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L11n329 L11n331

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X13,18,14,19 X9,13,10,22 X11,21,12,20 X19,5,20,12 X21,11,22,10 X17,1,18,4

Gauss Code:

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

Jones Polynomial:

- q-5 + 2q-4 - 3q-3 + 5q-2 - 5q-1 + 7 - 5q + 4q2 - 2q3 + 2q4

A2 (sl(3)) Invariant:

- q-16 - q-10 + 2q-8 + q-6 + q-4 + 3q-2 + 2 + 4q2 + 2q4 + 3q6 + 4q8 + 2q10 + 3q12 + 2q14

HOMFLY-PT Polynomial:

a-4z-2 + 2a-4 - 2a-2z-2 - 5a-2 - 3a-2z2 + z-2 + 2 + z2 + z4 + 2a2 + 2a2z2 + a2z4 - a4 - a4z2

Kauffman Polynomial:

- a-4z-2 + 6a-4 - 10a-4z2 + 3a-4z4 + 2a-3z-1 - 5a-3z + 4a-3z3 - 3a-3z5 + a-3z7 - 2a-2z-2 + 13a-2 - 30a-2z2 + 27a-2z4 - 11a-2z6 + 2a-2z8 + 2a-1z-1 - 8a-1z + 9a-1z3 + a-1z5 - 3a-1z7 + a-1z9 - z-2 + 9 - 27z2 + 39z4 - 20z6 + 4z8 - 3az + 9az3 - 2az5 - 2az7 + az9 - 4a2z2 + 9a2z4 - 7a2z6 + 2a2z8 + a3z + a3z3 - 5a3z5 + 2a3z7 - a4 + 3a4z2 - 6a4z4 + 2a4z6 + a5z - 3a5z3 + a5z5

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                   2 j = 7                 1 1 j = 5               3 1   j = 3             2 1     j = 1           5 3       j = -1         3 5         j = -3       2 2           j = -5     1 3             j = -7   1 2               j = -9   1                 j = -11 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, 330]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 330]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[3, 8, 4, 9], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[13, 18, 14, 19], X[9, 13, 10, 22], X[11, 21, 12, 20], > X[19, 5, 20, 12], X[21, 11, 22, 10], X[17, 1, 18, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 11}, {-2, -1, 5, 3, -7, 10, -8, 9}, > {-6, 2, 4, -5, -11, 6, -9, 8, -10, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-5 2 3 5 5 2 3 4 7 - q + -- - -- + -- - - - 5 q + 4 q - 2 q + 2 q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -10 2 -6 -4 3 2 4 6 8 10 2 - q - q + -- + q + q + -- + 4 q + 2 q + 3 q + 4 q + 2 q + 8 2 q q 12 14 > 3 q + 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 330]][a, z]
Out[8]=	2 2 5 2 4 -2 1 2 2 3 z 2 2 4 2 2 + -- - -- + 2 a - a + z + ----- - ----- + z - ---- + 2 a z - a z + 4 2 4 2 2 2 2 a a a z a z a 4 2 4 > z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 330]][a, z]
Out[9]=	6 13 4 -2 1 2 2 2 5 z 8 z 9 + -- + -- - a - z - ----- - ----- + ---- + --- - --- - --- - 3 a z + 4 2 4 2 2 2 3 a z 3 a a a a z a z a z a 2 2 3 3 3 5 2 10 z 30 z 2 2 4 2 4 z 9 z > a z + a z - 27 z - ----- - ----- - 4 a z + 3 a z + ---- + ---- + 4 2 3 a a a a 4 4 3 3 3 5 3 4 3 z 27 z 2 4 4 4 > 9 a z + a z - 3 a z + 39 z + ---- + ----- + 9 a z - 6 a z - 4 2 a a 5 5 6 3 z z 5 3 5 5 5 6 11 z 2 6 4 6 > ---- + -- - 2 a z - 5 a z + a z - 20 z - ----- - 7 a z + 2 a z + 3 a 2 a a 7 7 8 9 z 3 z 7 3 7 8 2 z 2 8 z 9 > -- - ---- - 2 a z + 2 a z + 4 z + ---- + 2 a z + -- + a z 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 1 1 2 1 3 2 2 3 - + 5 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 > 3 q t + 2 q t + q t + 3 q t + q t + q t + q t + 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n330

L11n329 L11n331