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L8n7 L9a1

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X2536 X16,11,13,12 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X12,15,9,16

Gauss Code:

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

Jones Polynomial:

- q-7/2 - q-3/2 - 2q-1/2 - 2q1/2 - q3/2 - q7/2

A2 (sl(3)) Invariant:

q-12 + 2q-10 + 4q-8 + 6q-6 + 9q-4 + 12q-2 + 13 + 12q2 + 9q4 + 6q6 + 4q8 + 2q10 + q12

HOMFLY-PT Polynomial:

- a-3z-3 - 2a-3z-1 - a-3z + 3a-1z-3 + 6a-1z-1 + 5a-1z + a-1z3 - 3az-3 - 6az-1 - 5az - az3 + a3z-3 + 2a3z-1 + a3z

Kauffman Polynomial:

- a-3z-3 + 4a-3z-1 - 6a-3z + 5a-3z3 - a-3z5 + 3a-2z-2 - 8a-2 + 6a-2z2 - a-2z4 - 3a-1z-3 + 9a-1z-1 - 14a-1z + 7a-1z3 - a-1z5 + 6z-2 - 15 + 12z2 - 2z4 - 3az-3 + 9az-1 - 14az + 7az3 - az5 + 3a2z-2 - 8a2 + 6a2z2 - a2z4 - a3z-3 + 4a3z-1 - 6a3z + 5a3z3 - a3z5

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L8n8

L8n7 L9a1