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L11n346 L11n348

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DrawMorseLink

PD Presentation:

X6172 X3,15,4,14 X11,20,12,21 X7,18,8,19 X17,22,18,13 X16,9,17,10 X10,15,11,16 X19,12,20,5 X21,8,22,9 X2536 X13,1,14,4

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-28 + q-26 + 3q-24 + 2q-22 + 3q-20 + 3q-18 + 2q-16 + 4q-14 + q-12 + 3q-10 + q-8 + q-6 + q-4 + 1

HOMFLY-PT Polynomial:

2a2 + 3a2z2 + a2z4 + a4z-2 + a4 - 3a4z2 - 4a4z4 - a4z6 - 2a6z-2 - 5a6 - 2a6z2 + a8z-2 + 2a8 + a8z2

Kauffman Polynomial:

- 2a2 + 5a2z2 - 4a2z4 + a2z6 + a3z + 4a3z3 - 7a3z5 + 2a3z7 - a4z-2 + 3a4 + 3a4z2 - 6a4z4 - a4z6 + a4z8 + 2a5z-1 - 8a5z + 12a5z3 - 11a5z5 + 3a5z7 - 2a6z-2 + 9a6 - 13a6z2 + 8a6z4 - 4a6z6 + a6z8 + 2a7z-1 - 8a7z + 10a7z3 - 5a7z5 + a7z7 - a8z-2 + 3a8 - 3a8z2 + 4a8z4 - a8z6 + a9z + 2a9z3 - a9z5 - 2a10 + 8a10z2 - 6a10z4 + a10z6

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                     1 j = -1                   1   j = -3                 3 1   j = -5               2 3     j = -7           1 3 1       j = -9           1 2         j = -11       1 4 3           j = -13         2             j = -15     1 2               j = -17 1                     j = -19 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, 347]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 347]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 15, 4, 14], X[11, 20, 12, 21], X[7, 18, 8, 19], > X[17, 22, 18, 13], X[16, 9, 17, 10], X[10, 15, 11, 16], X[19, 12, 20, 5], > X[21, 8, 22, 9], X[2, 5, 3, 6], X[13, 1, 14, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -4, 9, 6, -7, -3, 8}, > {-11, 2, 7, -6, -5, 4, -8, 3, -9, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-9 -7 3 3 4 3 4 2 1 + q - q + -- - -- + -- - -- + -- - - 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 3 2 3 3 2 4 -12 3 -8 -6 1 + q + q + --- + --- + --- + --- + --- + --- + q + --- + q + q + 24 22 20 18 16 14 10 q q q q q q q -4 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 347]][a, z]
Out[8]=	4 6 8 2 4 6 8 a 2 a a 2 2 4 2 6 2 2 a + a - 5 a + 2 a + -- - ---- + -- + 3 a z - 3 a z - 2 a z + 2 2 2 z z z 8 2 2 4 4 4 4 6 > a z + a z - 4 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 347]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 10 a 2 a a 2 a 2 a 3 -2 a + 3 a + 9 a + 3 a - 2 a - -- - ---- - -- + ---- + ---- + a z - 2 2 2 z z z z z 5 7 9 2 2 4 2 6 2 8 2 > 8 a z - 8 a z + a z + 5 a z + 3 a z - 13 a z - 3 a z + 10 2 3 3 5 3 7 3 9 3 2 4 4 4 > 8 a z + 4 a z + 12 a z + 10 a z + 2 a z - 4 a z - 6 a z + 6 4 8 4 10 4 3 5 5 5 7 5 9 5 > 8 a z + 4 a z - 6 a z - 7 a z - 11 a z - 5 a z - a z + 2 6 4 6 6 6 8 6 10 6 3 7 5 7 7 7 > a z - a z - 4 a z - a z + a z + 2 a z + 3 a z + a z + 4 8 6 8 > a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 1 1 2 1 2 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 19 8 17 8 15 6 15 5 11 5 13 4 11 4 q q q t q t q t q t q t q t q t 3 1 1 2 3 1 2 t t 2 > ------ + ----- + ----- + ----- + ----- + ---- + ---- + -- + - + q t 11 3 9 3 7 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n347

L11n346 L11n348