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L11n218 L11n220

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DrawMorseLink

PD Presentation:

X10,1,11,2 X8,9,1,10 X3,12,4,13 X22,16,9,15 X17,3,18,2 X4,22,5,21 X14,5,15,6 X13,21,14,20 X16,12,17,11 X19,7,20,6 X7,19,8,18

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:

- q-1/2 - q3/2 + q5/2 - 2q7/2 + q9/2 - q11/2 + q13/2

A2 (sl(3)) Invariant:

q-2 + 2 + 2q2 + 2q4 + q6 + 2q8 + q10 + q12 - q16 - q18 - q20

HOMFLY-PT Polynomial:

a-5z-1 + 3a-5z + a-5z3 - 3a-3z-1 - 7a-3z - 5a-3z3 - a-3z5 + 2a-1z-1 + 4a-1z + a-1z3

Kauffman Polynomial:

- a-6 + 10a-6z2 - 15a-6z4 + 7a-6z6 - a-6z8 + a-5z-1 - 6a-5z + 15a-5z3 - 16a-5z5 + 7a-5z7 - a-5z9 - 3a-4 + 16a-4z2 - 25a-4z4 + 13a-4z6 - 2a-4z8 + 3a-3z-1 - 13a-3z + 21a-3z3 - 17a-3z5 + 7a-3z7 - a-3z9 - 3a-2 + 7a-2z2 - 10a-2z4 + 6a-2z6 - a-2z8 + 2a-1z-1 - 7a-1z + 6a-1z3 - a-1z5 + z2

Khovanov Homology:

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 14               1 j = 12                 j = 10           1 1   j = 8         1       j = 6         1       j = 4     1 1         j = 2 1               j = 0 2 1             j = -2 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, 219]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 219]]
Out[4]=	PD[X[10, 1, 11, 2], X[8, 9, 1, 10], X[3, 12, 4, 13], X[22, 16, 9, 15], > X[17, 3, 18, 2], X[4, 22, 5, 21], X[14, 5, 15, 6], X[13, 21, 14, 20], > X[16, 12, 17, 11], X[19, 7, 20, 6], X[7, 19, 8, 18]]
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]=	1 3/2 5/2 7/2 9/2 11/2 13/2 -(-------) - q + q - 2 q + q - q + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 2 4 6 8 10 12 16 18 20 2 + q + 2 q + 2 q + q + 2 q + q + q - q - q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 219]][a, z]
Out[8]=	3 3 3 5 1 3 2 3 z 7 z 4 z z 5 z z z ---- - ---- + --- + --- - --- + --- + -- - ---- + -- - -- 5 3 a z 5 3 a 5 3 a 3 a z a z a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 219]][a, z]
Out[9]=	2 2 -6 3 3 1 3 2 6 z 13 z 7 z 2 10 z 16 z -a - -- - -- + ---- + ---- + --- - --- - ---- - --- + z + ----- + ----- + 4 2 5 3 a z 5 3 a 6 4 a a a z a z a a a a 2 3 3 3 4 4 4 5 5 5 7 z 15 z 21 z 6 z 15 z 25 z 10 z 16 z 17 z z > ---- + ----- + ----- + ---- - ----- - ----- - ----- - ----- - ----- - -- + 2 5 3 a 6 4 2 5 3 a a a a a a a a a 6 6 6 7 7 8 8 8 9 9 7 z 13 z 6 z 7 z 7 z z 2 z z z z > ---- + ----- + ---- + ---- + ---- - -- - ---- - -- - -- - -- 6 4 2 5 3 6 4 2 5 3 a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	-2 2 4 2 4 3 6 4 8 4 10 5 10 6 14 7 2 + q + q + t + q t + q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n219

L11n218 L11n220