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L11n328 L11n330

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X13,18,14,19 X9,13,10,22 X11,21,12,20 X19,5,20,12 X21,11,22,10 X4,17,1,18

Gauss Code:

{{1, 4, -3, -11}, {-2, -1, 5, 3, -7, 10, -8, 9}, {-6, 2, -4, -5, 11, 6, -9, 8, -10, 7}}

Jones Polynomial:

- q-7 + 2q-6 - 2q-5 + 3q-4 - 2q-3 + 4q-2 - 2q-1 + 2 - q + q2

A2 (sl(3)) Invariant:

- q-26 + q-18 + q-16 + 2q-14 + q-12 + 3q-10 + 4q-8 + 4q-6 + 4q-4 + 3q-2 + 2 + q2 + q4 + q6

HOMFLY-PT Polynomial:

z-2 + 3 + 4z2 + z4 - 2a2z-2 - 5a2 - 7a2z2 - 5a2z4 - a2z6 + a4z-2 + a4 - 2a4z2 - 4a4z4 - a4z6 + 2a6 + 4a6z2 + a6z4 - a8

Kauffman Polynomial:

- z-2 + 6 - 14z2 + 16z4 - 7z6 + z8 + 2az-1 - 5az - 2az3 + 10az5 - 6az7 + az9 - 2a2z-2 + 13a2 - 35a2z2 + 40a2z4 - 19a2z6 + 3a2z8 + 2a3z-1 - 8a3z + 8a3z3 + a3z5 - 4a3z7 + a3z9 - a4z-2 + 9a4 - 18a4z2 + 17a4z4 - 10a4z6 + 2a4z8 - 3a5z + 8a5z3 - 8a5z5 + 2a5z7 + 5a6z2 - 7a6z4 + 2a6z6 + a7z - 2a7z3 + a7z5 - a8 + 2a8z2 + a9z

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 5                   1 j = 3                     j = 1               2 1   j = -1           1 1       j = -3           4 2       j = -5       1 2 3         j = -7       2 1           j = -9   1 2 2             j = -11   2 2               j = -13   1                 j = -15 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[11, NonAlternating, 329]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 329]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[3, 8, 4, 9], X[15, 2, 16, 3], > X[16, 7, 17, 8], X[13, 18, 14, 19], X[9, 13, 10, 22], X[11, 21, 12, 20], > X[19, 5, 20, 12], X[21, 11, 22, 10], X[4, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -11}, {-2, -1, 5, 3, -7, 10, -8, 9}, > {-6, 2, -4, -5, 11, 6, -9, 8, -10, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-7 2 2 3 2 4 2 2 2 - q + -- - -- + -- - -- + -- - - - q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -18 -16 2 -12 3 4 4 4 3 2 4 6 2 - q + q + q + --- + q + --- + -- + -- + -- + -- + q + q + q 14 10 8 6 4 2 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 329]][a, z]
Out[8]=	2 4 2 4 6 8 -2 2 a a 2 2 2 4 2 3 - 5 a + a + 2 a - a + z - ---- + -- + 4 z - 7 a z - 2 a z + 2 2 z z 6 2 4 2 4 4 4 6 4 2 6 4 6 > 4 a z + z - 5 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 329]][a, z]
Out[9]=	2 4 3 2 4 8 -2 2 a a 2 a 2 a 3 6 + 13 a + 9 a - a - z - ---- - -- + --- + ---- - 5 a z - 8 a z - 2 2 z z z z 5 7 9 2 2 2 4 2 6 2 8 2 > 3 a z + a z + a z - 14 z - 35 a z - 18 a z + 5 a z + 2 a z - 3 3 3 5 3 7 3 4 2 4 4 4 > 2 a z + 8 a z + 8 a z - 2 a z + 16 z + 40 a z + 17 a z - 6 4 5 3 5 5 5 7 5 6 2 6 4 6 > 7 a z + 10 a z + a z - 8 a z + a z - 7 z - 19 a z - 10 a z + 6 6 7 3 7 5 7 8 2 8 4 8 9 3 9 > 2 a z - 6 a z - 4 a z + 2 a z + z + 3 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 1 1 2 1 2 2 2 -- + -- + - + ------ + ------ + ------ + ----- + ------ + ----- + ----- + 5 3 q 15 5 13 4 11 4 9 4 11 3 9 3 9 2 q q q t q t q t q t q t q t q t 2 1 1 2 2 t t 2 3 5 4 > ----- + ----- + ---- + ---- + --- + - + 2 q t + q t + q t 7 2 5 2 7 5 3 q q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n329

L11n328 L11n330