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DrawMorseLink

PD Presentation:

X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X16,10,11,9 X10,12,5,11 X4,15,1,16

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-6 + q-4 + q-2 + 3 + 5q2 + 3q4 + 6q6 + 3q8 + 4q10 + 2q12 + q16 - q18

HOMFLY-PT Polynomial:

a-4z-2 - 2a-4z2 - a-4z4 - 2a-2z-2 - a-2 + 4a-2z2 + 4a-2z4 + a-2z6 + z-2 + 1 - 2z2 - z4

Kauffman Polynomial:

a-7z3 - 2a-6z2 + 3a-6z4 - 3a-5z3 + 4a-5z5 - a-4z-2 + a-4 + 2a-4z2 - 5a-4z4 + 4a-4z6 + 2a-3z-1 - a-3z - 3a-3z3 + 2a-3z7 - 2a-2z-2 + a-2 + 10a-2z2 - 17a-2z4 + 7a-2z6 + 2a-1z-1 - a-1z - a-1z3 - 3a-1z5 + 2a-1z7 - z-2 + 1 + 6z2 - 9z4 + 3z6 - 2az3 + az5

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L8a16

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