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L10n57 L10n59

Knotscape

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DrawMorseLink

PD Presentation:

X12,1,13,2 X7,17,8,16 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 X15,9,16,8

Gauss Code:

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

Jones Polynomial:

- 3q3/2 + 4q5/2 - 8q7/2 + 8q9/2 - 8q11/2 + 8q13/2 - 5q15/2 + 3q17/2 - q19/2

A2 (sl(3)) Invariant:

3q4 + q6 + 5q8 + 5q10 + 2q12 + 3q14 - 3q16 - 2q18 - 4q20 - 3q22 + q24 - q26 + q28 + q30

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-11z + 2a-11z3 - a-11z5 + 2a-10 - 5a-10z2 + 7a-10z4 - 3a-10z6 - a-9z-1 - a-9z + 4a-9z3 + 3a-9z5 - 3a-9z7 + 9a-8 - 24a-8z2 + 26a-8z4 - 8a-8z6 - a-8z8 - 5a-7z-1 + 9a-7z - 12a-7z3 + 14a-7z5 - 7a-7z7 + 14a-6 - 31a-6z2 + 25a-6z4 - 8a-6z6 - a-6z8 - 8a-5z-1 + 19a-5z - 20a-5z3 + 10a-5z5 - 4a-5z7 + 8a-4 - 12a-4z2 + 6a-4z4 - 3a-4z6 - 4a-3z-1 + 10a-3z - 6a-3z3

Khovanov Homology:

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 20                 1 j = 18               2   j = 16             3 1   j = 14           5 2     j = 12         3 3       j = 10       5 5         j = 8     3 3           j = 6   1 5             j = 4 2 3               j = 2 3

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, 58]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 58]]
Out[4]=	PD[X[12, 1, 13, 2], X[7, 17, 8, 16], 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[15, 9, 16, 8]]
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]=	3/2 5/2 7/2 9/2 11/2 13/2 15/2 17/2 -3 q + 4 q - 8 q + 8 q - 8 q + 8 q - 5 q + 3 q - 19/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	4 6 8 10 12 14 16 18 20 22 3 q + q + 5 q + 5 q + 2 q + 3 q - 3 q - 2 q - 4 q - 3 q + 24 26 28 30 > q - q + q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 58]][a, z]
Out[8]=	3 3 3 5 1 5 8 4 z 7 z 13 z 7 z 3 z 8 z 3 z 2 z -(----) + ---- - ---- + ---- - -- + --- - ---- + --- + ---- - ---- + ---- - ---- 9 7 5 3 9 7 5 3 7 5 3 5 a z a z a z a z a a a a a a a a
In[9]:=	Kauffman[Link[10, NonAlternating, 58]][a, z]
Out[9]=	2 9 14 8 1 5 8 4 z z 9 z 19 z 10 z --- + -- + -- + -- - ---- - ---- - ---- - ---- - --- - -- + --- + ---- + ---- - 10 8 6 4 9 7 5 3 11 9 7 5 3 a a a a a z a z a z a z a a a a a 2 2 2 2 3 3 3 3 3 4 5 z 24 z 31 z 12 z 2 z 4 z 12 z 20 z 6 z 7 z > ---- - ----- - ----- - ----- + ---- + ---- - ----- - ----- - ---- + ---- + 10 8 6 4 11 9 7 5 3 10 a a a a a a a a a a 4 4 4 5 5 5 5 6 6 6 26 z 25 z 6 z z 3 z 14 z 10 z 3 z 8 z 8 z > ----- + ----- + ---- - --- + ---- + ----- + ----- - ---- - ---- - ---- - 8 6 4 11 9 7 5 10 8 6 a a a a a a a a a a 6 7 7 7 8 8 3 z 3 z 7 z 4 z z z > ---- - ---- - ---- - ---- - -- - -- 4 9 7 5 8 6 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 4 6 6 2 8 2 8 3 10 3 3 q + 2 q + 3 q t + q t + 5 q t + 3 q t + 3 q t + 5 q t + 10 4 12 4 12 5 14 5 14 6 16 6 16 7 > 5 q t + 3 q t + 3 q t + 5 q t + 2 q t + 3 q t + q t + 18 7 20 8 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n58

L10n57 L10n59