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L11n334 L11n336

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X11,16,12,17 X8493 X2,18,3,17 X5,14,6,15 X18,7,19,8 X15,12,16,5 X20,14,21,13 X22,9,13,10 X10,21,11,22 X4,19,1,20

Gauss Code:

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

Jones Polynomial:

3q-7 - 6q-6 + 12q-5 - 14q-4 + 17q-3 - 16q-2 + 13q-1 - 9 + 5q - q2

A2 (sl(3)) Invariant:

q-24 + 5q-22 + 3q-20 + 3q-18 + 8q-16 + q-14 + 4q-12 + q-10 - 2q-8 + 2q-6 - 4q-4 + 4q-2 - q2 + 3q4 - q6

HOMFLY-PT Polynomial:

1 - z4 + 2a2z2 + 2a2z4 + a2z6 + a4z-2 - 4a4z2 - 3a4z4 - 2a6z-2 - a6 + 2a6z2 + a8z-2

Kauffman Polynomial:

a-1z5 + 1 - 6z4 + 5z6 + 5az3 - 15az5 + 9az7 + 2a2z2 - 9a2z4 - 4a2z6 + 7a2z8 + 19a3z3 - 36a3z5 + 15a3z7 + 2a3z9 - a4z-2 + 3a4 - 3a4z2 + 7a4z4 - 18a4z6 + 12a4z8 + 2a5z-1 - 9a5z + 21a5z3 - 23a5z5 + 9a5z7 + 2a5z9 - 2a6z-2 + 9a6 - 16a6z2 + 16a6z4 - 9a6z6 + 5a6z8 + 2a7z-1 - 9a7z + 7a7z3 - 3a7z5 + 3a7z7 - a8z-2 + 6a8 - 11a8z2 + 6a8z4

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                   1 j = 3                 4   j = 1               5 1   j = -1             8 4     j = -3           10 7       j = -5         7 6         j = -7       7 10           j = -9     5 7             j = -11   1 7               j = -13 2 5                 j = -15 3

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, 335]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 335]]
Out[4]=	PD[X[6, 1, 7, 2], X[11, 16, 12, 17], X[8, 4, 9, 3], X[2, 18, 3, 17], > X[5, 14, 6, 15], X[18, 7, 19, 8], X[15, 12, 16, 5], X[20, 14, 21, 13], > X[22, 9, 13, 10], X[10, 21, 11, 22], X[4, 19, 1, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11}, {-5, -1, 6, -3, 9, -10, -2, 7}, > {8, 5, -7, 2, 4, -6, 11, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	3 6 12 14 17 16 13 2 -9 + -- - -- + -- - -- + -- - -- + -- + 5 q - q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 5 3 3 8 -14 4 -10 2 2 4 4 2 q + --- + --- + --- + --- + q + --- + q - -- + -- - -- + -- - q + 22 20 18 16 12 8 6 4 2 q q q q q q q q q 4 6 > 3 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 335]][a, z]
Out[8]=	4 6 8 6 a 2 a a 2 2 4 2 6 2 4 2 4 1 - a + -- - ---- + -- + 2 a z - 4 a z + 2 a z - z + 2 a z - 2 2 2 z z z 4 4 2 6 > 3 a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 335]][a, z]
Out[9]=	4 6 8 5 7 4 6 8 a 2 a a 2 a 2 a 5 7 1 + 3 a + 9 a + 6 a - -- - ---- - -- + ---- + ---- - 9 a z - 9 a z + 2 2 2 z z z z z 2 2 4 2 6 2 8 2 3 3 3 5 3 > 2 a z - 3 a z - 16 a z - 11 a z + 5 a z + 19 a z + 21 a z + 5 7 3 4 2 4 4 4 6 4 8 4 z 5 > 7 a z - 6 z - 9 a z + 7 a z + 16 a z + 6 a z + -- - 15 a z - a 3 5 5 5 7 5 6 2 6 4 6 6 6 > 36 a z - 23 a z - 3 a z + 5 z - 4 a z - 18 a z - 9 a z + 7 3 7 5 7 7 7 2 8 4 8 6 8 > 9 a z + 15 a z + 9 a z + 3 a z + 7 a z + 12 a z + 5 a z + 3 9 5 9 > 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 8 3 2 5 1 7 5 7 7 -- + - + ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t q t 10 7 6 10 4 t 2 3 2 5 3 > ----- + ----- + ---- + ---- + --- + 5 q t + q t + 4 q t + q t 7 2 5 2 5 3 q q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n335

L11n334 L11n336