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L10n93 L10n95

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X18,10,19,9 X6,18,1,17 X7,17,8,16 X3,10,4,11 X14,6,15,5 X4,14,5,13 X11,13,12,20 X15,7,16,12 X19,3,20,2

Gauss Code:

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

Jones Polynomial:

q2 + q3 + q4 + q9

A2 (sl(3)) Invariant:

q6 + 2q8 + 4q10 + 4q12 + 4q14 + 2q16 + q18 + q20 + 2q22 + 3q24 + 2q26 + q28

HOMFLY-PT Polynomial:

a-8z-2 + 3a-8 + a-8z2 - 2a-6z-2 - 9a-6 - 6a-6z2 - a-6z4 + a-4z-2 + 6a-4 + 5a-4z2 + a-4z4

Kauffman Polynomial:

- 2a-10 + 9a-10z2 - 6a-10z4 + a-10z6 - a-8z-2 + 3a-8 - a-8z2 + 2a-7z-1 - 9a-7z + 6a-7z3 - a-7z5 - 2a-6z-2 + 11a-6 - 15a-6z2 + 7a-6z4 - a-6z6 + 2a-5z-1 - 9a-5z + 6a-5z3 - a-5z5 - a-4z-2 + 7a-4 - 5a-4z2 + a-4z4

Khovanov Homology:

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                 1 j = 17                 1 j = 15           1 1     j = 13                   j = 11       1 2 1       j = 9     1             j = 7     3 1           j = 5 1   1             j = 3 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]=	3
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 94]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 94]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 10, 19, 9], X[6, 18, 1, 17], X[7, 17, 8, 16], > X[3, 10, 4, 11], X[14, 6, 15, 5], X[4, 14, 5, 13], X[11, 13, 12, 20], > X[15, 7, 16, 12], X[19, 3, 20, 2]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 10, -5, -7, 6, -3}, {-4, -1, 2, 5, -8, 9}, > {7, -6, -9, 4, 3, -2, -10, 8}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 9 q + q + q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	6 8 10 12 14 16 18 20 22 24 26 q + 2 q + 4 q + 4 q + 4 q + 2 q + q + q + 2 q + 3 q + 2 q + 28 > q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 94]][a, z]
Out[8]=	2 2 2 4 4 3 9 6 1 2 1 z 6 z 5 z z z -- - -- + -- + ----- - ----- + ----- + -- - ---- + ---- - -- + -- 8 6 4 8 2 6 2 4 2 8 6 4 6 4 a a a a z a z a z a a a a a
In[9]:=	Kauffman[Link[10, NonAlternating, 94]][a, z]
Out[9]=	2 -2 3 11 7 1 2 1 2 2 9 z 9 z 9 z --- + -- + -- + -- - ----- - ----- - ----- + ---- + ---- - --- - --- + ---- - 10 8 6 4 8 2 6 2 4 2 7 5 7 5 10 a a a a a z a z a z a z a z a a a 2 2 2 3 3 4 4 4 5 5 6 6 z 15 z 5 z 6 z 6 z 6 z 7 z z z z z z > -- - ----- - ---- + ---- + ---- - ---- + ---- + -- - -- - -- + --- - -- 8 6 4 7 5 10 6 4 7 5 10 6 a a a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 5 2 7 2 9 2 7 3 11 3 11 4 11 5 q + q + q t + 3 q t + q t + q t + q t + 2 q t + q t + 15 5 15 6 17 8 19 8 > q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n94

L10n93 L10n95