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L11n290 L11n292

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DrawMorseLink

PD Presentation:

X6172 X11,18,12,19 X3849 X15,2,16,3 X16,7,17,8 X9,11,10,22 X4,17,1,18 X19,5,20,10 X5,12,6,13 X21,15,22,14 X13,21,14,20

Gauss Code:

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

Jones Polynomial:

- q-6 + 4q-5 - 5q-4 + 9q-3 - 8q-2 + 9q-1 - 7 + 5q - 3q2 + q3

A2 (sl(3)) Invariant:

- q-22 + q-20 + 4q-16 + 4q-14 + 5q-12 + 7q-10 + 2q-8 + 5q-6 - q-4 + q-2 - q2 + q4 - q6 + q8

HOMFLY-PT Polynomial:

1 + 4z2 + 4z4 + z6 + a2z-2 - a2 - 9a2z2 - 12a2z4 - 6a2z6 - a2z8 - 2a4z-2 + a4 + 7a4z2 + 5a4z4 + a4z6 + a6z-2 - a6 - a6z2

Kauffman Polynomial:

a-2z2 - 3a-2z4 + a-2z6 - 2a-1z + 7a-1z3 - 10a-1z5 + 3a-1z7 + 2 - 7z2 + 13z4 - 14z6 + 4z8 - 7az + 18az3 - 12az5 - 2az7 + 2az9 + a2z-2 + 4a2 - 24a2z2 + 44a2z4 - 32a2z6 + 8a2z8 - 2a3z-1 - 7a3z + 19a3z3 - 7a3z5 - 3a3z7 + 2a3z9 + 2a4z-2 + 3a4 - 21a4z2 + 32a4z4 - 17a4z6 + 4a4z8 - 2a5z-1 - 3a5z + 9a5z3 - 5a5z5 + 2a5z7 + a6z-2 - 5a6z2 + 4a6z4 - a7z + a7z3

Khovanov Homology:

trqj 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                 2   j = 3               3 1   j = 1             4 2     j = -1           5 3       j = -3         4 5         j = -5       5 4           j = -7     2 6             j = -9   2 3               j = -11   3                 j = -13 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, 291]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 291]]
Out[4]=	PD[X[6, 1, 7, 2], X[11, 18, 12, 19], X[3, 8, 4, 9], X[15, 2, 16, 3], > X[16, 7, 17, 8], X[9, 11, 10, 22], X[4, 17, 1, 18], X[19, 5, 20, 10], > X[5, 12, 6, 13], X[21, 15, 22, 14], X[13, 21, 14, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -7}, {-9, -1, 5, 3, -6, 8}, > {-2, 9, -11, 10, -4, -5, 7, 2, -8, 11, -10, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-6 4 5 9 8 9 2 3 -7 - q + -- - -- + -- - -- + - + 5 q - 3 q + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 4 4 5 7 2 5 -4 -2 2 4 6 8 -q + q + --- + --- + --- + --- + -- + -- - q + q - q + q - q + q 16 14 12 10 8 6 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 291]][a, z]
Out[8]=	2 4 6 2 4 6 a 2 a a 2 2 2 4 2 6 2 4 1 - a + a - a + -- - ---- + -- + 4 z - 9 a z + 7 a z - a z + 4 z - 2 2 2 z z z 2 4 4 4 6 2 6 4 6 2 8 > 12 a z + 5 a z + z - 6 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 291]][a, z]
Out[9]=	2 4 6 3 5 2 4 a 2 a a 2 a 2 a 2 z 3 2 + 4 a + 3 a + -- + ---- + -- - ---- - ---- - --- - 7 a z - 7 a z - 2 2 2 z z a z z z 2 3 5 7 2 z 2 2 4 2 6 2 7 z > 3 a z - a z - 7 z + -- - 24 a z - 21 a z - 5 a z + ---- + 2 a a 4 3 3 3 5 3 7 3 4 3 z 2 4 4 4 > 18 a z + 19 a z + 9 a z + a z + 13 z - ---- + 44 a z + 32 a z + 2 a 5 6 6 4 10 z 5 3 5 5 5 6 z 2 6 > 4 a z - ----- - 12 a z - 7 a z - 5 a z - 14 z + -- - 32 a z - a 2 a 7 4 6 3 z 7 3 7 5 7 8 2 8 4 8 > 17 a z + ---- - 2 a z - 3 a z + 2 a z + 4 z + 8 a z + 4 a z + a 9 3 9 > 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 5 1 3 2 3 2 6 5 4 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q q t q t q t q t q t q t q t q t 4 3 t 2 3 2 3 3 5 3 7 4 > ---- + --- + 4 q t + 2 q t + 3 q t + q t + 2 q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n291

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