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L11n159 L11n161

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DrawMorseLink

PD Presentation:

X8192 X16,7,17,8 X10,4,11,3 X2,15,3,16 X14,10,15,9 X11,19,12,18 X5,13,6,12 X6,21,1,22 X20,14,21,13 X22,17,7,18 X19,4,20,5

Gauss Code:

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

Jones Polynomial:

- q-9/2 + 3q-7/2 - 6q-5/2 + 7q-3/2 - 9q-1/2 + 8q1/2 - 7q3/2 + 5q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-14 - q-12 + q-10 + 3q-8 + 3q-4 + 1 + q2 - q4 + 2q6 - q8 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - a-1z - 2a-1z3 - a-1z5 - az-1 - 2az - 2az3 - az5 + a3z-1 + a3z + a3z3

Kauffman Polynomial:

- a-4z2 + 3a-4z4 - a-4z6 + 2a-3z - 7a-3z3 + 10a-3z5 - 3a-3z7 + 4a-2z2 - 12a-2z4 + 14a-2z6 - 4a-2z8 + 3a-1z - 10a-1z3 + 7a-1z5 + 3a-1z7 - 2a-1z9 + 10z2 - 30z4 + 26z6 - 7z8 + az-1 - 3az3 - 3az5 + 5az7 - 2az9 - a2 + 8a2z2 - 18a2z4 + 11a2z6 - 3a2z8 + a3z-1 - a3z3 - a3z7 + 3a4z2 - 3a4z4 + a5z - a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                   1 j = 8                 2   j = 6               3 1   j = 4             4 2     j = 2           4 3       j = 0         5 4         j = -2       3 5           j = -4     3 4             j = -6   1 4               j = -8   2                 j = -10 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, 160]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 160]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 7, 17, 8], X[10, 4, 11, 3], X[2, 15, 3, 16], > X[14, 10, 15, 9], X[11, 19, 12, 18], X[5, 13, 6, 12], X[6, 21, 1, 22], > X[20, 14, 21, 13], X[22, 17, 7, 18], X[19, 4, 20, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, 11, -7, -8}, > {2, -1, 5, -3, -6, 7, 9, -5, 4, -2, 10, 6, -11, -9, 8, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 6 7 9 3/2 5/2 -q + ---- - ---- + ---- - ------- + 8 Sqrt[q] - 7 q + 5 q - 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 -10 3 3 2 4 6 8 12 14 1 + q - q + q + -- + -- + q - q + 2 q - q + q - q 8 4 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 160]][a, z]
Out[8]=	3 3 3 5 a a z z 3 z 2 z 3 3 3 z 5 -(-) + -- + -- - - - 2 a z + a z + -- - ---- - 2 a z + a z - -- - a z z z 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 160]][a, z]
Out[9]=	3 2 2 2 a a 2 z 3 z 5 2 z 4 z 2 2 4 2 -a + - + -- + --- + --- + a z + 10 z - -- + ---- + 8 a z + 3 a z - z z 3 a 4 2 a a a 3 3 4 4 7 z 10 z 3 3 3 5 3 4 3 z 12 z 2 4 > ---- - ----- - 3 a z - a z - a z - 30 z + ---- - ----- - 18 a z - 3 a 4 2 a a a 5 5 6 6 7 4 4 10 z 7 z 5 6 z 14 z 2 6 3 z > 3 a z + ----- + ---- - 3 a z + 26 z - -- + ----- + 11 a z - ---- + 3 a 4 2 3 a a a a 7 8 9 3 z 7 3 7 8 4 z 2 8 2 z 9 > ---- + 5 a z - a z - 7 z - ---- - 3 a z - ---- - 2 a z a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 2 1 4 3 4 3 2 5 + -- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + 4 t + 4 q t + 2 10 4 8 3 6 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t 2 2 4 2 4 3 6 3 6 4 8 4 10 5 > 3 q t + 4 q t + 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n160

L11n159 L11n161