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L11n149 L11n151

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DrawMorseLink

PD Presentation:

X8192 X18,9,19,10 X6718 X22,19,7,20 X12,5,13,6 X3,10,4,11 X15,5,16,4 X11,16,12,17 X20,13,21,14 X14,21,15,22 X2,18,3,17

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

A2 (sl(3)) Invariant:

- q-32 - q-28 - q-26 + q-24 - q-22 + q-16 + 3q-14 + 2q-12 + 3q-10 + 2q-8 + q-4

HOMFLY-PT Polynomial:

- a3z-1 - 2a3z - a3z3 - 3a5z - 2a5z3 + 2a7z-1 + 6a7z + 4a7z3 + a7z5 - a9z-1 - 2a9z - a9z3

Kauffman Polynomial:

- a3z-1 + 2a3z - a3z3 + a4 - a4z4 - 3a5z + 3a5z3 - 2a5z5 - 3a6 + 11a6z2 - 13a6z4 + 5a6z6 - a6z8 + 2a7z-1 - 8a7z + 13a7z3 - 12a7z5 + 5a7z7 - a7z9 - 5a8 + 20a8z2 - 25a8z4 + 14a8z6 - 3a8z8 + a9z-1 - 2a9z + 2a9z3 - 2a9z5 + 3a9z7 - a9z9 - 2a10 + 6a10z2 - 9a10z4 + 8a10z6 - 2a10z8 + a11z - 7a11z3 + 8a11z5 - 2a11z7 - 3a12z2 + 4a12z4 - 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                   1 j = -4                 1 1 j = -6               3     j = -8             2 2     j = -10           3 2       j = -12         2 2         j = -14       2 3           j = -16     1 2             j = -18   1 2               j = -20   1                 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, 150]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 150]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 9, 19, 10], X[6, 7, 1, 8], X[22, 19, 7, 20], > X[12, 5, 13, 6], 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[2, 18, 3, 17]]
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]=	-(21/2) 2 3 4 5 5 4 4 -(5/2) q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + q - 19/2 17/2 15/2 13/2 11/2 9/2 7/2 q q q q q q q -(3/2) > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -28 -26 -24 -22 -16 3 2 3 2 -4 -q - q - q + q - q + q + --- + --- + --- + -- + q 14 12 10 8 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 150]][a, z]
Out[8]=	3 7 9 a 2 a a 3 5 7 9 3 3 5 3 -(--) + ---- - -- - 2 a z - 3 a z + 6 a z - 2 a z - a z - 2 a z + z z z 7 3 9 3 7 5 > 4 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 150]][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 - -- + ---- + -- + 2 a z - 3 a z - 8 a z - 2 a z + z z z 11 6 2 8 2 10 2 12 2 3 3 5 3 > a z + 11 a z + 20 a z + 6 a z - 3 a z - a z + 3 a z + 7 3 9 3 11 3 4 4 6 4 8 4 10 4 > 13 a z + 2 a z - 7 a z - a z - 13 a z - 25 a z - 9 a z + 12 4 5 5 7 5 9 5 11 5 6 6 8 6 > 4 a z - 2 a z - 12 a z - 2 a z + 8 a z + 5 a z + 14 a z + 10 6 12 6 7 7 9 7 11 7 6 8 8 8 > 8 a z - a z + 5 a z + 3 a z - 2 a z - a z - 3 a z - 10 8 7 9 9 9 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 -2 1 1 1 2 1 2 2 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q t q t q t q t q t q t q t 3 2 2 3 2 2 2 3 1 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 4 q t q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n150

L11n149 L11n151