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L11n278 L11n280

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X18,10,19,9 X17,11,18,22 X11,21,12,20 X21,17,22,16 X15,1,16,4 X10,20,5,19

Gauss Code:

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

Jones Polynomial:

2q-1 - 2 + 5q - 4q2 + 6q3 - 4q4 + 3q5 - 2q6

A2 (sl(3)) Invariant:

2q-4 + 2q-2 + 4 + 7q2 + 5q4 + 7q6 + 4q8 + 3q10 + q12 - 2q14 - q16 - 3q18 - 2q20

HOMFLY-PT Polynomial:

- a-6z-2 - 2a-6 + 4a-4z-2 + 8a-4 + 4a-4z2 - 5a-2z-2 - 10a-2 - 7a-2z2 - 2a-2z4 + 2z-2 + 4 + 2z2

Kauffman Polynomial:

a-7z-1 - 5a-7z + 3a-7z3 - a-6z-2 + 3a-6 - 2a-6z2 + a-6z6 + 5a-5z-1 - 18a-5z + 20a-5z3 - 8a-5z5 + 2a-5z7 - 4a-4z-2 + 12a-4 - 14a-4z2 + 10a-4z4 - 3a-4z6 + a-4z8 + 9a-3z-1 - 24a-3z + 24a-3z3 - 11a-3z5 + 3a-3z7 - 5a-2z-2 + 15a-2 - 21a-2z2 + 13a-2z4 - 4a-2z6 + a-2z8 + 5a-1z-1 - 11a-1z + 7a-1z3 - 3a-1z5 + a-1z7 - 2z-2 + 7 - 9z2 + 3z4

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13               2 j = 11             1   j = 9           3 2   j = 7         3 1     j = 5       2 4       j = 3     3 2         j = 1   1 4           j = -1 1 1             j = -3 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 279]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 279]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[18, 10, 19, 9], X[17, 11, 18, 22], X[11, 21, 12, 20], > X[21, 17, 22, 16], X[15, 1, 16, 4], X[10, 20, 5, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 10}, {-2, -1, 5, 3, 6, -11}, > {-8, 2, 4, -5, -10, 9, -7, -6, 11, 8, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	2 2 3 4 5 6 -2 + - + 5 q - 4 q + 6 q - 4 q + 3 q - 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 2 2 4 6 8 10 12 14 16 18 4 + -- + -- + 7 q + 5 q + 7 q + 4 q + 3 q + q - 2 q - q - 3 q - 4 2 q q 20 > 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 279]][a, z]
Out[8]=	2 2 4 2 8 10 2 1 4 5 2 4 z 7 z 2 z 4 - -- + -- - -- + -- - ----- + ----- - ----- + 2 z + ---- - ---- - ---- 6 4 2 2 6 2 4 2 2 2 4 2 2 a a a z a z a z a z a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 279]][a, z]
Out[9]=	3 12 15 2 1 4 5 1 5 9 5 7 + -- + -- + -- - -- - ----- - ----- - ----- + ---- + ---- + ---- + --- - 6 4 2 2 6 2 4 2 2 2 7 5 3 a z a a a z a z a z a z a z a z a z 2 2 2 3 3 5 z 18 z 24 z 11 z 2 2 z 14 z 21 z 3 z 20 z > --- - ---- - ---- - ---- - 9 z - ---- - ----- - ----- + ---- + ----- + 7 5 3 a 6 4 2 7 5 a a a a a a a a 3 3 4 4 5 5 5 6 6 24 z 7 z 4 10 z 13 z 8 z 11 z 3 z z 3 z > ----- + ---- + 3 z + ----- + ----- - ---- - ----- - ---- + -- - ---- - 3 a 4 2 5 3 a 6 4 a a a a a a a 6 7 7 7 8 8 4 z 2 z 3 z z z z > ---- + ---- + ---- + -- + -- + -- 2 5 3 a 4 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 2 1 1 q 3 5 5 2 7 2 4 q + 3 q + ----- + ---- + --- + - + 2 q t + 2 q t + 4 q t + 3 q t + 3 2 2 q t t q t q t 7 3 9 3 9 4 11 4 13 5 > q t + 3 q t + 2 q t + q t + 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n279

L11n278 L11n280