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L10n13 L10n15

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X4,19,1,20 X9,14,10,15 X8493 X5,13,6,12 X13,5,14,20 X11,16,12,17 X15,10,16,11 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-13/2 + 2q-11/2 - 2q-9/2 + 2q-7/2 - 2q-5/2 + q-3/2 - q-1/2 - q1/2 + q3/2 - q5/2

A2 (sl(3)) Invariant:

q-20 + q-16 - q-12 - q-8 + q-6 + q-2 + 2 + q2 + 2q4 + q6 + q8

HOMFLY-PT Polynomial:

- 2a-1z-1 - 3a-1z - a-1z3 + 4az-1 + 9az + 6az3 + az5 - 3a3z-1 - 8a3z - 5a3z3 - a3z5 + a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

- 2a-1z-1 + 7a-1z - 10a-1z3 + 6a-1z5 - a-1z7 + 2 + z2 - 8z4 + 6z6 - z8 - 4az-1 + 19az - 30az3 + 15az5 - 2az7 + 3a2 - 5a2z2 - 6a2z4 + 6a2z6 - a2z8 - 3a3z-1 + 15a3z - 24a3z3 + 13a3z5 - 2a3z7 + 3a4 - 10a4z2 + 9a4z4 - 2a4z6 - a5z-1 + 2a5z - a5z3 + 3a5z5 - a5z7 + a6 - 4a6z2 + 7a6z4 - 2a6z6 - a7z + 3a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                     1 j = 4                       j = 2                 1 1   j = 0             3 1       j = -2           1 2 1       j = -4         2 2 1         j = -6       1 1             j = -8     1 2 1             j = -10   1 1                 j = -12   1                   j = -14 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, 14]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 14]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[4, 19, 1, 20], X[9, 14, 10, 15], > X[8, 4, 9, 3], X[5, 13, 6, 12], X[13, 5, 14, 20], X[11, 16, 12, 17], > X[15, 10, 16, 11], 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]=	-(13/2) 2 2 2 2 -(3/2) 1 3/2 -q + ----- - ---- + ---- - ---- + q - ------- - Sqrt[q] + q - 11/2 9/2 7/2 5/2 Sqrt[q] q q q q 5/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -16 -12 -8 -6 -2 2 4 6 8 2 + q + q - q - q + q + q + q + 2 q + q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 14]][a, z]
Out[8]=	3 5 3 -2 4 a 3 a a 3 z 3 5 z 3 3 3 --- + --- - ---- + -- - --- + 9 a z - 8 a z + 2 a z - -- + 6 a z - 5 a z + a z z z z a a 5 3 5 3 5 > a z + a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 14]][a, z]
Out[9]=	3 5 2 4 6 2 4 a 3 a a 7 z 3 2 + 3 a + 3 a + a - --- - --- - ---- - -- + --- + 19 a z + 15 a z + a z z z z a 3 5 7 2 2 2 4 2 6 2 10 z 3 > 2 a z - a z + z - 5 a z - 10 a z - 4 a z - ----- - 30 a z - a 5 3 3 5 3 7 3 4 2 4 4 4 6 4 6 z > 24 a z - a z + 3 a z - 8 z - 6 a z + 9 a z + 7 a z + ---- + a 5 3 5 5 5 7 5 6 2 6 4 6 6 6 > 15 a z + 13 a z + 3 a z - a z + 6 z + 6 a z - 2 a z - 2 a z - 7 z 7 3 7 5 7 8 2 8 > -- - 2 a z - 2 a z - a z - z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 1 1 1 1 2 1 3 + q + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 q q t q t q t q t q t q t q t 1 1 2 2 1 t 2 2 2 3 6 4 > ----- + ----- + ----- + ---- + ---- + t + -- + q t + q t + q t 8 2 6 2 4 2 4 2 2 q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n14

L10n13 L10n15