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L11n133 L11n135

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X18,11,19,12 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X9,16,10,17 X20,14,21,13 X22,16,7,15 X4,20,5,19 X14,22,15,21

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 3q-7/2 + 4q-5/2 - 5q-3/2 + 4q-1/2 - 5q1/2 + 3q3/2 - 2q5/2 + q7/2

A2 (sl(3)) Invariant:

q-16 + q-12 + q-6 + 3q-2 + 2 + 2q2 + q4 - q6 - q10

HOMFLY-PT Polynomial:

4a-1z + 4a-1z3 + a-1z5 - az-1 - 9az - 12az3 - 6az5 - az7 + a3z-1 + 4a3z + 4a3z3 + a3z5

Kauffman Polynomial:

- a-4z2 - 2a-3z3 + 2a-2z2 - 3a-2z4 - 6a-1z + 11a-1z3 - 5a-1z5 + 4z2 - 2z4 + 2z6 - z8 + az-1 - 11az + 19az3 - 10az5 + 4az7 - az9 - a2 + 6a2z2 - 13a2z4 + 12a2z6 - 3a2z8 + a3z-1 - 3a3z - a3z3 + 3a3z7 - a3z9 + 5a4z2 - 14a4z4 + 10a4z6 - 2a4z8 + 2a5z - 7a5z3 + 5a5z5 - 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 = 8                   1 j = 6                 1   j = 4               2 1   j = 2             3 1     j = 0           2 3       j = -2         3 2         j = -4       2 3           j = -6     1 2             j = -8   1 2               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, 134]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 134]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 11, 19, 12], X[3, 10, 4, 11], X[17, 3, 18, 2], > X[12, 5, 13, 6], X[6, 7, 1, 8], X[9, 16, 10, 17], X[20, 14, 21, 13], > X[22, 16, 7, 15], X[4, 20, 5, 19], X[14, 22, 15, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -10, 5, -6}, > {6, -1, -7, 3, 2, -5, 8, -11, 9, 7, -4, -2, 10, -8, 11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 3 4 5 4 3/2 5/2 -q + ---- - ---- + ---- - ---- + ------- - 5 Sqrt[q] + 3 q - 2 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 7/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -12 -6 3 2 4 6 10 2 + q + q + q + -- + 2 q + q - q - q 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 134]][a, z]
Out[8]=	3 3 5 a a 4 z 3 4 z 3 3 3 z 5 -(-) + -- + --- - 9 a z + 4 a z + ---- - 12 a z + 4 a z + -- - 6 a z + z z a a a 3 5 7 > a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 134]][a, z]
Out[9]=	3 2 2 2 a a 6 z 3 5 2 z 2 z 2 2 -a + - + -- - --- - 11 a z - 3 a z + 2 a z + 4 z - -- + ---- + 6 a z + z z a 4 2 a a 3 3 4 4 2 2 z 11 z 3 3 3 5 3 4 3 z > 5 a z - ---- + ----- + 19 a z - a z - 7 a z - 2 z - ---- - 3 a 2 a a 5 2 4 4 4 5 z 5 5 5 6 2 6 > 13 a z - 14 a z - ---- - 10 a z + 5 a z + 2 z + 12 a z + a 4 6 7 3 7 5 7 8 2 8 4 8 9 3 9 > 10 a z + 4 a z + 3 a z - a z - z - 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 2 1 2 2 3 3 + 3 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 3 2 2 2 4 4 2 6 2 8 3 > ----- + - + ---- + q t + 2 q t + q t + q t + q t 2 2 t 2 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n134

L11n133 L11n135