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L11n165 L11n167

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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X11,19,12,18 X19,7,20,22 X15,21,16,20 X21,17,22,16 X17,13,18,12 X6718 X4,13,5,14

Gauss Code:

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

Jones Polynomial:

- q-13/2 + q-11/2 - 2q-9/2 + 2q-7/2 - 2q-5/2 + q-3/2 - q-1/2 - q1/2 + q3/2 - q5/2 + q7/2

A2 (sl(3)) Invariant:

q-20 + 2q-16 + 2q-14 + q-12 + 2q-10 + q-6 + q-2 + 2 - q6 - q8 - q10

HOMFLY-PT Polynomial:

a-1z-1 + 6a-1z + 5a-1z3 + a-1z5 - 2az-1 - 12az - 15az3 - 7az5 - az7 + a3z + a3z3 + a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

- 2a-2 + 11a-2z2 - 15a-2z4 + 7a-2z6 - a-2z8 + a-1z-1 - 6a-1z + 13a-1z3 - 15a-1z5 + 7a-1z7 - a-1z9 - 5 + 25z2 - 33z4 + 15z6 - 2z8 + 2az-1 - 13az + 22az3 - 20az5 + 8az7 - az9 - 3a2 + 13a2z2 - 18a2z4 + 8a2z6 - a2z8 + a4 - 2a4z2 + 3a4z4 - a4z6 - a5z-1 + 4a5z - 5a5z3 + 4a5z5 - a5z7 - a6z2 + 3a6z4 - a6z6 - 3a7z + 4a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 8                       1 j = 6                         j = 4                   1 1   j = 2               1 1       j = 0             1   1       j = -2           2 3 1         j = -4         1               j = -6       1 2 1             j = -8     1 1                 j = -10     1                   j = -12 1 1                     j = -14 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[11, NonAlternating, 166]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 166]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 5, 15, 6], > X[11, 19, 12, 18], X[19, 7, 20, 22], X[15, 21, 16, 20], X[21, 17, 22, 16], > X[17, 13, 18, 12], X[6, 7, 1, 8], X[4, 13, 5, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -10}, > {10, -1, 2, -3, -5, 9, 11, -4, -7, 8, -9, 5, -6, 7, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) -(11/2) 2 2 2 -(3/2) 1 -q + q - ---- + ---- - ---- + q - ------- - Sqrt[q] + 9/2 7/2 5/2 Sqrt[q] q q q 3/2 5/2 7/2 > q - q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 2 -12 2 -6 -2 6 8 10 2 + q + --- + --- + q + --- + q + q - q - q - q 16 14 10 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 166]][a, z]
Out[8]=	5 3 1 2 a a 6 z 3 5 5 z 3 3 3 --- - --- + -- + --- - 12 a z + a z + 2 a z + ---- - 15 a z + a z + a z z z a a 5 5 3 z 5 7 > a z + -- - 7 a z - a z a
In[9]:=	Kauffman[Link[11, NonAlternating, 166]][a, z]
Out[9]=	5 2 2 4 1 2 a a 6 z 5 7 2 -5 - -- - 3 a + a + --- + --- - -- - --- - 13 a z + 4 a z - 3 a z + 25 z + 2 a z z z a a 2 3 11 z 2 2 4 2 6 2 13 z 3 5 3 7 3 > ----- + 13 a z - 2 a z - a z + ----- + 22 a z - 5 a z + 4 a z - 2 a a 4 5 4 15 z 2 4 4 4 6 4 15 z 5 5 5 > 33 z - ----- - 18 a z + 3 a z + 3 a z - ----- - 20 a z + 4 a z - 2 a a 6 7 7 5 6 7 z 2 6 4 6 6 6 7 z 7 5 7 > a z + 15 z + ---- + 8 a z - a z - a z + ---- + 8 a z - a z - 2 a a 8 9 8 z 2 8 z 9 > 2 z - -- - a z - -- - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 1 1 1 1 2 1 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 6 12 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 1 1 2 t 2 2 2 2 4 3 4 4 8 5 > ----- + ---- + ---- + -- + q t + t + q t + q t + q t + q t 4 2 6 2 2 q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n166

L11n165 L11n167