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L11n146 L11n148

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DrawMorseLink

PD Presentation:

X8192 X18,9,19,10 X6718 X22,19,7,20 X5,13,6,12 X3,10,4,11 X15,5,16,4 X11,16,12,17 X20,13,21,14 X14,21,15,22 X17,2,18,3

Gauss Code:

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

Jones Polynomial:

q-19/2 - 2q-17/2 + 4q-15/2 - 5q-13/2 + 5q-11/2 - 6q-9/2 + 4q-7/2 - 4q-5/2 + 2q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- 2q-28 - q-26 - 2q-24 + q-20 + q-18 + 4q-16 + q-14 + 3q-12 + q-10 + q-8 + q-6 + q-2

HOMFLY-PT Polynomial:

- a3z-1 - 4a3z - 4a3z3 - a3z5 + 3a5z + 7a5z3 + 5a5z5 + a5z7 + 2a7z-1 - 3a7z3 - a7z5 - a9z-1

Kauffman Polynomial:

- a3z-1 + 5a3z - 8a3z3 + 5a3z5 - a3z7 + a4 + a4z2 - 10a4z4 + 9a4z6 - 2a4z8 + 5a5z - 16a5z3 + 10a5z5 + a5z7 - a5z9 - 3a6 + 8a6z2 - 17a6z4 + 16a6z6 - 4a6z8 + 2a7z-1 - 2a7z - 3a7z3 + 5a7z5 + a7z7 - a7z9 - 5a8 + 13a8z2 - 11a8z4 + 7a8z6 - 2a8z8 + a9z-1 - 2a9z + 3a9z3 - a9z7 - 2a10 + 5a10z2 - 4a10z4 - 2a11z3 - a12z2

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                   1 j = -2                 1   j = -4               3 1   j = -6             2 2     j = -8           4 2       j = -10         2 3         j = -12       3 3           j = -14     1 2             j = -16   1 3               j = -18   1                 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, 147]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 147]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 9, 19, 10], X[6, 7, 1, 8], X[22, 19, 7, 20], > X[5, 13, 6, 12], X[3, 10, 4, 11], X[15, 5, 16, 4], X[11, 16, 12, 17], > X[20, 13, 21, 14], X[14, 21, 15, 22], X[17, 2, 18, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -6, 7, -5, -3}, > {3, -1, 2, 6, -8, 5, 9, -10, -7, 8, -11, -2, 4, -9, 10, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 2 4 5 5 6 4 4 2 1 q - ----- + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 -26 2 -20 -18 4 -14 3 -10 -8 -6 -2 --- - q - --- + q + q + --- + q + --- + q + q + q + q 28 24 16 12 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 147]][a, z]
Out[8]=	3 7 9 a 2 a a 3 5 3 3 5 3 7 3 3 5 -(--) + ---- - -- - 4 a z + 3 a z - 4 a z + 7 a z - 3 a z - a z + z z z 5 5 7 5 5 7 > 5 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 147]][a, z]
Out[9]=	3 7 9 4 6 8 10 a 2 a a 3 5 7 9 a - 3 a - 5 a - 2 a - -- + ---- + -- + 5 a z + 5 a z - 2 a z - 2 a z + z z z 4 2 6 2 8 2 10 2 12 2 3 3 5 3 > a z + 8 a z + 13 a z + 5 a z - a z - 8 a z - 16 a z - 7 3 9 3 11 3 4 4 6 4 8 4 10 4 > 3 a z + 3 a z - 2 a z - 10 a z - 17 a z - 11 a z - 4 a z + 3 5 5 5 7 5 4 6 6 6 8 6 3 7 > 5 a z + 10 a z + 5 a z + 9 a z + 16 a z + 7 a z - a z + 5 7 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > a z + a z - a z - 2 a z - 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 1 1 3 1 2 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 20 7 18 6 16 6 16 5 14 5 14 4 12 4 q q q t q t q t q t q t q t q t 3 2 3 4 2 2 t t 2 > ------ + ------ + ------ + ----- + ---- + ---- + -- + -- + t 12 3 10 3 10 2 8 2 8 6 4 2 q t q t q t q t q t q t q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n147

L11n146 L11n148