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L11n418 L11n420

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DrawMorseLink

PD Presentation:

X8192 X16,8,17,7 X3,10,4,11 X2,18,3,17 X9,19,10,18 X20,12,21,11 X5,14,6,15 X15,13,16,22 X13,6,14,1 X19,5,20,4 X12,22,7,21

Gauss Code:

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

Jones Polynomial:

2q-1 - 2 + 4q - 4q2 + 5q3 - 3q4 + 3q5 - q6

A2 (sl(3)) Invariant:

2q-4 + 2q-2 + 3 + 4q2 + q4 + 3q6 + 2q8 + 4q10 + 4q12 + 2q14 + 2q16 - q18 - q20

HOMFLY-PT Polynomial:

- a-6 + a-4z-2 + 5a-4 + 3a-4z2 - 2a-2z-2 - 8a-2 - 7a-2z2 - 2a-2z4 + z-2 + 4 + 2z2

Kauffman Polynomial:

- a-7z + a-7z3 + a-6 - 4a-6z2 + 3a-6z4 - 3a-5z + 7a-5z3 - 3a-5z5 + a-5z7 - a-4z-2 + 8a-4 - 18a-4z2 + 16a-4z4 - 5a-4z6 + a-4z8 + 2a-3z-1 - 8a-3z + 11a-3z3 - 6a-3z5 + 2a-3z7 - 2a-2z-2 + 12a-2 - 23a-2z2 + 16a-2z4 - 5a-2z6 + a-2z8 + 2a-1z-1 - 6a-1z + 5a-1z3 - 3a-1z5 + a-1z7 - z-2 + 6 - 9z2 + 3z4

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13               1 j = 11             2   j = 9           2 2   j = 7         3 1     j = 5       2 3       j = 3     2 2         j = 1   1 3           j = -1 1 1             j = -3 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 419]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 419]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 8, 17, 7], X[3, 10, 4, 11], X[2, 18, 3, 17], > X[9, 19, 10, 18], X[20, 12, 21, 11], X[5, 14, 6, 15], X[15, 13, 16, 22], > X[13, 6, 14, 1], X[19, 5, 20, 4], X[12, 22, 7, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 10, -7, 9}, {2, -1, -5, 3, 6, -11}, > {-9, 7, -8, -2, 4, 5, -10, -6, 11, 8}]
In[6]:=	Jones[L][q]
Out[6]=	2 2 3 4 5 6 -2 + - + 4 q - 4 q + 5 q - 3 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 2 2 4 6 8 10 12 14 16 18 3 + -- + -- + 4 q + q + 3 q + 2 q + 4 q + 4 q + 2 q + 2 q - q - 4 2 q q 20 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 419]][a, z]
Out[8]=	2 2 4 -6 5 8 -2 1 2 2 3 z 7 z 2 z 4 - a + -- - -- + z + ----- - ----- + 2 z + ---- - ---- - ---- 4 2 4 2 2 2 4 2 2 a a a z a z a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 419]][a, z]
Out[9]=	-6 8 12 -2 1 2 2 2 z 3 z 8 z 6 z 6 + a + -- + -- - z - ----- - ----- + ---- + --- - -- - --- - --- - --- - 4 2 4 2 2 2 3 a z 7 5 3 a a a a z a z a z a a a 2 2 2 3 3 3 3 4 2 4 z 18 z 23 z z 7 z 11 z 5 z 4 3 z > 9 z - ---- - ----- - ----- + -- + ---- + ----- + ---- + 3 z + ---- + 6 4 2 7 5 3 a 6 a a a a a a a 4 4 5 5 5 6 6 7 7 7 8 8 16 z 16 z 3 z 6 z 3 z 5 z 5 z z 2 z z z z > ----- + ----- - ---- - ---- - ---- - ---- - ---- + -- + ---- + -- + -- + -- 4 2 5 3 a 4 2 5 3 a 4 2 a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 2 1 1 q 3 5 5 2 7 2 3 q + 2 q + ----- + ---- + --- + - + 2 q t + 2 q t + 3 q t + 3 q t + 3 2 2 q t t q t q t 7 3 9 3 9 4 11 4 13 5 > q t + 2 q t + 2 q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n419

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