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L10n27 L10n29

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X7,14,8,15 X11,19,12,18 X15,20,16,5 X19,16,20,17 X17,13,18,12 X13,8,14,9 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-17/2 + q-15/2 - q-13/2 + q-9/2 - q-7/2 + q-5/2 - 2q-3/2 + q-1/2 - q1/2

A2 (sl(3)) Invariant:

q-28 + 2q-26 + q-24 + q-22 - q-20 - q-18 - q-16 - q-14 + q-12 + 2q-8 + q-6 + q-4 + q-2 + 1 + q2

HOMFLY-PT Polynomial:

- az-1 - 3az - az3 + a3z-1 + 3a3z + 4a3z3 + a3z5 + a5z-1 - 2a7z-1 - 2a7z + a9z-1

Kauffman Polynomial:

az-1 - 4az + 4az3 - az5 - a2 + 3a2z4 - a2z6 + a3z-1 - 4a3z + 7a3z3 - 2a3z5 - 2a4 + 4a4z2 - a4z4 - a5z-1 + 8a5z - 13a5z3 + 7a5z5 - a5z7 - 3a6 + 8a6z2 - 13a6z4 + 7a6z6 - a6z8 - 2a7z-1 + 14a7z - 26a7z3 + 14a7z5 - 2a7z7 - a8 + 4a8z2 - 9a8z4 + 6a8z6 - a8z8 - a9z-1 + 6a9z - 10a9z3 + 6a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                     1 j = 0                       j = -2                 2 1   j = -4             1 1 1     j = -6             1 1       j = -8         1 2 1         j = -10       1 1 1           j = -12       1 2             j = -14   1 1                 j = -16                       j = -18 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, 28]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 28]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[7, 14, 8, 15], X[11, 19, 12, 18], > X[15, 20, 16, 5], X[19, 16, 20, 17], X[17, 13, 18, 12], X[13, 8, 14, 9], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, -3, 8, 10, -2, -4, 7, -8, 3, -5, 6, -7, 4, > -6, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) -(15/2) -(13/2) -(9/2) -(7/2) -(5/2) 2 -q + q - q + q - q + q - ---- + 3/2 q 1 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 -24 -22 -20 -18 -16 -14 -12 2 -6 1 + q + --- + q + q - q - q - q - q + q + -- + q + 26 8 q q -4 -2 2 > q + q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 28]][a, z]
Out[8]=	3 5 7 9 a a a 2 a a 3 7 3 3 3 3 5 -(-) + -- + -- - ---- + -- - 3 a z + 3 a z - 2 a z - a z + 4 a z + a z z z z z z
In[9]:=	Kauffman[Link[10, NonAlternating, 28]][a, z]
Out[9]=	3 5 7 9 2 4 6 8 a a a 2 a a 3 5 -a - 2 a - 3 a - a + - + -- - -- - ---- - -- - 4 a z - 4 a z + 8 a z + z z z z z 7 9 4 2 6 2 8 2 3 3 3 > 14 a z + 6 a z + 4 a z + 8 a z + 4 a z + 4 a z + 7 a z - 5 3 7 3 9 3 2 4 4 4 6 4 8 4 > 13 a z - 26 a z - 10 a z + 3 a z - a z - 13 a z - 9 a z - 5 3 5 5 5 7 5 9 5 2 6 6 6 8 6 > a z - 2 a z + 7 a z + 14 a z + 6 a z - a z + 7 a z + 6 a z - 5 7 7 7 9 7 6 8 8 8 > a z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 1 1 1 1 2 1 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 8 14 7 14 6 12 5 10 5 12 4 10 4 q q t q t q t q t q t q t q t 1 1 2 1 1 1 1 1 t 2 2 > ----- + ------ + ----- + ----- + ----- + ----- + ---- + ---- + -- + q t 8 4 10 3 8 3 8 2 6 2 4 2 6 4 2 q t q t q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n28

L10n27 L10n29