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L11n322 L11n324

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X9,18,10,19 X11,21,12,20 X19,22,20,13 X13,12,14,5 X4,17,1,18 X21,11,22,10

Gauss Code:

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

Jones Polynomial:

- q-10 + q-9 + 2q-7 - q-6 + 2q-5 - q-4 + 2q-3 - q-2 + q-1

A2 (sl(3)) Invariant:

- q-34 - q-30 + 2q-26 + 3q-24 + 4q-22 + 3q-20 + 4q-18 + 3q-16 + 3q-14 + 2q-12 + 2q-10 + q-8 + q-6 + q-4

HOMFLY-PT Polynomial:

a4z-2 + 5a4 + 10a4z2 + 6a4z4 + a4z6 - 2a6z-2 - 9a6 - 16a6z2 - 16a6z4 - 7a6z6 - a6z8 + a8z-2 + 6a8 + 10a8z2 + 6a8z4 + a8z6 - 2a10 - a10z2

Kauffman Polynomial:

- a4z-2 + 6a4 - 15a4z2 + 16a4z4 - 7a4z6 + a4z8 + 2a5z-1 - 5a5z + 9a5z5 - 6a5z7 + a5z9 - 2a6z-2 + 13a6 - 35a6z2 + 42a6z4 - 20a6z6 + 3a6z8 + 2a7z-1 - 8a7z + 6a7z3 + 4a7z5 - 5a7z7 + a7z9 - a8z-2 + 9a8 - 24a8z2 + 27a8z4 - 13a8z6 + 2a8z8 - 3a9z + 6a9z3 - 5a9z5 + a9z7 - 3a10z2 + a10z4 + a11z - a12 + a12z2 + a13z

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = -1                   1 j = -3                     j = -5               2 1   j = -7           1 1 1     j = -9           2 1       j = -11       2 2 1         j = -13     1 4 2           j = -15   1 1 2             j = -17   2 1               j = -19 1 1                 j = -21 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, 323]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 323]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[3, 8, 4, 9], X[15, 2, 16, 3], > X[16, 7, 17, 8], X[9, 18, 10, 19], X[11, 21, 12, 20], X[19, 22, 20, 13], > X[13, 12, 14, 5], X[4, 17, 1, 18], X[21, 11, 22, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -10}, {-2, -1, 5, 3, -6, 11, -7, 9}, > {-9, 2, -4, -5, 10, 6, -8, 7, -11, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-10 -9 2 -6 2 -4 2 -2 1 -q + q + -- - q + -- - q + -- - q + - 7 5 3 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -30 2 3 4 3 4 3 3 2 2 -8 -q - q + --- + --- + --- + --- + --- + --- + --- + --- + --- + q + 26 24 22 20 18 16 14 12 10 q q q q q q q q q -6 -4 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 323]][a, z]
Out[8]=	4 6 8 4 6 8 10 a 2 a a 4 2 6 2 8 2 5 a - 9 a + 6 a - 2 a + -- - ---- + -- + 10 a z - 16 a z + 10 a z - 2 2 2 z z z 10 2 4 4 6 4 8 4 4 6 6 6 8 6 6 8 > a z + 6 a z - 16 a z + 6 a z + a z - 7 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 323]][a, z]
Out[9]=	4 6 8 5 7 4 6 8 12 a 2 a a 2 a 2 a 5 7 6 a + 13 a + 9 a - a - -- - ---- - -- + ---- + ---- - 5 a z - 8 a z - 2 2 2 z z z z z 9 11 13 4 2 6 2 8 2 10 2 > 3 a z + a z + a z - 15 a z - 35 a z - 24 a z - 3 a z + 12 2 7 3 9 3 4 4 6 4 8 4 10 4 > a z + 6 a z + 6 a z + 16 a z + 42 a z + 27 a z + a z + 5 5 7 5 9 5 4 6 6 6 8 6 5 7 > 9 a z + 4 a z - 5 a z - 7 a z - 20 a z - 13 a z - 6 a z - 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > 5 a z + a z + a z + 3 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 2 1 1 1 2 1 1 1 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 21 7 19 7 19 6 17 6 15 6 17 5 15 5 q q t q t q t q t q t q t q t 1 2 4 2 2 2 1 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 13 5 15 4 13 4 11 4 13 3 11 3 11 2 9 2 q t q t q t q t q t q t q t q t 2 1 1 1 t t > ----- + ---- + ---- + -- + -- 7 2 9 7 5 q q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n323

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