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L10n45 L10n47

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X9,19,10,18 X6718 X13,20,14,7 X12,5,13,6 X3,10,4,11 X4,15,5,16 X16,12,17,11 X19,14,20,15 X17,2,18,3

Gauss Code:

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

Jones Polynomial:

- 2q-15/2 + 2q-13/2 - 3q-11/2 + 4q-9/2 - 3q-7/2 + 2q-5/2 - 2q-3/2

A2 (sl(3)) Invariant:

q-28 + q-26 + 2q-24 + q-22 + q-18 - q-16 - q-12 + 2q-8 + q-6 + 2q-4

HOMFLY-PT Polynomial:

- a3z-1 - 5a3z - 2a3z3 + 2a5z-1 + 5a5z + 4a5z3 + a5z5 - 2a7z-1 - 3a7z - a7z3 + a9z-1

Kauffman Polynomial:

- a3z-1 + 6a3z - 3a3z3 - 2a4z2 + 2a4z4 - a4z6 - 2a5z-1 + 10a5z - 11a5z3 + 4a5z5 - a5z7 - a6 - 4a6z2 + 4a6z4 - 2a6z6 - 2a7z-1 + 10a7z - 11a7z3 + 4a7z5 - a7z7 - 2a8z2 + 2a8z4 - a8z6 - a9z-1 + 6a9z - 3a9z3

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2             2 j = -4           1 1 j = -6         2 1   j = -8       2 1     j = -10     1 2       j = -12   1 2         j = -14 1 1           j = -16 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[10, NonAlternating, 46]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 46]]
Out[4]=	PD[X[8, 1, 9, 2], X[9, 19, 10, 18], X[6, 7, 1, 8], X[13, 20, 14, 7], > X[12, 5, 13, 6], X[3, 10, 4, 11], X[4, 15, 5, 16], X[16, 12, 17, 11], > X[19, 14, 20, 15], X[17, 2, 18, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 10, -6, -7, 5, -3}, > {3, -1, -2, 6, 8, -5, -4, 9, 7, -8, -10, 2, -9, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-2 2 3 4 3 2 2 ----- + ----- - ----- + ---- - ---- + ---- - ---- 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 2 -22 -18 -16 -12 2 -6 2 q + q + --- + q + q - q - q + -- + q + -- 24 8 4 q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 46]][a, z]
Out[8]=	3 5 7 9 a 2 a 2 a a 3 5 7 3 3 5 3 -(--) + ---- - ---- + -- - 5 a z + 5 a z - 3 a z - 2 a z + 4 a z - z z z z 7 3 5 5 > a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 46]][a, z]
Out[9]=	3 5 7 9 6 a 2 a 2 a a 3 5 7 9 4 2 -a - -- - ---- - ---- - -- + 6 a z + 10 a z + 10 a z + 6 a z - 2 a z - z z z z 6 2 8 2 3 3 5 3 7 3 9 3 4 4 > 4 a z - 2 a z - 3 a z - 11 a z - 11 a z - 3 a z + 2 a z + 6 4 8 4 5 5 7 5 4 6 6 6 8 6 5 7 > 4 a z + 2 a z + 4 a z + 4 a z - a z - 2 a z - a z - a z - 7 7 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 2 1 1 1 2 1 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 16 6 14 6 14 5 12 5 12 4 10 4 10 3 q q t q t q t q t q t q t q t 2 1 2 1 1 > ----- + ----- + ----- + ---- + ---- 8 3 8 2 6 2 6 4 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n46

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