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L10n43 L10n45

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X9,19,10,18 X6718 X13,20,14,7 X5,13,6,12 X3,10,4,11 X15,5,16,4 X11,16,12,17 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:

q-15/2 - 2q-13/2 + 2q-11/2 - 3q-9/2 + 3q-7/2 - 3q-5/2 + 2q-3/2 - 2q-1/2

A2 (sl(3)) Invariant:

- q-24 + q-20 + q-16 + q-10 + q-8 + 2q-6 + q-4 + q-2 + 2

HOMFLY-PT Polynomial:

- az-1 - 2az + a3z-1 + a3z + a3z3 + a5z + a5z3 - a7z

Kauffman Polynomial:

az-1 - 2az - a2 + a2z2 - a2z4 + a3z-1 - 5a3z3 + 4a3z5 - a3z7 + 3a4z2 - 5a4z4 + 4a4z6 - a4z8 + 5a5z - 15a5z3 + 13a5z5 - 3a5z7 - a6z2 + 3a6z6 - a6z8 + 3a7z - 10a7z3 + 9a7z5 - 2a7z7 - 3a8z2 + 4a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = 0               2 j = -2             1 1 j = -4           2 1   j = -6         2 2     j = -8       1 1       j = -10     1 2         j = -12   1 1           j = -14   1             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[10, NonAlternating, 44]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 44]]
Out[4]=	PD[X[8, 1, 9, 2], X[9, 19, 10, 18], X[6, 7, 1, 8], X[13, 20, 14, 7], > X[5, 13, 6, 12], X[3, 10, 4, 11], X[15, 5, 16, 4], X[11, 16, 12, 17], > 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]=	-(15/2) 2 2 3 3 3 2 2 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]=	-24 -20 -16 -10 -8 2 -4 -2 2 - q + q + q + q + q + -- + q + q 6 q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 44]][a, z]
Out[8]=	3 a a 3 5 7 3 3 5 3 -(-) + -- - 2 a z + a z + a z - a z + a z + a z z z
In[9]:=	Kauffman[Link[10, NonAlternating, 44]][a, z]
Out[9]=	3 2 a a 5 7 2 2 4 2 6 2 8 2 -a + - + -- - 2 a z + 5 a z + 3 a z + a z + 3 a z - a z - 3 a z - z z 3 3 5 3 7 3 2 4 4 4 8 4 3 5 > 5 a z - 15 a z - 10 a z - a z - 5 a z + 4 a z + 4 a z + 5 5 7 5 4 6 6 6 8 6 3 7 5 7 > 13 a z + 9 a z + 4 a z + 3 a z - a z - a z - 3 a z - 7 7 4 8 6 8 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-2 1 1 1 1 1 2 1 1 2 + q + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q t q t q t q t q t q t q t q t 2 2 2 1 1 > ----- + ----- + ----- + ---- + ---- 6 3 6 2 4 2 4 2 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n44

L10n43 L10n45