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L11n443 L11n445

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DrawMorseLink

PD Presentation:

X6172 X2536 X11,19,12,18 X10,3,11,4 X4,9,1,10 X16,7,17,8 X8,15,5,16 X13,20,14,21 X19,15,20,22 X21,12,22,13 X17,9,18,14

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 2q-15/2 - 6q-13/2 + 6q-11/2 - 10q-9/2 + 8q-7/2 - 10q-5/2 + 6q-3/2 - 5q-1/2 + 2q1/2

A2 (sl(3)) Invariant:

q-28 + 3q-26 + 3q-24 + 6q-22 + 10q-20 + 9q-18 + 13q-16 + 11q-14 + 9q-12 + 9q-10 + 4q-8 + 5q-6 - q-2 + 1 - 2q2

HOMFLY-PT Polynomial:

az-1 + 3az + 2az3 - a3z-3 - 7a3z-1 - 13a3z - 8a3z3 - 2a3z5 + 3a5z-3 + 12a5z-1 + 14a5z + 5a5z3 - 3a7z-3 - 7a7z-1 - 4a7z + a9z-3 + a9z-1

Kauffman Polynomial:

1 - 3z2 - 2az-1 + 6az - 5az3 - az5 + 6a2 - 14a2z2 + 11a2z4 - 5a2z6 + a3z-3 - 11a3z-1 + 30a3z - 39a3z3 + 27a3z5 - 8a3z7 - 3a4z-2 + 18a4 - 28a4z2 + 11a4z4 + 9a4z6 - 5a4z8 + 3a5z-3 - 18a5z-1 + 49a5z - 74a5z3 + 54a5z5 - 11a5z7 - a5z9 - 6a6z-2 + 21a6 - 24a6z2 - 4a6z4 + 21a6z6 - 7a6z8 + 3a7z-3 - 14a7z-1 + 35a7z - 50a7z3 + 31a7z5 - 4a7z7 - a7z9 - 3a8z-2 + 9a8 - 7a8z2 - 4a8z4 + 7a8z6 - 2a8z8 + a9z-3 - 5a9z-1 + 10a9z - 10a9z3 + 5a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 2                   2 j = 0                 3   j = -2               4 3   j = -4             6 3 1   j = -6           5 7       j = -8         5 4 1       j = -10       4 8           j = -12     2 2             j = -14   1 5               j = -16   1                 j = -18 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 444]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 444]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[11, 19, 12, 18], X[10, 3, 11, 4], > X[4, 9, 1, 10], X[16, 7, 17, 8], X[8, 15, 5, 16], X[13, 20, 14, 21], > X[19, 15, 20, 22], X[21, 12, 22, 13], X[17, 9, 18, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, -3, 10, -8, 11}, > {7, -6, -11, 3, -9, 8, -10, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 2 6 6 10 8 10 6 5 -q + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q > 2 Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 3 3 6 10 9 13 11 9 9 4 5 1 + q + --- + --- + --- + --- + --- + --- + --- + --- + --- + -- + -- - 26 24 22 20 18 16 14 12 10 8 6 q q q q q q q q q q q -2 2 > q - 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 444]][a, z]
Out[8]=	3 5 7 9 3 5 7 9 a 3 a 3 a a a 7 a 12 a 7 a a 3 -(--) + ---- - ---- + -- + - - ---- + ----- - ---- + -- + 3 a z - 13 a z + 3 3 3 3 z z z z z z z z z 5 7 3 3 3 5 3 3 5 > 14 a z - 4 a z + 2 a z - 8 a z + 5 a z - 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 444]][a, z]
Out[9]=	3 5 7 9 4 6 8 2 4 6 8 a 3 a 3 a a 3 a 6 a 3 a 1 + 6 a + 18 a + 21 a + 9 a + -- + ---- + ---- + -- - ---- - ---- - ---- - 3 3 3 3 2 2 2 z z z z z z z 3 5 7 9 2 a 11 a 18 a 14 a 5 a 3 5 7 > --- - ----- - ----- - ----- - ---- + 6 a z + 30 a z + 49 a z + 35 a z + z z z z z 9 2 2 2 4 2 6 2 8 2 3 > 10 a z - 3 z - 14 a z - 28 a z - 24 a z - 7 a z - 5 a z - 3 3 5 3 7 3 9 3 2 4 4 4 6 4 > 39 a z - 74 a z - 50 a z - 10 a z + 11 a z + 11 a z - 4 a z - 8 4 5 3 5 5 5 7 5 9 5 2 6 > 4 a z - a z + 27 a z + 54 a z + 31 a z + 5 a z - 5 a z + 4 6 6 6 8 6 3 7 5 7 7 7 9 7 > 9 a z + 21 a z + 7 a z - 8 a z - 11 a z - 4 a z - a z - 4 8 6 8 8 8 5 9 7 9 > 5 a z - 7 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 3 1 1 1 5 2 2 4 3 + q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 8 16 7 14 7 14 6 12 6 12 5 10 5 q q t q t q t q t q t q t q t 8 5 4 5 1 7 6 3 4 > ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 10 4 8 4 8 3 6 3 8 2 6 2 4 2 4 2 q t q t q t q t q t q t q t q t q t 2 > 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n444

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