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L11n435 L11n437

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DrawMorseLink

PD Presentation:

X8192 X18,8,19,7 X5,16,6,17 X15,10,16,11 X3,15,4,14 X11,5,12,4 X2,20,3,19 X20,9,21,10 X13,7,14,12 X17,22,18,13 X6,21,1,22

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-16 + q-14 + q-8 + q-6 + 4q-4 + 4q-2 + 6 + 5q2 + 3q4 + 3q6 + q8 - q12

HOMFLY-PT Polynomial:

a-2z-2 - 3a-2z2 - a-2z4 - 2z-2 + 5z2 + 5z4 + z6 + a2z-2 - a2z2 - a4z2

Kauffman Polynomial:

5a-3z3 - 5a-3z5 + a-3z7 + a-2z-2 - 12a-2z2 + 20a-2z4 - 12a-2z6 + 2a-2z8 - 2a-1z-1 + 4a-1z3 + 3a-1z5 - 5a-1z7 + a-1z9 + 2z-2 + 1 - 20z2 + 36z4 - 20z6 + 3z8 - 2az-1 + 6az3 + 2az5 - 5az7 + az9 + a2z-2 - 4a2z2 + 8a2z4 - 6a2z6 + a2z8 + 4a3z3 - 5a3z5 + a3z7 + 4a4z2 - 8a4z4 + 2a4z6 - 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 r = 5 j = 9                     1 j = 7                   1   j = 5               1 1 1   j = 3               2 1     j = 1           4 2 1       j = -1         2 5 2         j = -3       1 2 2           j = -5     1 2 1             j = -7   1 1 1               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, 436]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 436]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 8, 19, 7], X[5, 16, 6, 17], X[15, 10, 16, 11], > X[3, 15, 4, 14], X[11, 5, 12, 4], X[2, 20, 3, 19], X[20, 9, 21, 10], > X[13, 7, 14, 12], X[17, 22, 18, 13], X[6, 21, 1, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, -5, 6, -3, -11}, {2, -1, 8, 4, -6, 9}, > {-9, 5, -4, 3, -10, -2, 7, -8, 11, 10}]
In[6]:=	Jones[L][q]
Out[6]=	-5 2 -3 -2 2 3 4 1 - q + -- - q + q + 2 q - q + 2 q - q 4 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 -8 -6 4 4 2 4 6 8 12 6 - q + q + q + q + -- + -- + 5 q + 3 q + 3 q + q - q 4 2 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 436]][a, z]
Out[8]=	2 2 4 -2 1 a 2 3 z 2 2 4 2 4 z 6 -- + ----- + -- + 5 z - ---- - a z - a z + 5 z - -- + z 2 2 2 2 2 2 z a z z a a
In[9]:=	Kauffman[Link[11, NonAlternating, 436]][a, z]
Out[9]=	2 2 3 2 1 a 2 2 a 2 12 z 2 2 4 2 5 z 1 + -- + ----- + -- - --- - --- - 20 z - ----- - 4 a z + 4 a z + ---- + 2 2 2 2 a z z 2 3 z a z z a a 3 4 4 z 3 3 3 5 3 4 20 z 2 4 4 4 > ---- + 6 a z + 4 a z - 3 a z + 36 z + ----- + 8 a z - 8 a z - a 2 a 5 5 6 5 z 3 z 5 3 5 5 5 6 12 z 2 6 > ---- + ---- + 2 a z - 5 a z + a z - 20 z - ----- - 6 a z + 3 a 2 a a 7 7 8 9 4 6 z 5 z 7 3 7 8 2 z 2 8 z 9 > 2 a z + -- - ---- - 5 a z + a z + 3 z + ---- + a z + -- + a z 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 5 1 1 1 1 1 1 2 1 -- + - + 4 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + 3 q 11 5 9 4 7 4 7 3 5 3 7 2 5 2 3 2 q q t q t q t q t q t q t q t q t 1 2 2 2 t 2 3 2 5 2 3 3 5 3 > ---- + ---- + --- + --- + 2 q t + q t + 2 q t + q t + q t + q t + 5 3 q t q q t q t 5 4 7 4 9 5 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n436

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