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L10n59 L10n61

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DrawMorseLink

PD Presentation:

X12,1,13,2 X16,7,17,8 X5,1,6,10 X3746 X9,5,10,4 X17,11,18,20 X13,19,14,18 X19,15,20,14 X2,11,3,12 X8,15,9,16

Gauss Code:

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

Jones Polynomial:

- q-5/2 - q-1/2 - 2q1/2 + 2q3/2 - 2q5/2 + 3q7/2 - 2q9/2 + 2q11/2 - q13/2

A2 (sl(3)) Invariant:

q-8 + 2q-6 + 4q-4 + 4q-2 + 5 + 2q2 - q4 - 2q6 - 4q8 - 2q10 - 2q12 + q16 + q20

HOMFLY-PT Polynomial:

- a-5z-1 - 2a-5z - a-5z3 + 5a-3z-1 + 9a-3z + 5a-3z3 + a-3z5 - 8a-1z-1 - 13a-1z - 7a-1z3 - a-1z5 + 4az-1 + 4az + az3

Kauffman Polynomial:

- a-7z + 3a-7z3 - a-7z5 + 2a-6 - 5a-6z2 + 7a-6z4 - 2a-6z6 - a-5z-1 - a-5z + 2a-5z3 + 2a-5z5 - a-5z7 + 9a-4 - 18a-4z2 + 13a-4z4 - 3a-4z6 - 5a-3z-1 + 8a-3z - 9a-3z3 + 5a-3z5 - a-3z7 + 14a-2 - 28a-2z2 + 14a-2z4 - 2a-2z6 - 8a-1z-1 + 21a-1z - 23a-1z3 + 9a-1z5 - a-1z7 + 8 - 15z2 + 8z4 - z6 - 4az-1 + 13az - 15az3 + 7az5 - 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 j = 14                     1 j = 12                   1   j = 10                 1 1   j = 8               2 1     j = 6           1 1 1       j = 4           2 2         j = 2       1 2 1           j = 0         3             j = -2     1                 j = -4 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]=	2
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 60]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 60]]
Out[4]=	PD[X[12, 1, 13, 2], X[16, 7, 17, 8], X[5, 1, 6, 10], X[3, 7, 4, 6], > X[9, 5, 10, 4], X[17, 11, 18, 20], X[13, 19, 14, 18], X[19, 15, 20, 14], > X[2, 11, 3, 12], X[8, 15, 9, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, -4, 5, -3, 4, 2, -10, -5, 3}, > {9, -1, -7, 8, 10, -2, -6, 7, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 1 3/2 5/2 7/2 9/2 11/2 -q - ------- - 2 Sqrt[q] + 2 q - 2 q + 3 q - 2 q + 2 q - Sqrt[q] 13/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 2 4 4 2 4 6 8 10 12 16 20 5 + q + -- + -- + -- + 2 q - q - 2 q - 4 q - 2 q - 2 q + q + q 6 4 2 q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 60]][a, z]
Out[8]=	3 3 3 1 5 8 4 a 2 z 9 z 13 z z 5 z 7 z -(----) + ---- - --- + --- - --- + --- - ---- + 4 a z - -- + ---- - ---- + 5 3 a z z 5 3 a 5 3 a a z a z a a a a 5 5 3 z z > a z + -- - -- 3 a a
In[9]:=	Kauffman[Link[10, NonAlternating, 60]][a, z]
Out[9]=	2 9 14 1 5 8 4 a z z 8 z 21 z 8 + -- + -- + -- - ---- - ---- - --- - --- - -- - -- + --- + ---- + 13 a z - 6 4 2 5 3 a z z 7 5 3 a a a a a z a z a a a 2 2 2 3 3 3 3 2 5 z 18 z 28 z 3 z 2 z 9 z 23 z 3 > 15 z - ---- - ----- - ----- + ---- + ---- - ---- - ----- - 15 a z + 6 4 2 7 5 3 a a a a a a a 4 4 4 5 5 5 5 4 7 z 13 z 14 z z 2 z 5 z 9 z 5 6 > 8 z + ---- + ----- + ----- - -- + ---- + ---- + ---- + 7 a z - z - 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 7 2 z 3 z 2 z z z z 7 > ---- - ---- - ---- - -- - -- - -- - a z 6 4 2 5 3 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 q 2 4 6 4 2 3 + 2 q + ----- + ----- + ----- + -- + q t + 2 q t + q t + 2 q t + 6 4 4 4 2 2 t q t q t q t 6 2 6 3 8 3 8 4 10 4 10 5 12 5 14 6 > q t + q t + 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 L10n60

L10n59 L10n61