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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X8,16,9,15 X14,8,15,7 X16,10,11,9 X10,12,5,11 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

1 - q + 3q2 - 2q3 + 4q4 - 3q5 + 3q6 - 2q7 + q8

A2 (sl(3)) Invariant:

1 + q2 + 2q4 + 3q6 + 3q8 + 5q10 + 3q12 + 4q14 + 2q16 + q18 + q20 + q24

HOMFLY-PT Polynomial:

a-6z-2 + 2a-6 + 3a-6z2 + a-6z4 - 2a-4z-2 - 6a-4 - 8a-4z2 - 5a-4z4 - a-4z6 + a-2z-2 + 4a-2 + 4a-2z2 + a-2z4

Kauffman Polynomial:

a-10z2 + 2a-9z3 + a-8 - 3a-8z2 + 3a-8z4 - 4a-7z3 + 3a-7z5 + a-6z-2 - 3a-6 + 6a-6z2 - 8a-6z4 + 3a-6z6 - 2a-5z-1 + 6a-5z - 7a-5z3 + a-5z7 + 2a-4z-2 - 8a-4 + 18a-4z2 - 16a-4z4 + 4a-4z6 - 2a-3z-1 + 6a-3z - a-3z3 - 3a-3z5 + a-3z7 + a-2z-2 - 5a-2 + 8a-2z2 - 5a-2z4 + a-2z6

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L8a18

L8a17 L8a19