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L11n362 L11n364

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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X4,13,1,14 X5,18,6,19 X8493 X9,21,10,20 X19,11,20,10 X15,22,16,17 X17,16,18,5 X21,14,22,15 X2,12,3,11

Gauss Code:

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

Jones Polynomial:

- q-8 + 2q-7 - 3q-6 + 4q-5 - 3q-4 + 4q-3 - q-2 + q-1 + 1 - q + q2

A2 (sl(3)) Invariant:

- q-24 - 2q-20 + 2q-16 + 2q-14 + 5q-12 + 3q-10 + 5q-8 + 3q-6 + 2q-4 + 3q-2 + 1 + 2q2 + q4 + q6

HOMFLY-PT Polynomial:

z-2 + 4 + 4z2 + z4 - 2a2z-2 - 10a2 - 14a2z2 - 7a2z4 - a2z6 + a4z-2 + 9a4 + 13a4z2 + 6a4z4 + a4z6 - 3a6 - 3a6z2 - a6z4

Kauffman Polynomial:

- z-2 + 6 - 13z2 + 15z4 - 7z6 + z8 + 2az-1 - 4az - 5az3 + 13az5 - 7az7 + az9 - 2a2z-2 + 15a2 - 41a2z2 + 43a2z4 - 17a2z6 + 2a2z8 + 2a3z-1 - 9a3z + 3a3z3 + 11a3z5 - 7a3z7 + a3z9 - a4z-2 + 13a4 - 34a4z2 + 37a4z4 - 15a4z6 + 2a4z8 - 6a5z + 14a5z3 - 9a5z5 + 2a5z7 + 2a6 - 3a6z2 + 3a6z4 - 3a6z6 + a6z8 + 3a7z3 - 6a7z5 + 2a7z7 - a8 + 3a8z2 - 6a8z4 + 2a8z6 + a9z - 3a9z3 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 5                       1 j = 3                         j = 1                   1 1   j = -1               3 1       j = -3             1 2 1       j = -5           4 2 1         j = -7         2 3             j = -9       2 2 1             j = -11     1 2                 j = -13   1 2                   j = -15   1                     j = -17 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 363]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 363]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[5, 18, 6, 19], > X[8, 4, 9, 3], X[9, 21, 10, 20], X[19, 11, 20, 10], X[15, 22, 16, 17], > X[17, 16, 18, 5], X[21, 14, 22, 15], X[2, 12, 3, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-9, 4, -7, 6, -10, 8}, > {-4, -1, 2, -5, -6, 7, 11, -2, 3, 10, -8, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-8 2 3 4 3 4 -2 1 2 1 - q + -- - -- + -- - -- + -- - q + - - q + q 7 6 5 4 3 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 2 2 5 3 5 3 2 3 2 4 6 1 - q - --- + --- + --- + --- + --- + -- + -- + -- + -- + 2 q + q + q 20 16 14 12 10 8 6 4 2 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 363]][a, z]
Out[8]=	2 4 2 4 6 -2 2 a a 2 2 2 4 2 4 - 10 a + 9 a - 3 a + z - ---- + -- + 4 z - 14 a z + 13 a z - 2 2 z z 6 2 4 2 4 4 4 6 4 2 6 4 6 > 3 a z + z - 7 a z + 6 a z - a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 363]][a, z]
Out[9]=	2 4 3 2 4 6 8 -2 2 a a 2 a 2 a 3 6 + 15 a + 13 a + 2 a - a - z - ---- - -- + --- + ---- - 4 a z - 9 a z - 2 2 z z z z 5 9 2 2 2 4 2 6 2 8 2 3 > 6 a z + a z - 13 z - 41 a z - 34 a z - 3 a z + 3 a z - 5 a z + 3 3 5 3 7 3 9 3 4 2 4 4 4 > 3 a z + 14 a z + 3 a z - 3 a z + 15 z + 43 a z + 37 a z + 6 4 8 4 5 3 5 5 5 7 5 9 5 6 > 3 a z - 6 a z + 13 a z + 11 a z - 9 a z - 6 a z + a z - 7 z - 2 6 4 6 6 6 8 6 7 3 7 5 7 > 17 a z - 15 a z - 3 a z + 2 a z - 7 a z - 7 a z + 2 a z + 7 7 8 2 8 4 8 6 8 9 3 9 > 2 a z + z + 2 a z + 2 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 2 3 1 1 1 2 1 2 2 q + -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q q t q t q t q t q t q t q t 2 2 1 3 4 2 1 t t 2 > ----- + ----- + ----- + ----- + ----- + ---- + ---- + -- + - + q t + 9 3 7 3 9 2 7 2 5 2 5 3 3 q q t q t q t q t q t q t q t q 3 5 4 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n363

L11n362 L11n364