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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 4q-3/2 + 5q-1/2 - 6q1/2 + 5q3/2 - 5q5/2 + 3q7/2 - q9/2

A2 (sl(3)) Invariant:

q-12 + q-10 + 2q-6 + 1 - q2 + 2q4 + 2q8 + q10 - q12 + q14

HOMFLY-PT Polynomial:

- a-3z-1 - a-3z - a-3z3 + 2a-1z-1 + 4a-1z + 3a-1z3 + a-1z5 - 2az-1 - 4az - 2az3 + a3z-1 + a3z

Kauffman Polynomial:

- a-5z3 + a-4z2 - 3a-4z4 + a-3z-1 - 3a-3z + 6a-3z3 - 5a-3z5 - a-2z2 + 4a-2z4 - 4a-2z6 + 2a-1z-1 - 10a-1z + 17a-1z3 - 7a-1z5 - a-1z7 + 1 - 5z2 + 12z4 - 6z6 + 2az-1 - 10az + 13az3 - 3az5 - az7 - 3a2z2 + 5a2z4 - 2a2z6 + a3z-1 - 3a3z + 3a3z3 - a3z5

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L8a2

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