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L11n185 L11n187

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X18,10,19,9 X19,22,20,7 X13,20,14,21 X21,14,22,15 X10,16,11,15 X16,6,17,5 X2738 X4,11,5,12 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

q-11/2 - 3q-9/2 + 5q-7/2 - 8q-5/2 + 8q-3/2 - 9q-1/2 + 7q1/2 - 5q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-16 + q-14 - q-12 + 2q-10 + 3q-8 + q-6 + 4q-4 - q-2 + 2 - q2 - q4 + q6 - q8 + q10

HOMFLY-PT Polynomial:

- 2a-1z - 3a-1z3 - a-1z5 - az-1 + 3az + 8az3 + 5az5 + az7 + a3z-1 - 2a3z - 3a3z3 - a3z5

Kauffman Polynomial:

2a-3z3 - a-3z5 - 2a-2z2 + 7a-2z4 - 3a-2z6 + 3a-1z - 7a-1z3 + 10a-1z5 - 4a-1z7 - 3z4 + 6z6 - 3z8 + az-1 + 4az - 14az3 + 10az5 - 2az7 - az9 - a2 + 4a2z2 - 14a2z4 + 11a2z6 - 4a2z8 + a3z-1 - 4a3z5 + 2a3z7 - a3z9 + 4a4z2 - 5a4z4 + 2a4z6 - a4z8 - a5z + 5a5z3 - 3a5z5 + 2a6z2 - a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                   1 j = 6                 2   j = 4               3 1   j = 2             4 2     j = 0           5 3       j = -2         4 5         j = -4       4 4           j = -6     2 5             j = -8   1 3               j = -10   2                 j = -12 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, 186]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 186]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[18, 10, 19, 9], X[19, 22, 20, 7], > X[13, 20, 14, 21], X[21, 14, 22, 15], X[10, 16, 11, 15], X[16, 6, 17, 5], > X[2, 7, 3, 8], X[4, 11, 5, 12], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 8, -11}, > {9, -1, 3, -7, 10, -2, -5, 6, 7, -8, 11, -3, -4, 5, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 5 8 8 9 3/2 5/2 q - ---- + ---- - ---- + ---- - ------- + 7 Sqrt[q] - 5 q + 3 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 7/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 -12 2 3 -6 4 -2 2 4 6 8 10 2 - q + q - q + --- + -- + q + -- - q - q - q + q - q + q 10 8 4 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 186]][a, z]
Out[8]=	3 3 5 a a 2 z 3 3 z 3 3 3 z 5 -(-) + -- - --- + 3 a z - 2 a z - ---- + 8 a z - 3 a z - -- + 5 a z - z z a a a 3 5 7 > a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 186]][a, z]
Out[9]=	3 2 3 2 a a 3 z 5 2 z 2 2 4 2 6 2 2 z -a + - + -- + --- + 4 a z - a z - ---- + 4 a z + 4 a z + 2 a z + ---- - z z a 2 3 a a 3 4 5 7 z 3 5 3 4 7 z 2 4 4 4 6 4 z > ---- - 14 a z + 5 a z - 3 z + ---- - 14 a z - 5 a z - a z - -- + a 2 3 a a 5 6 10 z 5 3 5 5 5 6 3 z 2 6 4 6 > ----- + 10 a z - 4 a z - 3 a z + 6 z - ---- + 11 a z + 2 a z - a 2 a 7 4 z 7 3 7 8 2 8 4 8 9 3 9 > ---- - 2 a z + 2 a z - 3 z - 4 a z - a z - a z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 2 1 3 2 5 4 4 5 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 4 2 2 2 4 2 4 3 6 3 8 4 > ---- + 3 t + 4 q t + 2 q t + 3 q t + q t + 2 q t + q t 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n186

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