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L10n14 L10n16

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X4,19,1,20 X14,10,15,9 X8493 X5,13,6,12 X13,5,14,20 X16,12,17,11 X10,16,11,15 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

q-3/2 - 3q-1/2 + 4q1/2 - 7q3/2 + 6q5/2 - 7q7/2 + 6q9/2 - 4q11/2 + 2q13/2

A2 (sl(3)) Invariant:

- q-4 + q-2 + 2q2 + 3q4 + 2q6 + 5q8 + 2q12 - q14 - q16 - 2q20 - q24

HOMFLY-PT Polynomial:

a-7z-1 + a-7z - 2a-5z-1 - 5a-5z - 4a-5z3 - a-5z5 + 5a-3z + 8a-3z3 + 5a-3z5 + a-3z7 + a-1z-1 - a-1z - 3a-1z3 - a-1z5

Kauffman Polynomial:

2a-8 - 3a-8z2 - a-7z-1 + 2a-7z - 3a-7z3 - a-7z5 + 5a-6 - 12a-6z2 + 9a-6z4 - 4a-6z6 - 2a-5z-1 + 9a-5z - 18a-5z3 + 15a-5z5 - 5a-5z7 + 3a-4 - 12a-4z2 + 12a-4z4 - 2a-4z8 + 10a-3z - 26a-3z3 + 27a-3z5 - 8a-3z7 - a-2 - 5a-2z2 + 6a-2z4 + 3a-2z6 - 2a-2z8 + a-1z-1 + 3a-1z - 11a-1z3 + 11a-1z5 - 3a-1z7 - 2z2 + 3z4 - z6

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 14                 2 j = 12               2   j = 10             4 2   j = 8           3 2     j = 6         3 4       j = 4       4 3         j = 2     2 5           j = 0   1 2             j = -2   2               j = -4 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, 15]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 15]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[4, 19, 1, 20], X[14, 10, 15, 9], > X[8, 4, 9, 3], X[5, 13, 6, 12], X[13, 5, 14, 20], X[16, 12, 17, 11], > X[10, 16, 11, 15], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {-6, -1, 2, -5, 4, -9, 8, 6, -7, -4, 9, -8, 10, -2, > 3, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 3 3/2 5/2 7/2 9/2 11/2 q - ------- + 4 Sqrt[q] - 7 q + 6 q - 7 q + 6 q - 4 q + Sqrt[q] 13/2 > 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 -2 2 4 6 8 12 14 16 20 24 -q + q + 2 q + 3 q + 2 q + 5 q + 2 q - q - q - 2 q - q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 15]][a, z]
Out[8]=	3 3 3 5 5 5 1 2 1 z 5 z 5 z z 4 z 8 z 3 z z 5 z z ---- - ---- + --- + -- - --- + --- - - - ---- + ---- - ---- - -- + ---- - -- + 7 5 a z 7 5 3 a 5 3 a 5 3 a a z a z a a a a a a a 7 z > -- 3 a
In[9]:=	Kauffman[Link[10, NonAlternating, 15]][a, z]
Out[9]=	2 2 5 3 -2 1 2 1 2 z 9 z 10 z 3 z 2 3 z -- + -- + -- - a - ---- - ---- + --- + --- + --- + ---- + --- - 2 z - ---- - 8 6 4 7 5 a z 7 5 3 a 8 a a a a z a z a a a a 2 2 2 3 3 3 3 4 4 12 z 12 z 5 z 3 z 18 z 26 z 11 z 4 9 z 12 z > ----- - ----- - ---- - ---- - ----- - ----- - ----- + 3 z + ---- + ----- + 6 4 2 7 5 3 a 6 4 a a a a a a a a 4 5 5 5 5 6 6 7 7 7 6 z z 15 z 27 z 11 z 6 4 z 3 z 5 z 8 z 3 z > ---- - -- + ----- + ----- + ----- - z - ---- + ---- - ---- - ---- - ---- - 2 7 5 3 a 6 2 5 3 a a a a a a a a a 8 8 2 z 2 z > ---- - ---- 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 -2 2 2 2 q 4 6 6 2 5 q + 4 q + ----- + t + ----- + - + ---- + 3 q t + 3 q t + 4 q t + 4 3 2 2 t t q t q t 8 2 8 3 10 3 10 4 12 4 14 5 > 3 q t + 2 q t + 4 q t + 2 q t + 2 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n15

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