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Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X3,12,4,13 X18,5,9,6 X6,9,7,10 X16,12,17,11 X7,14,8,15 X13,4,14,5 X15,8,16,1 X2,17,3,18

Gauss Code:

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

Jones Polynomial:

- q-19/2 - q-5/2

A2 (sl(3)) Invariant:

q-30 + q-28 + 2q-26 + 2q-24 + q-22 - q-18 + q-12 + q-10 + q-8

HOMFLY-PT Polynomial:

- 5a5z - 5a5z3 - a5z5 - a7z-1 + a9z-1 + a9z

Kauffman Polynomial:

- 5a5z + 5a5z3 - a5z5 - a7z-1 + a7z + a8 - a9z-1 + a9z - 5a11z + 5a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                 1 j = -6                 1 j = -8           1 1     j = -10                   j = -12       1 2 1       j = -14                   j = -16     1 1           j = -18 1                 j = -20 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[9, NonAlternating, 19]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 19]]
Out[4]=	PD[X[10, 1, 11, 2], X[3, 12, 4, 13], X[18, 5, 9, 6], X[6, 9, 7, 10], > X[16, 12, 17, 11], X[7, 14, 8, 15], X[13, 4, 14, 5], X[15, 8, 16, 1], > X[2, 17, 3, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, -2, 7, 3, -4, -6, 8}, {4, -1, 5, 2, -7, 6, -8, -5, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) -(5/2) -q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 2 2 -22 -18 -12 -10 -8 q + q + --- + --- + q - q + q + q + q 26 24 q q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 19]][a, z]
Out[8]=	7 9 a a 5 9 5 3 5 5 -(--) + -- - 5 a z + a z - 5 a z - a z z z
In[9]:=	Kauffman[Link[9, NonAlternating, 19]][a, z]
Out[9]=	7 9 8 a a 5 7 9 11 5 3 11 3 5 5 a - -- - -- - 5 a z + a z + a z - 5 a z + 5 a z + 5 a z - a z - z z 11 5 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 1 1 1 2 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 8 18 8 16 6 16 5 12 5 12 4 12 3 q t q t q t q t q t q t q t 1 1 > ----- + ----- 8 3 8 2 q t q t

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