z - 5 a z - a z - 3 9 5 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 2 1 3 1 5 3 5 5 6 + -- + ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 2 14 6 12 6 12 5 10 5 10 4 8 4 8 3 6 3 q q t q t q t q t q t q t q t q t 7 5 5 7 2 2 2 4 2 6 3 > ----- + ----- + ---- + ---- + 3 t + 4 q t + q t + 3 q t + q t 6 2 4 2 4 2 q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n196

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