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L11n261 L11n263

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X19,22,20,9 X15,20,16,21 X21,16,22,17 X17,12,18,13 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-13 + 2q-12 - 5q-11 + 6q-10 - 6q-9 + 7q-8 - 4q-7 + 5q-6 - q-5 + q-3

A2 (sl(3)) Invariant:

- q-42 - 3q-40 - 2q-38 - 2q-36 - 4q-34 + q-32 + q-30 + 4q-28 + 7q-26 + 6q-24 + 8q-22 + 3q-20 + 4q-18 + 3q-16 + q-12 + q-10

HOMFLY-PT Polynomial:

a6z-2 + 4a6 + 8a6z2 + 6a6z4 + a6z6 - a8z-2 - a8 + a8z4 - 2a10z-2 - 7a10 - 5a10z2 + 3a12z-2 + 4a12 - a14z-2

Kauffman Polynomial:

a6z-2 - 4a6 + 8a6z2 - 6a6z4 + a6z6 - 2a7z-1 + 4a7z - a7z5 + a8z-2 - 3a8 + 3a8z2 + 2a8z4 - a8z6 + 2a9z-1 - 10a9z + 22a9z3 - 15a9z5 + 3a9z7 - 2a10z-2 + 4a10 - 13a10z2 + 24a10z4 - 18a10z6 + 4a10z8 + 10a11z-1 - 34a11z + 52a11z3 - 33a11z5 + 4a11z7 + a11z9 - 3a12z-2 + 5a12 - 10a12z2 + 24a12z4 - 24a12z6 + 6a12z8 + 8a13z-1 - 27a13z + 39a13z3 - 24a13z5 + 2a13z7 + a13z9 - a14z-2 + a14 - 2a14z2 + 8a14z4 - 8a14z6 + 2a14z8 + 2a15z-1 - 7a15z + 9a15z3 - 5a15z5 + a15z7

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -5                       1 j = -7                       1 j = -9                 2 1     j = -11               4         j = -13             4 6 1       j = -15           4 1           j = -17         2 4 1           j = -19       4 4               j = -21     1 2                 j = -23   1 4                   j = -25   1                     j = -27 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, 262]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 262]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], > X[11, 18, 12, 19], X[19, 22, 20, 9], X[15, 20, 16, 21], X[21, 16, 22, 17], > X[17, 12, 18, 13], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -4}, > {11, -2, -5, 9, 4, -3, -7, 8, -9, 5, -6, 7, -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-13 2 5 6 6 7 4 5 -5 -3 -q + --- - --- + --- - -- + -- - -- + -- - q + q 12 11 10 9 8 7 6 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 3 2 2 4 -32 -30 4 7 6 8 3 -q - --- - --- - --- - --- + q + q + --- + --- + --- + --- + --- + 40 38 36 34 28 26 24 22 20 q q q q q q q q q 4 3 -12 -10 > --- + --- + q + q 18 16 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 262]][a, z]
Out[8]=	6 8 10 12 14 6 8 10 12 a a 2 a 3 a a 6 2 4 a - a - 7 a + 4 a + -- - -- - ----- + ----- - --- + 8 a z - 2 2 2 2 2 z z z z z 10 2 6 4 8 4 6 6 > 5 a z + 6 a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 262]][a, z]
Out[9]=	6 8 10 12 14 7 6 8 10 12 14 a a 2 a 3 a a 2 a -4 a - 3 a + 4 a + 5 a + a + -- + -- - ----- - ----- - --- - ---- + 2 2 2 2 2 z z z z z z 9 11 13 15 2 a 10 a 8 a 2 a 7 9 11 13 > ---- + ------ + ----- + ----- + 4 a z - 10 a z - 34 a z - 27 a z - z z z z 15 6 2 8 2 10 2 12 2 14 2 9 3 > 7 a z + 8 a z + 3 a z - 13 a z - 10 a z - 2 a z + 22 a z + 11 3 13 3 15 3 6 4 8 4 10 4 > 52 a z + 39 a z + 9 a z - 6 a z + 2 a z + 24 a z + 12 4 14 4 7 5 9 5 11 5 13 5 > 24 a z + 8 a z - a z - 15 a z - 33 a z - 24 a z - 15 5 6 6 8 6 10 6 12 6 14 6 9 7 > 5 a z + a z - a z - 18 a z - 24 a z - 8 a z + 3 a z + 11 7 13 7 15 7 10 8 12 8 14 8 11 9 > 4 a z + 2 a z + a z + 4 a z + 6 a z + 2 a z + a z + 13 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 -5 1 1 1 4 1 2 4 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 27 11 25 10 23 10 23 9 21 9 21 8 19 8 q t q t q t q t q t q t q t 4 2 4 4 1 1 4 6 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 19 7 17 7 17 6 15 6 17 5 15 5 13 5 13 4 q t q t q t q t q t q t q t q t 4 1 2 1 > ------ + ------ + ----- + ----- 11 4 13 3 9 3 9 2 q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n262

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