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L10n69 L10n71

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X9,18,10,19 X11,16,12,17 X17,20,18,11 X4,15,1,16 X19,10,20,5

Gauss Code:

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

Jones Polynomial:

q-9 - 2q-8 + 2q-7 - 2q-6 + 3q-5 - 2q-4 + 3q-3 + q-1

A2 (sl(3)) Invariant:

q-28 - q-20 + q-18 + 2q-16 + 4q-14 + 5q-12 + 5q-10 + 5q-8 + 3q-6 + q-4 + q-2

HOMFLY-PT Polynomial:

a2z-2 + 2a2 + a2z2 - 2a4z-2 - a4 + a4z2 + a6z-2 - 2a6 - 3a6z2 - a6z4 + a8 + a8z2

Kauffman Polynomial:

a2z-2 - 2a2 + a2z2 - 2a3z-1 + 2a3z + 2a4z-2 - 2a4 - a4z2 + a4z4 - 2a5z-1 - 2a5z + 8a5z3 - 5a5z5 + a5z7 + a6z-2 - a6 + a6z2 + 3a6z4 - 4a6z6 + a6z8 - 6a7z + 17a7z3 - 14a7z5 + 3a7z7 - a8 + 6a8z2 - 2a8z4 - 3a8z6 + a8z8 - 2a9z + 9a9z3 - 9a9z5 + 2a9z7 - a10 + 3a10z2 - 4a10z4 + a10z6

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -1                 1 j = -3               1 2 j = -5             2   1 j = -7           1 2     j = -9         3 2       j = -11       1 3 1       j = -13     1 2 1         j = -15   1 1             j = -17   1               j = -19 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]=	3
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 70]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 70]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[8, 4, 9, 3], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[9, 18, 10, 19], X[11, 16, 12, 17], X[17, 20, 18, 11], > X[4, 15, 1, 16], X[19, 10, 20, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -9}, {-2, -1, 5, -3, -6, 10}, > {-7, 2, 4, -5, 9, 7, -8, 6, -10, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-9 2 2 2 3 2 3 1 q - -- + -- - -- + -- - -- + -- + - 8 7 6 5 4 3 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -20 -18 2 4 5 5 5 3 -4 -2 q - q + q + --- + --- + --- + --- + -- + -- + q + q 16 14 12 10 8 6 q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 70]][a, z]
Out[8]=	2 4 6 2 4 6 8 a 2 a a 2 2 4 2 6 2 8 2 6 4 2 a - a - 2 a + a + -- - ---- + -- + a z + a z - 3 a z + a z - a z 2 2 2 z z z
In[9]:=	Kauffman[Link[10, NonAlternating, 70]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 10 a 2 a a 2 a 2 a 3 5 -2 a - 2 a - a - a - a + -- + ---- + -- - ---- - ---- + 2 a z - 2 a z - 2 2 2 z z z z z 7 9 2 2 4 2 6 2 8 2 10 2 5 3 > 6 a z - 2 a z + a z - a z + a z + 6 a z + 3 a z + 8 a z + 7 3 9 3 4 4 6 4 8 4 10 4 5 5 > 17 a z + 9 a z + a z + 3 a z - 2 a z - 4 a z - 5 a z - 7 5 9 5 6 6 8 6 10 6 5 7 7 7 > 14 a z - 9 a z - 4 a z - 3 a z + a z + a z + 3 a z + 9 7 6 8 8 8 > 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 2 1 1 1 1 1 1 2 1 q + -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 1 3 3 1 2 1 2 2 1 > ------ + ------ + ----- + ------ + ----- + ----- + ----- + ----- + ---- 13 4 11 4 9 4 11 3 9 3 7 3 7 2 5 2 3 q t q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n70

L10n69 L10n71