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L11n333 L11n335

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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 X13,20,14,21 X9,13,10,22 X21,11,22,10 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:

- q-7 + 2q-6 - 2q-5 + 2q-4 + q-2 + 2q-1 - 1 + 2q - 2q2 + q3

A2 (sl(3)) Invariant:

- q-22 + q-14 + q-12 + 5q-10 + 5q-8 + 6q-6 + 5q-4 + 2q-2 + 2 + q10

HOMFLY-PT Polynomial:

a-2 + a-2z2 + z-2 - 1 - 3z2 - z4 - 2a2z-2 - a2 + a4z-2 + 2a4 + 3a4z2 + a4z4 - a6 - a6z2

Kauffman Polynomial:

- a-2 + 3a-2z2 - 4a-2z4 + a-2z6 - a-1z + 8a-1z3 - 9a-1z5 + 2a-1z7 - z-2 + 1 + z2 + z4 - 4z6 + z8 + 2az-1 - 6az + 15az3 - 12az5 + 2az7 - 2a2z-2 + 7a2 - 20a2z2 + 26a2z4 - 14a2z6 + 2a2z8 + 2a3z-1 - 10a3z + 13a3z3 - 5a3z7 + a3z9 - a4z-2 + 9a4 - 27a4z2 + 37a4z4 - 20a4z6 + 3a4z8 - 7a5z + 12a5z3 - 2a5z5 - 4a5z7 + a5z9 + 3a6 - 9a6z2 + 16a6z4 - 11a6z6 + 2a6z8 - 2a7z + 6a7z3 - 5a7z5 + a7z7

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     1   j = 3                   1 1   j = 1               2 2 1     j = -1             1 3 1       j = -3           2 2 3         j = -5         2 4 1           j = -7       1 1 2             j = -9     1 2 1               j = -11   1 1                   j = -13   1                     j = -15 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, 334]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 334]]
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[13, 20, 14, 21], > X[9, 13, 10, 22], X[21, 11, 22, 10], 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]=	-7 2 2 2 -2 2 2 3 -1 - q + -- - -- + -- + q + - + 2 q - 2 q + q 6 5 4 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -14 -12 5 5 6 5 2 10 2 - q + q + q + --- + -- + -- + -- + -- + q 10 8 6 4 2 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 334]][a, z]
Out[8]=	2 4 2 -2 2 4 6 -2 2 a a 2 z 4 2 6 2 -1 + a - a + 2 a - a + z - ---- + -- - 3 z + -- + 3 a z - a z - 2 2 2 z z a 4 4 4 > z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 334]][a, z]
Out[9]=	2 4 3 -2 2 4 6 -2 2 a a 2 a 2 a z 1 - a + 7 a + 9 a + 3 a - z - ---- - -- + --- + ---- - - - 6 a z - 2 2 z z a z z 2 3 5 7 2 3 z 2 2 4 2 6 2 > 10 a z - 7 a z - 2 a z + z + ---- - 20 a z - 27 a z - 9 a z + 2 a 3 4 8 z 3 3 3 5 3 7 3 4 4 z 2 4 > ---- + 15 a z + 13 a z + 12 a z + 6 a z + z - ---- + 26 a z + a 2 a 5 6 4 4 6 4 9 z 5 5 5 7 5 6 z > 37 a z + 16 a z - ---- - 12 a z - 2 a z - 5 a z - 4 z + -- - a 2 a 7 2 6 4 6 6 6 2 z 7 3 7 5 7 > 14 a z - 20 a z - 11 a z + ---- + 2 a z - 5 a z - 4 a z + a 7 7 8 2 8 4 8 6 8 3 9 5 9 > a z + z + 2 a z + 3 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 1 1 1 1 2 1 -- + - + 2 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 q q t q t q t q t q t q t q t 1 1 2 2 4 2 1 2 1 t > ----- + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + - + 9 3 7 3 5 3 7 2 5 2 3 2 5 3 q t q q t q t q t q t q t q t q t q t 2 3 2 3 3 5 3 7 4 > 2 q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n334

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