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L11n191 L11n193

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X14,9,15,10 X4758 X16,6,17,5 X18,16,19,15 X6,18,1,17 X19,7,20,22 X11,20,12,21 X21,10,22,11 X2,14,3,13 X12,4,13,3

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 5q-5/2 - 7q-3/2 + 8q-1/2 - 9q1/2 + 7q3/2 - 6q5/2 + 3q7/2 - q9/2

A2 (sl(3)) Invariant:

- q-14 + q-12 - q-8 + 2q-6 - q-4 + q-2 + 1 + 3q4 + 3q8 + q10 - q12 + q14

HOMFLY-PT Polynomial:

- a-3z-1 - a-3z - a-3z3 + a-1z-1 + 2a-1z + 2a-1z3 + a-1z5 + az + 2az3 + az5 - a3z - a3z3

Kauffman Polynomial:

a-5z - a-5z3 + 2a-4z2 - 3a-4z4 + a-3z-1 - 2a-3z3 - a-3z7 - a-2 + 4a-2z2 - 9a-2z4 + 5a-2z6 - 2a-2z8 + a-1z-1 - 6a-1z3 + 6a-1z5 - a-1z7 - a-1z9 + 4z2 - 12z4 + 14z6 - 5z8 + 3az - 13az3 + 16az5 - 3az7 - az9 - 3a2z4 + 8a2z6 - 3a2z8 + 2a3z - 8a3z3 + 10a3z5 - 3a3z7 - 2a4z2 + 3a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 10                   1 j = 8                 2   j = 6               4 1   j = 4             4 3     j = 2           5 3       j = 0         4 5         j = -2       3 4           j = -4     2 4             j = -6   1 3               j = -8   2                 j = -10 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, 192]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 192]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 9, 15, 10], X[4, 7, 5, 8], X[16, 6, 17, 5], > X[18, 16, 19, 15], X[6, 18, 1, 17], X[19, 7, 20, 22], X[11, 20, 12, 21], > X[21, 10, 22, 11], X[2, 14, 3, 13], X[12, 4, 13, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 11, -3, 4, -6}, > {3, -1, 2, 9, -8, -11, 10, -2, 5, -4, 6, -5, -7, 8, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 5 7 8 3/2 5/2 7/2 q - ---- + ---- - ---- + ------- - 9 Sqrt[q] + 7 q - 6 q + 3 q - 7/2 5/2 3/2 Sqrt[q] q q q 9/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 -8 2 -4 -2 4 8 10 12 14 1 - q + q - q + -- - q + q + 3 q + 3 q + q - q + q 6 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 192]][a, z]
Out[8]=	3 3 5 1 1 z 2 z 3 z 2 z 3 3 3 z 5 -(----) + --- - -- + --- + a z - a z - -- + ---- + 2 a z - a z + -- + a z 3 a z 3 a 3 a a a z a a
In[9]:=	Kauffman[Link[11, NonAlternating, 192]][a, z]
Out[9]=	2 2 3 -2 1 1 z 3 2 2 z 4 z 4 2 z -a + ---- + --- + -- + 3 a z + 2 a z + 4 z + ---- + ---- - 2 a z - -- - 3 a z 5 4 2 5 a z a a a a 3 3 4 4 2 z 6 z 3 3 3 4 3 z 9 z 2 4 4 4 > ---- - ---- - 13 a z - 8 a z - 12 z - ---- - ---- - 3 a z + 3 a z + 3 a 4 2 a a a 5 6 7 7 6 z 5 3 5 6 5 z 2 6 4 6 z z > ---- + 16 a z + 10 a z + 14 z + ---- + 8 a z - a z - -- - -- - a 2 3 a a a 8 9 7 3 7 8 2 z 2 8 z 9 > 3 a z - 3 a z - 5 z - ---- - 3 a z - -- - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 3 2 4 3 4 4 5 + 5 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 10 4 > 3 q t + 4 q t + 3 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n192

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