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L11n14 L11n16

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X5,10,6,11 X8493 X11,22,12,5 X13,20,14,21 X19,14,20,15 X21,12,22,13 X9,18,10,19 X2,16,3,15

Gauss Code:

{{1, -11, 5, -3}, {-4, -1, 2, -5, -10, 4, -6, 9, -7, 8, 11, -2, 3, 10, -8, 7, -9, 6}}

Jones Polynomial:

q-19/2 - q-17/2 + q-15/2 - q-13/2 + q-11/2 - q-9/2 - q-5/2 - q-1/2

A2 (sl(3)) Invariant:

- q-30 - q-28 - q-26 - q-24 + q-18 + q-16 + q-14 + q-12 + q-10 + 2q-8 + 2q-6 + 2q-4 + q-2 + 1

HOMFLY-PT Polynomial:

- az-1 - az + a3z-1 - a5z-1 - a5z + 2a7z-1 + 3a7z + a7z3 - a9z-1 - a9z

Kauffman Polynomial:

az-1 - az - a2 + a3z-1 - a3z + a5z-1 - 5a5z + 5a5z3 - a5z5 - 4a6 + 15a6z2 - 16a6z4 + 7a6z6 - a6z8 + 2a7z-1 - 8a7z + 16a7z3 - 16a7z5 + 7a7z7 - a7z9 - 7a8 + 26a8z2 - 31a8z4 + 14a8z6 - 2a8z8 + a9z-1 - 3a9z + 11a9z3 - 15a9z5 + 7a9z7 - a9z9 - 3a10 + 11a10z2 - 15a10z4 + 7a10z6 - a10z8

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = 0                   1 j = -2                 1 2 j = -4               1 1 1 j = -6             1 2     j = -8           1 1 1     j = -10         1 2 1       j = -12       1 1           j = -14       1 1           j = -16   1 1               j = -18                     j = -20 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[11, NonAlternating, 15]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 15]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[5, 10, 6, 11], > X[8, 4, 9, 3], X[11, 22, 12, 5], X[13, 20, 14, 21], X[19, 14, 20, 15], > X[21, 12, 22, 13], X[9, 18, 10, 19], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-4, -1, 2, -5, -10, 4, -6, 9, -7, 8, 11, -2, 3, 10, > -8, 7, -9, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) -(17/2) -(15/2) -(13/2) -(11/2) -(9/2) -(5/2) q - q + q - q + q - q - q - 1 > ------- Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 -26 -24 -18 -16 -14 -12 -10 2 2 1 - q - q - q - q + q + q + q + q + q + -- + -- + 8 6 q q 2 -2 > -- + q 4 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 15]][a, z]
Out[8]=	3 5 7 9 a a a 2 a a 5 7 9 7 3 -(-) + -- - -- + ---- - -- - a z - a z + 3 a z - a z + a z z z z z z
In[9]:=	Kauffman[Link[11, NonAlternating, 15]][a, z]
Out[9]=	3 5 7 9 2 6 8 10 a a a 2 a a 3 5 -a - 4 a - 7 a - 3 a + - + -- + -- + ---- + -- - a z - a z - 5 a z - z z z z z 7 9 6 2 8 2 10 2 5 3 7 3 > 8 a z - 3 a z + 15 a z + 26 a z + 11 a z + 5 a z + 16 a z + 9 3 6 4 8 4 10 4 5 5 7 5 9 5 > 11 a z - 16 a z - 31 a z - 15 a z - a z - 16 a z - 15 a z + 6 6 8 6 10 6 7 7 9 7 6 8 8 8 > 7 a z + 14 a z + 7 a z + 7 a z + 7 a z - a z - 2 a z - 10 8 7 9 9 9 > a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 1 1 1 1 1 1 1 + q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 20 9 16 8 16 7 14 6 12 6 14 5 12 5 q q t q t q t q t q t q t q t 1 2 1 1 1 1 1 2 1 > ------ + ------ + ----- + ------ + ----- + ----- + ----- + ----- + ----- + 10 5 10 4 8 4 10 3 8 3 6 3 8 2 6 2 4 2 q t q t q t q t q t q t q t q t q t 1 1 > ---- + ---- 4 2 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n15

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