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L11n207 L11n209

Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X5,14,6,15 X22,18,9,17 X19,5,20,4 X21,6,22,7 X7,17,8,16 X8,9,1,10 X18,14,19,13 X15,21,16,20

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-16 + q-14 + q-12 + q-10 + q-8 + q-6 + q-4 + 2q-2 + 2 + q2 - q6 - q8 - q10

HOMFLY-PT Polynomial:

a-1z-1 + 6a-1z + 5a-1z3 + a-1z5 - 3az-1 - 14az - 16az3 - 7az5 - az7 + 2a3z-1 + 6a3z + 5a3z3 + a3z5

Kauffman Polynomial:

- a-2 + 10a-2z2 - 15a-2z4 + 7a-2z6 - a-2z8 + a-1z-1 - 7a-1z + 15a-1z3 - 16a-1z5 + 7a-1z7 - a-1z9 - 3 + 18z2 - 26z4 + 13z6 - 2z8 + 3az-1 - 16az + 24az3 - 18az5 + 7az7 - az9 - 3a2 + 11a2z2 - 12a2z4 + 6a2z6 - a2z8 + 2a3z-1 - 7a3z + 8a3z3 - 2a3z5 + 2a4z2 - a4z4 + a5z - a5z3 - a6z2 - a7z

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 8                   1 j = 6                     j = 4               1 1   j = 2             1       j = 0             1       j = -2         2 1         j = -4         1           j = -6     1 1             j = -8                     j = -10 1 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, 208]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 208]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[5, 14, 6, 15], > X[22, 18, 9, 17], X[19, 5, 20, 4], X[21, 6, 22, 7], X[7, 17, 8, 16], > X[8, 9, 1, 10], X[18, 14, 19, 13], X[15, 21, 16, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, 6, -4, 7, -8, -9}, > {9, -1, 2, -3, 10, 4, -11, 8, 5, -10, -6, 11, -7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) -(9/2) -(7/2) -(5/2) 2 1 3/2 -q + q - q + q - ---- + ------- - 2 Sqrt[q] + q - 3/2 Sqrt[q] q 5/2 7/2 > q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 -12 -10 -8 -6 -4 2 2 6 8 10 2 + q + q + q + q + q + q + q + -- + q - q - q - q 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 208]][a, z]
Out[8]=	3 3 5 1 3 a 2 a 6 z 3 5 z 3 3 3 z --- - --- + ---- + --- - 14 a z + 6 a z + ---- - 16 a z + 5 a z + -- - a z z z a a a 5 3 5 7 > 7 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 208]][a, z]
Out[9]=	3 -2 2 1 3 a 2 a 7 z 3 5 7 -3 - a - 3 a + --- + --- + ---- - --- - 16 a z - 7 a z + a z - a z + a z z z a 2 3 2 10 z 2 2 4 2 6 2 15 z 3 3 3 > 18 z + ----- + 11 a z + 2 a z - a z + ----- + 24 a z + 8 a z - 2 a a 4 5 5 3 4 15 z 2 4 4 4 16 z 5 3 5 > a z - 26 z - ----- - 12 a z - a z - ----- - 18 a z - 2 a z + 2 a a 6 7 8 9 6 7 z 2 6 7 z 7 8 z 2 8 z 9 > 13 z + ---- + 6 a z + ---- + 7 a z - 2 z - -- - a z - -- - a z 2 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 1 1 1 1 t 2 2 2 4 3 q + -- + ------ + ------ + ------ + ----- + ---- + -- + t + q t + q t + 2 12 4 10 4 10 3 6 2 6 2 q q t q t q t q t q t q 4 4 8 5 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n208

L11n207 L11n209