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L11n421 L11n423

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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 X14,6,15,5 X22,15,13,16 X6,14,1,13 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:

- 1 + 4q - 5q2 + 8q3 - 8q4 + 8q5 - 6q6 + 5q7 - 2q8 + q9

A2 (sl(3)) Invariant:

- 1 + 2q2 + 2q6 + 3q8 + 3q12 + q14 + 4q16 + 4q18 + 2q20 + 4q22 + q24 + q26 + q28

HOMFLY-PT Polynomial:

a-8z-2 + 2a-8 + a-8z2 - 2a-6z-2 - 6a-6 - 6a-6z2 - 2a-6z4 + a-4z-2 + 3a-4 + 6a-4z2 + 4a-4z4 + a-4z6 + a-2 - a-2z2 - a-2z4

Kauffman Polynomial:

- a-10 + 4a-10z2 - 4a-10z4 + a-10z6 + 3a-9z3 - 6a-9z5 + 2a-9z7 - a-8z-2 + 5a-8 - 12a-8z2 + 14a-8z4 - 11a-8z6 + 3a-8z8 + 2a-7z-1 - 9a-7z + 18a-7z3 - 15a-7z5 + 2a-7z7 + a-7z9 - 2a-6z-2 + 11a-6 - 23a-6z2 + 25a-6z4 - 17a-6z6 + 5a-6z8 + 2a-5z-1 - 9a-5z + 14a-5z3 - 8a-5z5 + a-5z7 + a-5z9 - a-4z-2 + 5a-4 - 10a-4z2 + 11a-4z4 - 5a-4z6 + 2a-4z8 + a-3z5 + a-3z7 - a-2 - 3a-2z2 + 4a-2z4 + a-1z3

Khovanov Homology:

trqj r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                   1 j = 17                 2 1 j = 15               3     j = 13             3 2     j = 11           5 3       j = 9         3 3         j = 7       5 5           j = 5     2 5             j = 3   2 3               j = 1   3                 j = -1 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 422]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 422]]
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[14, 6, 15, 5], X[22, 15, 13, 16], > X[6, 14, 1, 13], 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 3 4 5 6 7 8 9 -1 + 4 q - 5 q + 8 q - 8 q + 8 q - 6 q + 5 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 6 8 12 14 16 18 20 22 24 -1 + 2 q + 2 q + 3 q + 3 q + q + 4 q + 4 q + 2 q + 4 q + q + 26 28 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 422]][a, z]
Out[8]=	2 2 2 2 4 2 6 3 -2 1 2 1 z 6 z 6 z z 2 z -- - -- + -- + a + ----- - ----- + ----- + -- - ---- + ---- - -- - ---- + 8 6 4 8 2 6 2 4 2 8 6 4 2 6 a a a a z a z a z a a a a a 4 4 6 4 z z z > ---- - -- + -- 4 2 4 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 422]][a, z]
Out[9]=	-10 5 11 5 -2 1 2 1 2 2 9 z 9 z -a + -- + -- + -- - a - ----- - ----- - ----- + ---- + ---- - --- - --- + 8 6 4 8 2 6 2 4 2 7 5 7 5 a a a a z a z a z a z a z a a 2 2 2 2 2 3 3 3 3 4 4 z 12 z 23 z 10 z 3 z 3 z 18 z 14 z z 4 z > ---- - ----- - ----- - ----- - ---- + ---- + ----- + ----- + -- - ---- + 10 8 6 4 2 9 7 5 a 10 a a a a a a a a a 4 4 4 4 5 5 5 5 6 6 14 z 25 z 11 z 4 z 6 z 15 z 8 z z z 11 z > ----- + ----- + ----- + ---- - ---- - ----- - ---- + -- + --- - ----- - 8 6 4 2 9 7 5 3 10 8 a a a a a a a a a a 6 6 7 7 7 7 8 8 8 9 9 17 z 5 z 2 z 2 z z z 3 z 5 z 2 z z z > ----- - ---- + ---- + ---- + -- + -- + ---- + ---- + ---- + -- + -- 6 4 9 7 5 3 8 6 4 7 5 a a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 3 5 5 2 7 2 7 3 9 3 3 q + 2 q + --- + 3 q t + 2 q t + 5 q t + 5 q t + 5 q t + 3 q t + q t 9 4 11 4 11 5 13 5 13 6 15 6 17 7 > 3 q t + 5 q t + 3 q t + 3 q t + 2 q t + 3 q t + 2 q t + 17 8 19 8 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n422

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