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L11n458 L2a1

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X7,21,8,20 X19,5,20,10 X13,19,14,22 X21,11,22,18 X17,15,18,14 X9,17,10,16 X15,9,16,8 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-5/2 + q-3/2 - 3q-1/2 - 2q3/2 - q5/2 - 2q9/2 + q11/2 - 2q13/2 + q15/2

A2 (sl(3)) Invariant:

q-8 + q-6 + 3q-4 + 5q-2 + 8 + 10q2 + 10q4 + 12q6 + 10q8 + 8q10 + 6q12 + 3q14 + 3q16 + q18 + q20 - q24

HOMFLY-PT Polynomial:

a-7z - a-5z-3 - 2a-5z-1 - 2a-5z - a-5z3 + 3a-3z-3 + 7a-3z-1 + 6a-3z + a-3z3 - 3a-1z-3 - 8a-1z-1 - 8a-1z - 5a-1z3 - a-1z5 + az-3 + 3az-1 + 3az + az3

Kauffman Polynomial:

- 2a-8z2 + 4a-8z4 - a-8z6 + 6a-7z - 12a-7z3 + 10a-7z5 - 2a-7z7 - 2a-6z2 + 4a-6z6 - a-6z8 + a-5z-3 - 5a-5z-1 + 20a-5z - 39a-5z3 + 25a-5z5 - 4a-5z7 - 3a-4z-2 + 10a-4 - 23a-4z4 + 18a-4z6 - 3a-4z8 + 3a-3z-3 - 12a-3z-1 + 27a-3z - 35a-3z3 + 12a-3z5 + 3a-3z7 - a-3z9 - 6a-2z-2 + 19a-2 - 10a-2z2 - 18a-2z4 + 17a-2z6 - 3a-2z8 + 3a-1z-3 - 12a-1z-1 + 23a-1z - 19a-1z3 + 3a-1z5 + 4a-1z7 - a-1z9 - 3z-2 + 10 - 10z2 + z4 + 4z6 - z8 + az-3 - 5az-1 + 10az - 11az3 + 6az5 - az7

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 = 16                       1 j = 14                     1   j = 12                 1 1 1   j = 10               1 3 1     j = 8             2 2 2       j = 6           3 6 2         j = 4         1 2 4           j = 2       2 5 1             j = 0     1 1 3               j = -2     2                   j = -4 1 1                     j = -6 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 459]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 459]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[7, 21, 8, 20], X[19, 5, 20, 10], > X[13, 19, 14, 22], X[21, 11, 22, 18], X[17, 15, 18, 14], X[9, 17, 10, 16], > X[15, 9, 16, 8], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {-4, 3, -6, 5}, {10, -1, -3, 9, -8, 4}, > {11, -2, -5, 7, -9, 8, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) -(3/2) 3 3/2 5/2 9/2 11/2 13/2 15/2 -q + q - ------- - 2 q - q - 2 q + q - 2 q + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 3 5 2 4 6 8 10 12 8 + q + q + -- + -- + 10 q + 10 q + 12 q + 10 q + 8 q + 6 q + 4 2 q q 14 16 18 20 24 > 3 q + 3 q + q + q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 459]][a, z]
Out[8]=	1 3 3 a 2 7 8 3 a z 2 z 6 z 8 z -(-----) + ----- - ---- + -- - ---- + ---- - --- + --- + -- - --- + --- - --- + 5 3 3 3 3 3 5 3 a z z 7 5 3 a a z a z a z z a z a z a a a 3 3 3 5 z z 5 z 3 z > 3 a z - -- + -- - ---- + a z - -- 5 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 459]][a, z]
Out[9]=	10 19 1 3 3 a 3 3 6 5 12 10 + -- + -- + ----- + ----- + ---- + -- - -- - ----- - ----- - ---- - ---- - 4 2 5 3 3 3 3 3 2 4 2 2 2 5 3 a a a z a z a z z z a z a z a z a z 2 2 12 5 a 6 z 20 z 27 z 23 z 2 2 z 2 z > --- - --- + --- + ---- + ---- + ---- + 10 a z - 10 z - ---- - ---- - a z z 7 5 3 a 8 6 a a a a a 2 3 3 3 3 4 4 10 z 12 z 39 z 35 z 19 z 3 4 4 z 23 z > ----- - ----- - ----- - ----- - ----- - 11 a z + z + ---- - ----- - 2 7 5 3 a 8 4 a a a a a a 4 5 5 5 5 6 6 6 18 z 10 z 25 z 12 z 3 z 5 6 z 4 z 18 z > ----- + ----- + ----- + ----- + ---- + 6 a z + 4 z - -- + ---- + ----- + 2 7 5 3 a 8 6 4 a a a a a a a 6 7 7 7 7 8 8 8 9 9 17 z 2 z 4 z 3 z 4 z 7 8 z 3 z 3 z z z > ----- - ---- - ---- + ---- + ---- - a z - z - -- - ---- - ---- - -- - -- 2 7 5 3 a 6 4 2 3 a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 2 1 2 q 2 3 + 5 q + q + ----- + ----- + ----- + t + ----- + - + ---- + q t + 6 4 4 4 4 3 2 2 t t q t q t q t q t 4 6 4 2 6 2 8 2 6 3 8 3 > 2 q t + 3 q t + 4 q t + 6 q t + 2 q t + 2 q t + 2 q t + 10 3 8 4 10 4 12 4 10 5 12 5 12 6 14 6 > q t + 2 q t + 3 q t + q t + q t + q t + q t + q t + 16 7 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n459

L11n458 L2a1