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L11n398 L11n400

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X22,16,19,15 X7,20,8,21 X19,8,20,9 X13,18,14,5 X11,14,12,15 X17,12,18,13 X16,22,17,21 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-10 + q-9 - q-8 + 3q-6 - 2q-5 + 4q-4 - 3q-3 + 4q-2 - 2q-1 + 1

A2 (sl(3)) Invariant:

- q-32 - 2q-30 - 2q-28 - 3q-26 + q-22 + 5q-20 + 7q-18 + 6q-16 + 7q-14 + 2q-12 + 3q-10 + q-8 + q-6 + q-4 + 1

HOMFLY-PT Polynomial:

2a2 + 3a2z2 + a2z4 + 2a4z-2 + a4 - 3a4z2 - 4a4z4 - a4z6 - 5a6z-2 - 7a6 - 2a6z2 + 4a8z-2 + 5a8 + 2a8z2 - a10z-2 - a10

Kauffman Polynomial:

- 2a2 + 5a2z2 - 4a2z4 + a2z6 + a3z + 4a3z3 - 7a3z5 + 2a3z7 - 2a4z-2 + 4a4 + 2a4z2 - 6a4z4 - a4z6 + a4z8 + 5a5z-1 - 13a5z + 15a5z3 - 12a5z5 + 3a5z7 - 5a6z-2 + 16a6 - 19a6z2 + 11a6z4 - 5a6z6 + a6z8 + 9a7z-1 - 30a7z + 34a7z3 - 15a7z5 + 2a7z7 - 4a8z-2 + 15a8 - 22a8z2 + 22a8z4 - 9a8z6 + a8z8 + 5a9z-1 - 22a9z + 33a9z3 - 16a9z5 + 2a9z7 - a10z-2 + 4a10 - 6a10z2 + 9a10z4 - 6a10z6 + a10z8 + a11z-1 - 6a11z + 10a11z3 - 6a11z5 + a11z7

Khovanov Homology:

trqj r = -9 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 1 2         j = -11         1 5 3           j = -13       1   2             j = -15       1 2               j = -17   1 1                   j = -19                         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, 399]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 399]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[22, 16, 19, 15], X[7, 20, 8, 21], > X[19, 8, 20, 9], X[13, 18, 14, 5], X[11, 14, 12, 15], X[17, 12, 18, 13], > X[16, 22, 17, 21], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {-5, 4, 9, -3}, > {10, -1, -4, 5, 11, -2, -7, 8, -6, 7, 3, -9, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-10 -9 -8 3 2 4 3 4 2 1 - q + q - q + -- - -- + -- - -- + -- - - 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 2 2 3 -22 5 7 6 7 2 3 -8 1 - q - --- - --- - --- + q + --- + --- + --- + --- + --- + --- + q + 30 28 26 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, 399]][a, z]
Out[8]=	4 6 8 10 2 4 6 8 10 2 a 5 a 4 a a 2 2 4 2 2 a + a - 7 a + 5 a - a + ---- - ---- + ---- - --- + 3 a z - 3 a z - 2 2 2 2 z z z z 6 2 8 2 2 4 4 4 4 6 > 2 a z + 2 a z + a z - 4 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 399]][a, z]
Out[9]=	4 6 8 10 5 7 2 4 6 8 10 2 a 5 a 4 a a 5 a 9 a -2 a + 4 a + 16 a + 15 a + 4 a - ---- - ---- - ---- - --- + ---- + ---- + 2 2 2 2 z z z z z z 9 11 5 a a 3 5 7 9 11 2 2 > ---- + --- + a z - 13 a z - 30 a z - 22 a z - 6 a z + 5 a z + z z 4 2 6 2 8 2 10 2 3 3 5 3 7 3 > 2 a z - 19 a z - 22 a z - 6 a z + 4 a z + 15 a z + 34 a z + 9 3 11 3 2 4 4 4 6 4 8 4 10 4 > 33 a z + 10 a z - 4 a z - 6 a z + 11 a z + 22 a z + 9 a z - 3 5 5 5 7 5 9 5 11 5 2 6 4 6 > 7 a z - 12 a z - 15 a z - 16 a z - 6 a z + a z - a z - 6 6 8 6 10 6 3 7 5 7 7 7 9 7 > 5 a z - 9 a z - 6 a z + 2 a z + 3 a z + 2 a z + 2 a z + 11 7 4 8 6 8 8 8 10 8 > a z + a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 1 1 1 1 2 1 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 9 17 8 17 7 15 6 13 6 15 5 11 5 q q q t q t q t q t q t q t q t 2 5 1 3 1 1 2 3 1 > ------ + ------ + ----- + ------ + ----- + ----- + ----- + ----- + ---- + 13 4 11 4 9 4 11 3 9 3 7 3 9 2 7 2 7 q t q t q t q t q t q t q t q t q t 2 t t 2 > ---- + -- + - + q t 5 3 q q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n399

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