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L9n16 L9n18

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X7,14,8,15 X13,18,14,7 X17,1,18,6 X16,11,17,12 X5,12,6,13 X4,16,5,15

Gauss Code:

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

Jones Polynomial:

q-17/2 - 2q-15/2 + 3q-13/2 - 4q-11/2 + 4q-9/2 - 4q-7/2 + 2q-5/2 - 2q-3/2

A2 (sl(3)) Invariant:

- q-26 - q-22 - q-20 + q-18 + 2q-14 + q-12 + 2q-10 + 3q-8 + q-6 + 2q-4

HOMFLY-PT Polynomial:

- 2a3z-1 - 5a3z - 2a3z3 + 3a5z-1 + 6a5z + 4a5z3 + a5z5 - a7z-1 - 2a7z - a7z3

Kauffman Polynomial:

- 2a3z-1 + 6a3z - 3a3z3 + 3a4 - 4a4z2 + 2a4z4 - a4z6 - 3a5z-1 + 9a5z - 9a5z3 + 3a5z5 - a5z7 + 3a6 - 9a6z2 + 7a6z4 - 3a6z6 - a7z-1 + 2a7z - 2a7z3 + a7z5 - a7z7 + a8 - 3a8z2 + 4a8z4 - 2a8z6 - a9z + 4a9z3 - 2a9z5 + 2a10z2 - a10z4

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n17

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