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L11n342 L11n344

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,19,12,18 X7,14,8,15 X13,8,14,9 X15,13,16,22 X17,21,18,20 X21,17,22,16 X19,5,20,12 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-6 - q-5 + 4q-4 - 5q-3 + 8q-2 - 8q-1 + 8 - 6q + 5q2 - 2q3

A2 (sl(3)) Invariant:

q-20 + 2q-18 + 2q-16 + 5q-14 + 5q-12 + 2q-10 + 4q-8 + q-6 + q-4 + 2q-2 + 3q2 - q4 + q6 + q8 - 2q10

HOMFLY-PT Polynomial:

- a-2 - 2a-2z2 + 3 + 4z2 + 2z4 + a2z-2 + a2z4 - 2a4z-2 - 3a4 - 2a4z2 + a6z-2 + a6

Kauffman Polynomial:

- 2a-3z + 3a-3z3 + 2a-2 - 7a-2z2 + 4a-2z4 + a-2z6 - 4a-1z + 9a-1z3 - 6a-1z5 + 3a-1z7 + 5 - 20z2 + 23z4 - 10z6 + 3z8 - az3 + az9 + a2z-2 - a2 - 8a2z2 + 15a2z4 - 12a2z6 + 4a2z8 - 2a3z-1 + 8a3z - 10a3z3 + 4a3z5 - 2a3z7 + a3z9 + 2a4z-2 - 8a4 + 13a4z2 - 9a4z4 + a4z8 - 2a5z-1 + 6a5z - 3a5z3 - 2a5z5 + a5z7 + a6z-2 - 5a6 + 8a6z2 - 5a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 7                   2 j = 5                 3   j = 3               3 2   j = 1             5 3     j = -1           5 5       j = -3         3 3         j = -5       2 5           j = -7     2 3             j = -9   1 4               j = -11                     j = -13 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, 343]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 343]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 19, 12, 18], X[7, 14, 8, 15], > X[13, 8, 14, 9], X[15, 13, 16, 22], X[17, 21, 18, 20], X[21, 17, 22, 16], > X[19, 5, 20, 12], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9}, > {-5, 4, -6, 8, -7, 3, -9, 7, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-6 -5 4 5 8 8 2 3 8 + q - q + -- - -- + -- - - - 6 q + 5 q - 2 q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 2 5 5 2 4 -6 -4 2 2 4 6 q + --- + --- + --- + --- + --- + -- + q + q + -- + 3 q - q + q + 18 16 14 12 10 8 2 q q q q q q q 8 10 > q - 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 343]][a, z]
Out[8]=	2 4 6 2 -2 4 6 a 2 a a 2 2 z 4 2 4 2 4 3 - a - 3 a + a + -- - ---- + -- + 4 z - ---- - 2 a z + 2 z + a z 2 2 2 2 z z z a
In[9]:=	Kauffman[Link[11, NonAlternating, 343]][a, z]
Out[9]=	2 4 6 3 5 2 2 4 6 a 2 a a 2 a 2 a 2 z 4 z 3 5 + -- - a - 8 a - 5 a + -- + ---- + -- - ---- - ---- - --- - --- + 8 a z + 2 2 2 2 z z 3 a a z z z a 2 3 3 5 2 7 z 2 2 4 2 6 2 3 z 9 z 3 > 6 a z - 20 z - ---- - 8 a z + 13 a z + 8 a z + ---- + ---- - a z - 2 3 a a a 4 5 3 3 5 3 4 4 z 2 4 4 4 6 4 6 z > 10 a z - 3 a z + 23 z + ---- + 15 a z - 9 a z - 5 a z - ---- + 2 a a 6 7 3 5 5 5 6 z 2 6 6 6 3 z 3 7 > 4 a z - 2 a z - 10 z + -- - 12 a z + a z + ---- - 2 a z + 2 a a 5 7 8 2 8 4 8 9 3 9 > a z + 3 z + 4 a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 1 4 2 3 2 5 3 - + 5 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 9 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t q t 3 5 3 3 2 5 2 7 3 > ---- + --- + 3 q t + 3 q t + 2 q t + 3 q t + 2 q t 3 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n343

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