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L11n286 L11n288

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X4,15,1,16 X21,18,22,19 X9,21,10,20 X19,5,20,10 X11,16,12,17 X17,22,18,11

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-22 + q-14 + q-10 + q-8 + 2q-6 + 3q-4 + 4q-2 + 5 + 5q2 + 3q4 + 2q6 + q8

HOMFLY-PT Polynomial:

a-2z-2 + a-2 - 2z-2 - 2 - z2 + a2z-2 - a2z2 + 2a4 + 3a4z2 + a4z4 - a6 - a6z2

Kauffman Polynomial:

a-2z-2 - 2a-2 + a-2z2 - 2a-1z-1 + 2a-1z + 2z-2 - 3 + 2z2 - 2az-1 + 4az - 4az3 + az5 + a2z-2 - 10a2z2 + 12a2z4 - 6a2z6 + a2z8 - 2a3z3 + 6a3z5 - 5a3z7 + a3z9 + 4a4 - 18a4z2 + 28a4z4 - 17a4z6 + 3a4z8 - 4a5z + 8a5z3 - 4a5z7 + a5z9 + 2a6 - 7a6z2 + 16a6z4 - 11a6z6 + 2a6z8 - 2a7z + 6a7z3 - 5a7z5 + a7z7

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 5                   1 j = 3                   1 j = 1               3 1   j = -1             2 4 1   j = -3           2 2 2     j = -5         2 3         j = -7       1 2 1         j = -9     1 2 1           j = -11   1 1               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, 287]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 287]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[8, 4, 9, 3], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[4, 15, 1, 16], X[21, 18, 22, 19], X[9, 21, 10, 20], > X[19, 5, 20, 10], X[11, 16, 12, 17], X[17, 22, 18, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -6}, {-2, -1, 5, -3, -8, 9}, > {-10, 2, 4, -5, 6, 10, -11, 7, -9, 8, -7, 11}]
In[6]:=	Jones[L][q]
Out[6]=	-7 2 2 2 2 3 1 2 2 - q + -- - -- + -- - -- + -- - - + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -14 -10 -8 2 3 4 2 4 6 8 5 - q + q + q + q + -- + -- + -- + 5 q + 3 q + 2 q + q 6 4 2 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 287]][a, z]
Out[8]=	2 -2 4 6 2 1 a 2 2 2 4 2 6 2 4 4 -2 + a + 2 a - a - -- + ----- + -- - z - a z + 3 a z - a z + a z 2 2 2 2 z a z z
In[9]:=	Kauffman[Link[11, NonAlternating, 287]][a, z]
Out[9]=	2 2 4 6 2 1 a 2 2 a 2 z 5 -3 - -- + 4 a + 2 a + -- + ----- + -- - --- - --- + --- + 4 a z - 4 a z - 2 2 2 2 2 a z z a a z a z z 2 7 2 z 2 2 4 2 6 2 3 3 3 > 2 a z + 2 z + -- - 10 a z - 18 a z - 7 a z - 4 a z - 2 a z + 2 a 5 3 7 3 2 4 4 4 6 4 5 3 5 > 8 a z + 6 a z + 12 a z + 28 a z + 16 a z + a z + 6 a z - 7 5 2 6 4 6 6 6 3 7 5 7 7 7 > 5 a z - 6 a z - 17 a z - 11 a z - 5 a z - 4 a z + a z + 2 8 4 8 6 8 3 9 5 9 > a z + 3 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 1 1 1 1 2 1 -- + - + 3 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 q q t q t q t q t q t q t q t 1 2 2 1 3 2 2 2 t > ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + - + q t + 9 3 7 3 5 3 7 2 5 2 3 2 3 q t q q t q t q t q t q t q t q t 3 2 5 2 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n287

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