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L11n220 L11n222

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DrawMorseLink

PD Presentation:

X10,1,11,2 X8,9,1,10 X12,4,13,3 X22,16,9,15 X2,17,3,18 X21,4,22,5 X5,15,6,14 X13,21,14,20 X16,12,17,11 X6,19,7,20 X18,7,19,8

Gauss Code:

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

Jones Polynomial:

- 2q-9/2 + 5q-7/2 - 9q-5/2 + 10q-3/2 - 12q-1/2 + 11q1/2 - 9q3/2 + 6q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

a-3z + a-3z3 + a-1z-1 + a-1z - a-1z3 - a-1z5 - 3az-1 - 7az - 6az3 - 2az5 + 2a3z-1 + 3a3z + 2a3z3

Kauffman Polynomial:

- 2a-4z2 + 3a-4z4 - a-4z6 + a-3z - 7a-3z3 + 9a-3z5 - 3a-3z7 - a-2 + 3a-2z2 - 8a-2z4 + 11a-2z6 - 4a-2z8 + a-1z-1 - 2a-1z - 2a-1z3 + 8a-1z5 - 2a-1z9 - 3 + 12z2 - 18z4 + 20z6 - 8z8 + 3az-1 - 10az + 11az3 - az5 - 2az9 - 3a2 + 9a2z2 - 11a2z4 + 7a2z6 - 4a2z8 + 2a3z-1 - 5a3z + 3a3z3 - 3a3z7 + 2a4z2 - 4a4z4 - a4z6 + 2a5z - 3a5z3

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               4 1   j = 4             5 2     j = 2           6 4       j = 0         6 5         j = -2       5 7           j = -4     4 5             j = -6   1 5               j = -8 1 4                 j = -10 2

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, 221]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 221]]
Out[4]=	PD[X[10, 1, 11, 2], X[8, 9, 1, 10], X[12, 4, 13, 3], X[22, 16, 9, 15], > X[2, 17, 3, 18], X[21, 4, 22, 5], X[5, 15, 6, 14], X[13, 21, 14, 20], > X[16, 12, 17, 11], X[6, 19, 7, 20], X[18, 7, 19, 8]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, 3, 6, -7, -10, 11, -2}, > {2, -1, 9, -3, -8, 7, 4, -9, 5, -11, 10, 8, -6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-2 5 9 10 12 3/2 5/2 7/2 ---- + ---- - ---- + ---- - ------- + 11 Sqrt[q] - 9 q + 6 q - 3 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 9/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 -12 2 4 4 -2 4 6 8 12 14 1 + --- - q + --- + -- + -- - q - 2 q + 2 q - 2 q + q - q 14 10 8 4 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 221]][a, z]
Out[8]=	3 3 3 5 1 3 a 2 a z z 3 z z 3 3 3 z --- - --- + ---- + -- + - - 7 a z + 3 a z + -- - -- - 6 a z + 2 a z - -- - a z z z 3 a 3 a a a a 5 > 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 221]][a, z]
Out[9]=	3 -2 2 1 3 a 2 a z 2 z 3 5 -3 - a - 3 a + --- + --- + ---- + -- - --- - 10 a z - 5 a z + 2 a z + a z z z 3 a a 2 2 3 3 2 2 z 3 z 2 2 4 2 7 z 2 z 3 3 3 > 12 z - ---- + ---- + 9 a z + 2 a z - ---- - ---- + 11 a z + 3 a z - 4 2 3 a a a a 4 4 5 5 5 3 4 3 z 8 z 2 4 4 4 9 z 8 z 5 > 3 a z - 18 z + ---- - ---- - 11 a z - 4 a z + ---- + ---- - a z + 4 2 3 a a a a 6 6 7 8 6 z 11 z 2 6 4 6 3 z 3 7 8 4 z > 20 z - -- + ----- + 7 a z - a z - ---- - 3 a z - 8 z - ---- - 4 2 3 2 a a a a 9 2 8 2 z 9 > 4 a z - ---- - 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	7 2 1 4 1 5 4 5 5 6 + -- + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 5 t + 2 10 4 8 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 q t 2 2 2 4 2 4 3 6 3 6 4 8 4 10 5 > 6 q t + 4 q t + 5 q t + 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n221

L11n220 L11n222