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L11n192 L11n194

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X9,21,10,20 X4758 X16,5,17,6 X6,15,1,16 X22,17,7,18 X18,13,19,14 X14,21,15,22 X2,11,3,12 X12,3,13,4 X19,11,20,10

Gauss Code:

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

Jones Polynomial:

q-21/2 - 4q-19/2 + 7q-17/2 - 10q-15/2 + 12q-13/2 - 13q-11/2 + 11q-9/2 - 9q-7/2 + 5q-5/2 - 2q-3/2

A2 (sl(3)) Invariant:

- q-32 + 2q-30 - 2q-26 + 3q-24 - q-22 + 2q-20 + 2q-18 + 3q-14 - 2q-12 + 2q-10 + q-8 - 2q-6 + 2q-4

HOMFLY-PT Polynomial:

- 2a3z - 2a3z3 - a5z-1 - 2a5z + a5z5 + a7z-1 + a7z + a7z3 + a7z5 - a9z3

Kauffman Polynomial:

2a3z - 3a3z3 + 2a4z2 - 4a4z4 - a4z6 + a5z-1 - 3a5z + a5z3 - 3a5z7 - a6 + 4a6z2 - 6a6z4 + 5a6z6 - 4a6z8 + a7z-1 - 3a7z - a7z3 + 9a7z5 - 3a7z7 - 2a7z9 + a8z2 - 8a8z4 + 19a8z6 - 9a8z8 + 2a9z - 12a9z3 + 20a9z5 - 4a9z7 - 2a9z9 - 2a10z2 - 4a10z4 + 12a10z6 - 5a10z8 - 7a11z3 + 11a11z5 - 4a11z7 - a12z2 + 2a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                   2 j = -4                 4 1 j = -6               5 1   j = -8             6 4     j = -10           7 5       j = -12         5 6         j = -14       5 7           j = -16     3 6             j = -18   1 4               j = -20   3                 j = -22 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, 193]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 193]]
Out[4]=	PD[X[8, 1, 9, 2], X[9, 21, 10, 20], X[4, 7, 5, 8], X[16, 5, 17, 6], > X[6, 15, 1, 16], X[22, 17, 7, 18], X[18, 13, 19, 14], X[14, 21, 15, 22], > X[2, 11, 3, 12], X[12, 3, 13, 4], X[19, 11, 20, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 10, -3, 4, -5}, > {3, -1, -2, 11, 9, -10, 7, -8, 5, -4, 6, -7, -11, 2, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 4 7 10 12 13 11 9 5 2 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 2 2 3 -22 2 2 3 2 2 -8 2 2 -q + --- - --- + --- - q + --- + --- + --- - --- + --- + q - -- + -- 30 26 24 20 18 14 12 10 6 4 q q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 193]][a, z]
Out[8]=	5 7 a a 3 5 7 3 3 7 3 9 3 5 5 7 5 -(--) + -- - 2 a z - 2 a z + a z - 2 a z + a z - a z + a z + a z z z
In[9]:=	Kauffman[Link[11, NonAlternating, 193]][a, z]
Out[9]=	5 7 6 a a 3 5 7 9 4 2 6 2 8 2 -a + -- + -- + 2 a z - 3 a z - 3 a z + 2 a z + 2 a z + 4 a z + a z - z z 10 2 12 2 3 3 5 3 7 3 9 3 11 3 > 2 a z - a z - 3 a z + a z - a z - 12 a z - 7 a z - 4 4 6 4 8 4 10 4 12 4 7 5 9 5 > 4 a z - 6 a z - 8 a z - 4 a z + 2 a z + 9 a z + 20 a z + 11 5 4 6 6 6 8 6 10 6 12 6 5 7 > 11 a z - a z + 5 a z + 19 a z + 12 a z - a z - 3 a z - 7 7 9 7 11 7 6 8 8 8 10 8 7 9 > 3 a z - 4 a z - 4 a z - 4 a z - 9 a z - 5 a z - 2 a z - 9 9 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 3 1 4 3 6 5 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q t q t q t q t q t q t q t 7 5 6 7 5 6 4 5 1 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 4 > ---- 4 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n193

L11n192 L11n194