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L11n327 L11n329

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X13,18,14,19 X9,21,10,20 X19,5,20,12 X11,13,12,22 X21,11,22,10 X4,17,1,18

Gauss Code:

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

Jones Polynomial:

- q-4 + 2q-3 - 3q-2 + 5q-1 - 4 + 6q - 4q2 + 4q3 - 2q4 + q5

A2 (sl(3)) Invariant:

- q-12 - q-8 + 2q-4 + 2q-2 + 5 + 4q2 + 4q4 + 4q6 + 2q8 + 3q10 + q12 + q14 + q16

HOMFLY-PT Polynomial:

a-4z-2 + 2a-4 + a-4z2 - 2a-2z-2 - 7a-2 - 7a-2z2 - 2a-2z4 + z-2 + 7 + 9z2 + 5z4 + z6 - 2a2 - 3a2z2 - a2z4

Kauffman Polynomial:

- a-6 + a-6z2 - a-5z + 2a-5z3 - a-4z-2 + a-4 + 3a-4z2 - 2a-4z4 + a-4z6 + 2a-3z-1 - 6a-3z + 10a-3z3 - 7a-3z5 + 2a-3z7 - 2a-2z-2 + 7a-2 - 10a-2z2 + 9a-2z4 - 7a-2z6 + 2a-2z8 + 2a-1z-1 - 10a-1z + 14a-1z3 - 8a-1z5 - a-1z7 + a-1z9 - z-2 + 9 - 21z2 + 26z4 - 18z6 + 4z8 - 7az + 13az3 - 6az5 - 2az7 + az9 + 3a2 - 9a2z2 + 15a2z4 - 10a2z6 + 2a2z8 - 2a3z + 7a3z3 - 5a3z5 + a3z7

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 11                   1 j = 9                 1   j = 7               3 1   j = 5             3 3     j = 3           3 2 1     j = 1         3 5         j = -1       2 2 1         j = -3     1 3             j = -5   1 2               j = -7   1                 j = -9 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, 328]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 328]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[8, 4, 9, 3], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[13, 18, 14, 19], X[9, 21, 10, 20], X[19, 5, 20, 12], > X[11, 13, 12, 22], X[21, 11, 22, 10], X[4, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11}, {-2, -1, 5, -3, -7, 10, -9, 8}, > {-6, 2, 4, -5, 11, 6, -8, 7, -10, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-4 2 3 5 2 3 4 5 -4 - q + -- - -- + - + 6 q - 4 q + 4 q - 2 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -8 2 2 2 4 6 8 10 12 14 16 5 - q - q + -- + -- + 4 q + 4 q + 4 q + 2 q + 3 q + q + q + q 4 2 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 328]][a, z]
Out[8]=	2 2 2 7 2 -2 1 2 2 z 7 z 2 2 4 7 + -- - -- - 2 a + z + ----- - ----- + 9 z + -- - ---- - 3 a z + 5 z - 4 2 4 2 2 2 4 2 a a a z a z a a 4 2 z 2 4 6 > ---- - a z + z 2 a
In[9]:=	Kauffman[Link[11, NonAlternating, 328]][a, z]
Out[9]=	-6 -4 7 2 -2 1 2 2 2 z 6 z 9 - a + a + -- + 3 a - z - ----- - ----- + ---- + --- - -- - --- - 2 4 2 2 2 3 a z 5 3 a a z a z a z a a 2 2 2 3 10 z 3 2 z 3 z 10 z 2 2 2 z > ---- - 7 a z - 2 a z - 21 z + -- + ---- - ----- - 9 a z + ---- + a 6 4 2 5 a a a a 3 3 4 4 5 10 z 14 z 3 3 3 4 2 z 9 z 2 4 7 z > ----- + ----- + 13 a z + 7 a z + 26 z - ---- + ---- + 15 a z - ---- - 3 a 4 2 3 a a a a 5 6 6 7 7 8 z 5 3 5 6 z 7 z 2 6 2 z z > ---- - 6 a z - 5 a z - 18 z + -- - ---- - 10 a z + ---- - -- - a 4 2 3 a a a a 8 9 7 3 7 8 2 z 2 8 z 9 > 2 a z + a z + 4 z + ---- + 2 a z + -- + a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	1 3 1 1 1 2 1 3 2 2 - + 5 q + 3 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 3 q 3 5 3 2 5 2 7 2 7 3 9 3 11 4 > --- + 2 q t + 3 q t + q t + 3 q t + 3 q t + q t + q t + q t t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n328

L11n327 L11n329