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L11n166 L11n168

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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X11,19,12,18 X22,19,7,20 X20,15,21,16 X16,21,17,22 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-23/2 - 3q-21/2 + 5q-19/2 - 8q-17/2 + 9q-15/2 - 9q-13/2 + 8q-11/2 - 6q-9/2 + 3q-7/2 - 2q-5/2

A2 (sl(3)) Invariant:

- q-34 + q-32 - q-30 + q-28 + 2q-26 + 3q-22 - 2q-20 + q-18 + 3q-12 + 2q-8

HOMFLY-PT Polynomial:

- a5z-1 - 8a5z - 8a5z3 - 2a5z5 + a7z-1 + 7a7z + 9a7z3 + 5a7z5 + a7z7 - 2a9z - 3a9z3 - a9z5

Kauffman Polynomial:

a5z-1 - 8a5z + 10a5z3 - 3a5z5 - a6 + 4a6z2 - 2a6z4 + 2a6z6 - a6z8 + a7z-1 - 7a7z + 11a7z3 - 5a7z5 + 2a7z7 - a7z9 + 6a8z2 - 11a8z4 + 10a8z6 - 4a8z8 - 3a9z3 + 5a9z5 - 2a9z7 - a9z9 + a10z2 - 4a10z4 + 4a10z6 - 3a10z8 - 2a11z + 4a11z5 - 4a11z7 + 4a12z4 - 4a12z6 - a13z + 4a13z3 - 3a13z5 + a14z2 - a14z4

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                   2 j = -6                 2 1 j = -8               4 1   j = -10             4 2     j = -12           5 4       j = -14         4 4         j = -16       4 5           j = -18     2 5             j = -20   1 3               j = -22   2                 j = -24 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, 167]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 167]]
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[22, 19, 7, 20], X[20, 15, 21, 16], X[16, 21, 17, 22], > 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]=	-(23/2) 3 5 8 9 9 8 6 3 2 q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -32 -30 -28 2 3 2 -18 3 2 -q + q - q + q + --- + --- - --- + q + --- + -- 26 22 20 12 8 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 167]][a, z]
Out[8]=	5 7 a a 5 7 9 5 3 7 3 9 3 5 5 -(--) + -- - 8 a z + 7 a z - 2 a z - 8 a z + 9 a z - 3 a z - 2 a z + z z 7 5 9 5 7 7 > 5 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 167]][a, z]
Out[9]=	5 7 6 a a 5 7 11 13 6 2 8 2 -a + -- + -- - 8 a z - 7 a z - 2 a z - a z + 4 a z + 6 a z + z z 10 2 14 2 5 3 7 3 9 3 13 3 6 4 > a z + a z + 10 a z + 11 a z - 3 a z + 4 a z - 2 a z - 8 4 10 4 12 4 14 4 5 5 7 5 9 5 > 11 a z - 4 a z + 4 a z - a z - 3 a z - 5 a z + 5 a z + 11 5 13 5 6 6 8 6 10 6 12 6 7 7 > 4 a z - 3 a z + 2 a z + 10 a z + 4 a z - 4 a z + 2 a z - 9 7 11 7 6 8 8 8 10 8 7 9 9 9 > 2 a z - 4 a z - a z - 4 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 2 1 2 1 3 2 5 4 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q q t q t q t q t q t q t q t 5 4 4 5 4 4 2 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 q t q t q t q t q t q t q t q t 1 2 > ---- + ---- 8 6 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n167

L11n166 L11n168