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L11n181 L11n183

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DrawMorseLink

PD Presentation:

X8192 X9,21,10,20 X14,5,15,6 X16,8,17,7 X3,10,4,11 X22,14,7,13 X18,12,19,11 X12,18,13,17 X6,15,1,16 X4,21,5,22 X19,2,20,3

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 7q-9/2 + 9q-7/2 - 12q-5/2 + 12q-3/2 - 11q-1/2 + 8q1/2 - 5q3/2 + 2q5/2

A2 (sl(3)) Invariant:

q-20 - q-18 + q-16 + 4q-14 + 4q-10 + q-8 + 2q-4 - 2q-2 + 3 - 2q2 - q4 + q6 - 2q8

HOMFLY-PT Polynomial:

a-1z-1 + 3a-1z + 2a-1z3 - 2az-1 - 7az - 6az3 - 2az5 - a3z3 - a3z5 + a5z-1 + a5z + a5z3

Kauffman Polynomial:

- 2a-2 + 5a-2z2 - 3a-2z4 + a-1z-1 - 3a-1z + 7a-1z3 - 3a-1z5 - a-1z7 - 5 + 20z2 - 19z4 + 8z6 - 3z8 + 2az-1 - 9az + 15az3 - 10az5 + 3az7 - 2az9 - 3a2 + 14a2z2 - 27a2z4 + 21a2z6 - 8a2z8 + 2a3z - 10a3z3 + 10a3z5 - 2a3z7 - 2a3z9 + a4 - a4z2 - 6a4z4 + 10a4z6 - 5a4z8 - a5z-1 + 8a5z - 16a5z3 + 16a5z5 - 6a5z7 + 5a6z4 - 3a6z6 + 2a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                   2 j = 4                 3   j = 2               5 2   j = 0             6 3     j = -2           7 6       j = -4         5 5         j = -6       4 7           j = -8     3 5             j = -10     4               j = -12 1 3                 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, 182]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 182]]
Out[4]=	PD[X[8, 1, 9, 2], X[9, 21, 10, 20], X[14, 5, 15, 6], X[16, 8, 17, 7], > X[3, 10, 4, 11], X[22, 14, 7, 13], X[18, 12, 19, 11], X[12, 18, 13, 17], > X[6, 15, 1, 16], X[4, 21, 5, 22], X[19, 2, 20, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -10, 3, -9}, > {4, -1, -2, 5, 7, -8, 6, -3, 9, -4, 8, -7, -11, 2, 10, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 7 9 12 12 11 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 8 Sqrt[q] - 5 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 > 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 4 4 -8 2 2 2 4 6 8 3 + q - q + q + --- + --- + q + -- - -- - 2 q - q + q - 2 q 14 10 4 2 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 182]][a, z]
Out[8]=	5 3 1 2 a a 3 z 5 2 z 3 3 3 5 3 5 --- - --- + -- + --- - 7 a z + a z + ---- - 6 a z - a z + a z - 2 a z - a z z z a a 3 5 > a z
In[9]:=	Kauffman[Link[11, NonAlternating, 182]][a, z]
Out[9]=	5 2 2 4 1 2 a a 3 z 3 5 2 -5 - -- - 3 a + a + --- + --- - -- - --- - 9 a z + 2 a z + 8 a z + 20 z + 2 a z z z a a 2 3 5 z 2 2 4 2 7 z 3 3 3 5 3 7 3 > ---- + 14 a z - a z + ---- + 15 a z - 10 a z - 16 a z + 2 a z - 2 a a 4 5 4 3 z 2 4 4 4 6 4 3 z 5 3 5 > 19 z - ---- - 27 a z - 6 a z + 5 a z - ---- - 10 a z + 10 a z + 2 a a 7 5 5 7 5 6 2 6 4 6 6 6 z 7 > 16 a z - a z + 8 z + 21 a z + 10 a z - 3 a z - -- + 3 a z - a 3 7 5 7 8 2 8 4 8 9 3 9 > 2 a z - 6 a z - 3 z - 8 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 1 3 4 3 5 4 7 6 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 5 5 7 2 2 2 4 2 6 3 > ----- + ---- + ---- + 3 t + 5 q t + 2 q t + 3 q t + 2 q t 4 2 4 2 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n182

L11n181 L11n183