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L11n100 L11n102

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X13,21,14,20 X16,7,17,8 X8,19,9,20 X18,9,19,10 X10,17,11,18 X15,5,16,22 X21,15,22,14 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- 2q-13/2 + 4q-11/2 - 7q-9/2 + 10q-7/2 - 12q-5/2 + 11q-3/2 - 10q-1/2 + 7q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

q-22 + 3q-20 + q-16 + q-14 - 4q-12 + q-10 + 2q-6 + 3q-4 - q-2 + 3 - 2q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 - az-1 - az - az3 - az5 + 2a3z-1 - a3z3 - a3z5 - 2a5z-1 - a5z + a5z3 + a7z-1

Kauffman Polynomial:

- a-2z4 + 3a-1z3 - 4a-1z5 - z2 + 8z4 - 7z6 + az-1 - 2az - az3 + 8az5 - 7az7 - 2a2z2 + 6a2z4 - 4a2z8 + 2a3z-1 - 6a3z - a3z3 + 13a3z5 - 7a3z7 - a3z9 + a4 - 4a4z2 + a4z4 + 7a4z6 - 5a4z8 + 2a5z-1 - 9a5z + 11a5z3 - 2a5z5 - a5z9 - 3a6z2 + 4a6z4 - a6z8 + a7z-1 - 5a7z + 8a7z3 - 3a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                   1 j = 4                 3   j = 2               4 1   j = 0             6 3     j = -2           6 5       j = -4         6 5         j = -6       4 6           j = -8     3 6             j = -10   2 5               j = -12   2                 j = -14 2

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, 101]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 101]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[13, 21, 14, 20], X[16, 7, 17, 8], > X[8, 19, 9, 20], X[18, 9, 19, 10], X[10, 17, 11, 18], X[15, 5, 16, 22], > X[21, 15, 22, 14], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -5, 6, -7, 11, -2, -3, 9, -8, -4, 7, > -6, 5, 3, -9, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-2 4 7 10 12 11 10 3/2 5/2 ----- + ----- - ---- + ---- - ---- + ---- - ------- + 7 Sqrt[q] - 4 q + q 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 3 -16 -14 4 -10 2 3 -2 2 6 8 3 + q + --- + q + q - --- + q + -- + -- - q - 2 q + 2 q - q 20 12 6 4 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 101]][a, z]
Out[8]=	3 5 7 3 a 2 a 2 a a 5 z 3 3 3 5 3 5 3 5 -(-) + ---- - ---- + -- - a z - a z + -- - a z - a z + a z - a z - a z z z z z a
In[9]:=	Kauffman[Link[11, NonAlternating, 101]][a, z]
Out[9]=	3 5 7 4 a 2 a 2 a a 3 5 7 2 2 2 a + - + ---- + ---- + -- - 2 a z - 6 a z - 9 a z - 5 a z - z - 2 a z - z z z z 3 4 4 2 6 2 3 z 3 3 3 5 3 7 3 4 z > 4 a z - 3 a z + ---- - a z - a z + 11 a z + 8 a z + 8 z - -- + a 2 a 5 2 4 4 4 6 4 4 z 5 3 5 5 5 7 5 > 6 a z + a z + 4 a z - ---- + 8 a z + 13 a z - 2 a z - 3 a z - a 6 4 6 7 3 7 2 8 4 8 6 8 3 9 > 7 z + 7 a z - 7 a z - 7 a z - 4 a z - 5 a z - a z - a z - 5 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 2 2 2 5 3 6 4 6 6 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 6 5 6 2 2 2 4 2 6 3 > ----- + ---- + ---- + 3 t + 4 q t + q t + 3 q t + q t 4 2 4 2 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n101

L11n100 L11n102