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L9n17 L9n19

Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,5,9,6 X7,14,8,15 X13,16,14,17 X15,8,16,1 X6,9,7,10 X4,17,5,18

Gauss Code:

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

Jones Polynomial:

- q-21/2 + q-15/2 - q-11/2 - q-7/2

A2 (sl(3)) Invariant:

- q-36 + q-34 + 2q-32 + 2q-30 + q-28 - q-26 - q-24 - q-22 + q-20 + 2q-18 + 2q-16 + q-14 + q-12

HOMFLY-PT Polynomial:

- a7z-1 - 11a7z - 15a7z3 - 7a7z5 - a7z7 + a9z-1 + 8a9z + 6a9z3 + a9z5 - a11z

Kauffman Polynomial:

- a7z-1 + 11a7z - 15a7z3 + 7a7z5 - a7z7 + a8 - 8a8z2 + 6a8z4 - a8z6 - a9z-1 + 9a9z - 14a9z3 + 7a9z5 - a9z7 - 8a10z2 + 6a10z4 - a10z6 + a11z + 3a13z - a13z3

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6                 1 j = -8                 1 j = -10             1     j = -12         1         j = -14       1 1 1       j = -16       1           j = -18     1 1           j = -20 1                 j = -22 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, 18]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 18]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[18, 5, 9, 6], > X[7, 14, 8, 15], X[13, 16, 14, 17], X[15, 8, 16, 1], X[6, 9, 7, 10], > X[4, 17, 5, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -9, 4, -8, -5, 7}, {8, -1, 2, -3, -6, 5, -7, 6, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) -(15/2) -(11/2) -(7/2) -q + q - q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-36 -34 2 2 -28 -26 -24 -22 -20 2 2 -q + q + --- + --- + q - q - q - q + q + --- + --- + 32 30 18 16 q q q q -14 -12 > q + q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 18]][a, z]
Out[8]=	7 9 a a 7 9 11 7 3 9 3 7 5 9 5 -(--) + -- - 11 a z + 8 a z - a z - 15 a z + 6 a z - 7 a z + a z - z z 7 7 > a z
In[9]:=	Kauffman[Link[9, NonAlternating, 18]][a, z]
Out[9]=	7 9 8 a a 7 9 11 13 8 2 10 2 a - -- - -- + 11 a z + 9 a z + a z + 3 a z - 8 a z - 8 a z - z z 7 3 9 3 13 3 8 4 10 4 7 5 9 5 > 15 a z - 14 a z - a z + 6 a z + 6 a z + 7 a z + 7 a z - 8 6 10 6 7 7 9 7 > a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 1 1 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 22 8 20 8 18 6 18 5 16 5 14 5 14 4 q t q t q t q t q t q t q t 1 1 1 > ------ + ------ + ------ 12 4 14 3 10 2 q t q t q t

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