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L11n416 L11n418

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X15,21,16,20 X5,15,6,14 X13,5,14,4 X19,7,20,12 X11,19,12,18 X17,13,18,22 X21,17,22,16 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

q-2 - q-1 + 4 - 3q + 5q2 - 4q3 + 4q4 - 3q5 + 2q6 - q7

A2 (sl(3)) Invariant:

q-6 + q-4 + 3q-2 + 5 + 6q2 + 7q4 + 5q6 + 4q8 + q10 - 2q12 - q14 - 2q16 - q26

HOMFLY-PT Polynomial:

- a-8 - a-6z-2 + 2a-6 + 4a-6z2 + a-6z4 + 4a-4z-2 + 4a-4 - 2a-4z2 - 4a-4z4 - a-4z6 - 5a-2z-2 - 10a-2 - 9a-2z2 - 5a-2z4 - a-2z6 + 2z-2 + 5 + 4z2 + z4

Kauffman Polynomial:

a-9z - a-8 + 2a-8z2 + a-7z-1 - 2a-7z - a-7z3 + a-7z5 - a-6z-2 + 6a-6z2 - 10a-6z4 + 3a-6z6 + 5a-5z-1 - 19a-5z + 31a-5z3 - 23a-5z5 + 5a-5z7 - 4a-4z-2 + 13a-4 - 21a-4z2 + 27a-4z4 - 19a-4z6 + 4a-4z8 + 9a-3z-1 - 35a-3z + 50a-3z3 - 25a-3z5 + a-3z7 + a-3z9 - 5a-2z-2 + 22a-2 - 46a-2z2 + 55a-2z4 - 29a-2z6 + 5a-2z8 + 5a-1z-1 - 19a-1z + 18a-1z3 - a-1z5 - 4a-1z7 + a-1z9 - 2z-2 + 11 - 21z2 + 18z4 - 7z6 + z8

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 15                   1 j = 13                 1   j = 11               3 2   j = 9             2 2 1   j = 7           3 3       j = 5         2 2 1       j = 3       2 4           j = 1     2 1             j = -1   1 4               j = -3                     j = -5 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, 417]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 417]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[15, 21, 16, 20], X[5, 15, 6, 14], > X[13, 5, 14, 4], X[19, 7, 20, 12], X[11, 19, 12, 18], X[17, 13, 18, 22], > X[21, 17, 22, 16], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, 5, -4, -11}, {10, -1, 11, -2, -7, 6}, > {-5, 4, -3, 9, -8, 7, -6, 3, -9, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-2 1 2 3 4 5 6 7 4 + q - - - 3 q + 5 q - 4 q + 4 q - 3 q + 2 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -4 3 2 4 6 8 10 12 14 16 26 5 + q + q + -- + 6 q + 7 q + 5 q + 4 q + q - 2 q - q - 2 q - q 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 417]][a, z]
Out[8]=	2 2 -8 2 4 10 2 1 4 5 2 4 z 2 z 5 - a + -- + -- - -- + -- - ----- + ----- - ----- + 4 z + ---- - ---- - 6 4 2 2 6 2 4 2 2 2 6 4 a a a z a z a z a z a a 2 4 4 4 6 6 9 z 4 z 4 z 5 z z z > ---- + z + -- - ---- - ---- - -- - -- 2 6 4 2 4 2 a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 417]][a, z]
Out[9]=	-8 13 22 2 1 4 5 1 5 9 5 11 - a + -- + -- - -- - ----- - ----- - ----- + ---- + ---- + ---- + --- + 4 2 2 6 2 4 2 2 2 7 5 3 a z a a z a z a z a z a z a z a z 2 2 2 2 3 z 2 z 19 z 35 z 19 z 2 2 z 6 z 21 z 46 z z > -- - --- - ---- - ---- - ---- - 21 z + ---- + ---- - ----- - ----- - -- + 9 7 5 3 a 8 6 4 2 7 a a a a a a a a a 3 3 3 4 4 4 5 5 31 z 50 z 18 z 4 10 z 27 z 55 z z 23 z > ----- + ----- + ----- + 18 z - ----- + ----- + ----- + -- - ----- - 5 3 a 6 4 2 7 5 a a a a a a a 5 5 6 6 6 7 7 7 8 25 z z 6 3 z 19 z 29 z 5 z z 4 z 8 4 z > ----- - -- - 7 z + ---- - ----- - ----- + ---- + -- - ---- + z + ---- + 3 a 6 4 2 5 3 a 4 a a a a a a a 8 9 9 5 z z z > ---- + -- + -- 2 3 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 1 4 2 q q 2 q 5 7 5 2 4 q + 2 q + ----- + ---- + ---- + --- + - + ---- + 2 q t + 3 q t + q t + 5 4 3 2 2 t t q t q t q t t 7 2 9 2 9 3 11 3 9 4 11 4 13 4 15 5 > 3 q t + 2 q t + 2 q t + 3 q t + q t + 2 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n417

L11n416 L11n418