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L11a495 L11a497

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X14,8,15,7 X16,10,17,9 X8,16,9,15 X22,17,19,18 X20,12,21,11 X10,20,11,19 X18,21,5,22 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

q-2 - 2q-1 + 6 - 7q + 11q2 - 11q3 + 13q4 - 11q5 + 8q6 - 6q7 + 3q8 - q9

A2 (sl(3)) Invariant:

q-6 + q-4 + 2q-2 + 5 + 4q2 + 7q4 + 5q6 + 4q8 + 5q10 - q12 + 2q14 - 3q16 - 2q18 - q20 - 2q22 + q24 - q26

HOMFLY-PT Polynomial:

- a-6z-2 - 4a-6 - 5a-6z2 - 4a-6z4 - a-6z6 + 4a-4z-2 + 13a-4 + 18a-4z2 + 14a-4z4 + 6a-4z6 + a-4z8 - 5a-2z-2 - 14a-2 - 17a-2z2 - 10a-2z4 - 2a-2z6 + 2z-2 + 5 + 4z2 + z4

Kauffman Polynomial:

a-11z3 + 3a-10z4 + 2a-9z - 5a-9z3 + 6a-9z5 + 2a-8z2 - 10a-8z4 + 8a-8z6 + a-7z-1 - 4a-7z + 5a-7z3 - 16a-7z5 + 9a-7z7 - a-6z-2 + 7a-6 - 14a-6z2 + 19a-6z4 - 24a-6z6 + 9a-6z8 + 5a-5z-1 - 25a-5z + 41a-5z3 - 22a-5z5 - 7a-5z7 + 5a-5z9 - 4a-4z-2 + 22a-4 - 57a-4z2 + 88a-4z4 - 60a-4z6 + 11a-4z8 + a-4z10 + 9a-3z-1 - 31a-3z + 34a-3z3 + 10a-3z5 - 25a-3z7 + 7a-3z9 - 5a-2z-2 + 23a-2 - 57a-2z2 + 70a-2z4 - 34a-2z6 + 3a-2z8 + a-2z10 + 5a-1z-1 - 12a-1z + 4a-1z3 + 10a-1z5 - 9a-1z7 + 2a-1z9 - 2z-2 + 9 - 16z2 + 14z4 - 6z6 + z8

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 19                       1 j = 17                     2   j = 15                   4 1   j = 13                 4 2     j = 11               7 4       j = 9             6 4         j = 7           5 7           j = 5         6 6             j = 3       5 9               j = 1     1 2                 j = -1   1 5                   j = -3   1                     j = -5 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, Alternating, 496]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 496]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 8, 15, 7], X[16, 10, 17, 9], > X[8, 16, 9, 15], X[22, 17, 19, 18], X[20, 12, 21, 11], X[10, 20, 11, 19], > X[18, 21, 5, 22], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {8, -7, 9, -6}, > {10, -1, 3, -5, 4, -8, 7, -2, 11, -3, 5, -4, 6, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-2 2 2 3 4 5 6 7 8 9 6 + q - - - 7 q + 11 q - 11 q + 13 q - 11 q + 8 q - 6 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -4 2 2 4 6 8 10 12 14 16 5 + q + q + -- + 4 q + 7 q + 5 q + 4 q + 5 q - q + 2 q - 3 q - 2 q 18 20 22 24 26 > 2 q - q - 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 496]][a, z]
Out[8]=	2 2 2 4 13 14 2 1 4 5 2 5 z 18 z 17 z 5 - -- + -- - -- + -- - ----- + ----- - ----- + 4 z - ---- + ----- - ----- + 6 4 2 2 6 2 4 2 2 2 6 4 2 a a a z a z a z a z a a a 4 4 4 6 6 6 8 4 4 z 14 z 10 z z 6 z 2 z z > z - ---- + ----- - ----- - -- + ---- - ---- + -- 6 4 2 6 4 2 4 a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 496]][a, z]
Out[9]=	7 22 23 2 1 4 5 1 5 9 5 9 + -- + -- + -- - -- - ----- - ----- - ----- + ---- + ---- + ---- + --- + 6 4 2 2 6 2 4 2 2 2 7 5 3 a z a a a z a z a z a z a z a z a z 2 2 2 2 2 z 4 z 25 z 31 z 12 z 2 2 z 14 z 57 z 57 z > --- - --- - ---- - ---- - ---- - 16 z + ---- - ----- - ----- - ----- + 9 7 5 3 a 8 6 4 2 a a a a a a a a 3 3 3 3 3 3 4 4 4 z 5 z 5 z 41 z 34 z 4 z 4 3 z 10 z 19 z > --- - ---- + ---- + ----- + ----- + ---- + 14 z + ---- - ----- + ----- + 11 9 7 5 3 a 10 8 6 a a a a a a a a 4 4 5 5 5 5 5 6 88 z 70 z 6 z 16 z 22 z 10 z 10 z 6 8 z > ----- + ----- + ---- - ----- - ----- + ----- + ----- - 6 z + ---- - 4 2 9 7 5 3 a 8 a a a a a a a 6 6 6 7 7 7 7 8 8 24 z 60 z 34 z 9 z 7 z 25 z 9 z 8 9 z 11 z > ----- - ----- - ----- + ---- - ---- - ----- - ---- + z + ---- + ----- + 6 4 2 7 5 3 a 6 4 a a a a a a a a 8 9 9 9 10 10 3 z 5 z 7 z 2 z z z > ---- + ---- + ---- + ---- + --- + --- 2 5 3 a 4 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 1 1 5 q 2 q 5 q 5 7 9 q + 6 q + ----- + ----- + ---- + ---- + -- + --- + ---- + 6 q t + 5 q t + 5 4 3 3 3 2 2 t t q t q t q t q t t 7 2 9 2 9 3 11 3 11 4 13 4 13 5 > 7 q t + 6 q t + 4 q t + 7 q t + 4 q t + 4 q t + 2 q t + 15 5 15 6 17 6 19 7 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a496

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