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L11n144 L11n146

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DrawMorseLink

PD Presentation:

X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X15,20,16,21 X13,22,14,7 X21,14,22,15 X4,20,5,19

Gauss Code:

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

Jones Polynomial:

- q-15/2 + 4q-13/2 - 7q-11/2 + 9q-9/2 - 11q-7/2 + 10q-5/2 - 9q-3/2 + 6q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

q-22 - 2q-20 + q-18 - q-16 + 3q-12 - q-10 + 4q-8 - q-6 + 2q-4 + 2q-2 + 2q2 - q4

HOMFLY-PT Polynomial:

- az-1 - az + 2az3 + az5 + a3z-1 - 4a3z3 - 4a3z5 - a3z7 + 2a5z3 + a5z5

Kauffman Polynomial:

2z4 - z6 + az-1 - az - 7az3 + 12az5 - 4az7 - a2 + 2a2z2 - 10a2z4 + 15a2z6 - 5a2z8 + a3z-1 - a3z - 10a3z3 + 15a3z5 - a3z7 - 2a3z9 + 6a4z2 - 20a4z4 + 23a4z6 - 8a4z8 + 2a5z7 - 2a5z9 + 6a6z2 - 12a6z4 + 7a6z6 - 3a6z8 + 2a7z3 - 3a7z5 - a7z7 + 2a8z2 - 4a8z4 - a9z3

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                   1 j = 2                 3   j = 0               3 1   j = -2             6 3     j = -4           5 4       j = -6         6 5         j = -8       4 6           j = -10     3 5             j = -12   1 4               j = -14   3                 j = -16 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, 145]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 145]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 11, 19, 12], X[10, 4, 11, 3], X[2, 17, 3, 18], > X[12, 5, 13, 6], X[6, 7, 1, 8], X[16, 10, 17, 9], X[15, 20, 16, 21], > X[13, 22, 14, 7], X[21, 14, 22, 15], X[4, 20, 5, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11, 5, -6}, > {6, -1, 7, -3, 2, -5, -9, 10, -8, -7, 4, -2, 11, 8, -10, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 7 9 11 10 9 6 -q + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 4 Sqrt[q] + 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 -18 -16 3 -10 4 -6 2 2 2 4 q - --- + q - q + --- - q + -- - q + -- + -- + 2 q - q 20 12 8 4 2 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 145]][a, z]
Out[8]=	3 a a 3 3 3 5 3 5 3 5 5 5 3 7 -(-) + -- - a z + 2 a z - 4 a z + 2 a z + a z - 4 a z + a z - a z z z
In[9]:=	Kauffman[Link[11, NonAlternating, 145]][a, z]
Out[9]=	3 2 a a 3 2 2 4 2 6 2 8 2 3 -a + - + -- - a z - a z + 2 a z + 6 a z + 6 a z + 2 a z - 7 a z - z z 3 3 7 3 9 3 4 2 4 4 4 6 4 > 10 a z + 2 a z - a z + 2 z - 10 a z - 20 a z - 12 a z - 8 4 5 3 5 7 5 6 2 6 4 6 > 4 a z + 12 a z + 15 a z - 3 a z - z + 15 a z + 23 a z + 6 6 7 3 7 5 7 7 7 2 8 4 8 6 8 > 7 a z - 4 a z - a z + 2 a z - a z - 5 a z - 8 a z - 3 a z - 3 9 5 9 > 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 6 1 3 1 4 3 5 4 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 4 2 16 6 14 5 12 5 12 4 10 4 10 3 8 3 8 2 q q q t q t q t q t q t q t q t q t 6 5 5 3 t 2 2 2 4 3 > ----- + ---- + ---- + 3 t + --- + t + 3 q t + q t 6 2 6 4 2 q t q t q t q

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