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L11n147 L11n149

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DrawMorseLink

PD Presentation:

X8192 X9,19,10,18 X6718 X19,7,20,22 X12,5,13,6 X3,10,4,11 X4,15,5,16 X16,12,17,11 X13,21,14,20 X21,15,22,14 X17,2,18,3

Gauss Code:

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

Jones Polynomial:

- q-11/2 - q-5/2 + q-3/2 - 2q-1/2 + q1/2 - q3/2 + q5/2

A2 (sl(3)) Invariant:

q-18 + q-16 + 2q-14 + 2q-12 + 2q-10 + 2q-8 + q-4 + 1 - q4 - q6 - q8

HOMFLY-PT Polynomial:

a-1z-1 + 3a-1z + a-1z3 - 2az-1 - 7az - 5az3 - az5 + a5z-1 + a5z

Kauffman Polynomial:

- 2a-2 + 3a-2z2 - a-2z4 + a-1z-1 - 3a-1z + 3a-1z3 - a-1z5 - 5 + 14z2 - 8z4 + z6 + 2az-1 - 9az + 13az3 - 7az5 + az7 - 3a2 + 8a2z2 - 6a2z4 + a2z6 + 2a3z - 4a3z3 + a3z5 + a4 - 3a4z2 + a4z4 - a5z-1 + 8a5z - 14a5z3 + 7a5z5 - a5z7

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                   1 j = 4                     j = 2               1 1   j = 0           1 2       j = -2           1 2       j = -4       1 2 1         j = -6         1           j = -8     1 1             j = -10 1                   j = -12 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, 148]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 148]]
Out[4]=	PD[X[8, 1, 9, 2], X[9, 19, 10, 18], X[6, 7, 1, 8], X[19, 7, 20, 22], > X[12, 5, 13, 6], X[3, 10, 4, 11], X[4, 15, 5, 16], X[16, 12, 17, 11], > X[13, 21, 14, 20], X[21, 15, 22, 14], X[17, 2, 18, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -6, -7, 5, -3}, > {3, -1, -2, 6, 8, -5, -9, 10, 7, -8, -11, 2, -4, 9, -10, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) -(5/2) -(3/2) 2 3/2 5/2 -q - q + q - ------- + Sqrt[q] - q + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 2 2 2 2 -4 4 6 8 1 + q + q + --- + --- + --- + -- + q - q - q - q 14 12 10 8 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 148]][a, z]
Out[8]=	5 3 1 2 a a 3 z 5 z 3 5 --- - --- + -- + --- - 7 a z + a z + -- - 5 a z - a z a z z z a a
In[9]:=	Kauffman[Link[11, NonAlternating, 148]][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 + 14 z + 2 a z z z a a 2 3 3 z 2 2 4 2 3 z 3 3 3 5 3 4 > ---- + 8 a z - 3 a z + ---- + 13 a z - 4 a z - 14 a z - 8 z - 2 a a 4 5 z 2 4 4 4 z 5 3 5 5 5 6 2 6 7 > -- - 6 a z + a z - -- - 7 a z + a z + 7 a z + z + a z + a z - 2 a a 5 7 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 1 1 1 2 1 1 2 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + - + ---- + 2 12 6 10 6 8 4 8 3 4 3 6 2 4 2 t 4 q q t q t q t q t q t q t q t q t 1 2 2 2 6 3 > ---- + q t + q t + q t 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n148

L11n147 L11n149