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L10n44 L10n46

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X18,9,19,10 X6718 X20,14,7,13 X5,13,6,12 X3,10,4,11 X15,5,16,4 X11,16,12,17 X14,20,15,19 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-7/2 - q-3/2 + q-1/2 - q1/2 + q3/2 - q5/2

A2 (sl(3)) Invariant:

q-16 + q-14 + q-12 + q-10 + q-8 + q-6 + q4 + q6 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 3a-1z - a-1z3 + 2az-1 + 6az + 5az3 + az5 - 2a3z-1 - 4a3z - a3z3 + a5z-1

Kauffman Polynomial:

- a-1z-1 + 5a-1z - 10a-1z3 + 6a-1z5 - a-1z7 + 5z2 - 10z4 + 6z6 - z8 - 2az-1 + 11az - 20az3 + 12az5 - 2az7 - a2 + 6a2z2 - 10a2z4 + 6a2z6 - a2z8 - 2a3z-1 + 7a3z - 10a3z3 + 6a3z5 - a3z7 + a4z2 - a5z-1 + a5z

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6             1 j = 4               j = 2         1 1   j = 0     1 1       j = -2     1 1       j = -4 1 1 1         j = -6 1             j = -8 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, 45]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 45]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 9, 19, 10], X[6, 7, 1, 8], X[20, 14, 7, 13], > X[5, 13, 6, 12], X[3, 10, 4, 11], X[15, 5, 16, 4], X[11, 16, 12, 17], > X[14, 20, 15, 19], 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]=	-(7/2) -(3/2) 1 3/2 5/2 -q - q + ------- - Sqrt[q] + q - q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 -12 -10 -8 -6 4 6 8 q + q + q + q + q + q + q + q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 45]][a, z]
Out[8]=	3 5 3 1 2 a 2 a a 3 z 3 z 3 3 3 5 -(---) + --- - ---- + -- - --- + 6 a z - 4 a z - -- + 5 a z - a z + a z a z z z z a a
In[9]:=	Kauffman[Link[10, NonAlternating, 45]][a, z]
Out[9]=	3 5 2 1 2 a 2 a a 5 z 3 5 2 2 2 -a - --- - --- - ---- - -- + --- + 11 a z + 7 a z + a z + 5 z + 6 a z + a z z z z a 3 5 4 2 10 z 3 3 3 4 2 4 6 z 5 > a z - ----- - 20 a z - 10 a z - 10 z - 10 a z + ---- + 12 a z + a a 7 3 5 6 2 6 z 7 3 7 8 2 8 > 6 a z + 6 z + 6 a z - -- - 2 a z - a z - z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	-4 -2 1 1 1 1 t 2 2 2 3 6 4 1 + q + q + ----- + ----- + ----- + ---- + t + -- + q t + q t + q t 8 2 6 2 4 2 4 2 q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n45

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