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L11a258 L11a260

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DrawMorseLink

PD Presentation:

X10,1,11,2 X14,5,15,6 X12,3,13,4 X18,8,19,7 X20,15,21,16 X22,17,9,18 X16,21,17,22 X4,13,5,14 X6,20,7,19 X2,9,3,10 X8,11,1,12

Gauss Code:

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

Jones Polynomial:

q-23/2 - 2q-21/2 + 4q-19/2 - 7q-17/2 + 9q-15/2 - 11q-13/2 + 10q-11/2 - 9q-9/2 + 7q-7/2 - 5q-5/2 + 2q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-34 - q-30 - q-28 + q-26 - q-24 + 3q-22 + q-20 + 2q-18 + 3q-16 - q-14 + 2q-12 - q-10 + q-8 + q-6 + q-2

HOMFLY-PT Polynomial:

- 4a3z - 4a3z3 - a3z5 - 2a5z-1 - 2a5z + 3a5z3 + 4a5z5 + a5z7 + 3a7z-1 + 8a7z + 9a7z3 + 5a7z5 + a7z7 - a9z-1 - 4a9z - 4a9z3 - a9z5

Kauffman Polynomial:

4a3z - 8a3z3 + 5a3z5 - a3z7 + a4z2 - 8a4z4 + 8a4z6 - 2a4z8 + 2a5z-1 - 6a5z + 3a5z3 - 3a5z5 + 6a5z7 - 2a5z9 - 3a6 + 6a6z2 - 8a6z4 + 8a6z6 - a6z10 + 3a7z-1 - 15a7z + 30a7z3 - 29a7z5 + 19a7z7 - 5a7z9 - 3a8 + 14a8z2 - 20a8z4 + 14a8z6 - 2a8z8 - a8z10 + a9z-1 - a9z + 3a9z3 - 9a9z5 + 8a9z7 - 3a9z9 - a10 + 4a10z2 - 14a10z4 + 11a10z6 - 4a10z8 + 4a11z - 13a11z3 + 10a11z5 - 4a11z7 - 3a12z2 + 5a12z4 - 3a12z6 + 3a13z3 - 2a13z5 + 2a14z2 - a14z4

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 = 0                       1 j = -2                     1   j = -4                   4 1   j = -6                 4 2     j = -8               5 3       j = -10             5 4         j = -12           6 5           j = -14         4 6             j = -16       3 5               j = -18     1 4                 j = -20   1 3                   j = -22   1                     j = -24 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 259]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 259]]
Out[4]=	PD[X[10, 1, 11, 2], X[14, 5, 15, 6], X[12, 3, 13, 4], X[18, 8, 19, 7], > X[20, 15, 21, 16], X[22, 17, 9, 18], X[16, 21, 17, 22], X[4, 13, 5, 14], > X[6, 20, 7, 19], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 3, -8, 2, -9, 4, -11}, > {10, -1, 11, -3, 8, -2, 5, -7, 6, -4, 9, -5, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 2 4 7 9 11 10 9 7 5 q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q q 2 1 > ---- - ------- 3/2 Sqrt[q] q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -30 -28 -26 -24 3 -20 2 3 -14 2 -q - q - q + q - q + --- + q + --- + --- - q + --- - 22 18 16 12 q q q q -10 -8 -6 -2 > q + q + q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 259]][a, z]
Out[8]=	5 7 9 -2 a 3 a a 3 5 7 9 3 3 5 3 ----- + ---- - -- - 4 a z - 2 a z + 8 a z - 4 a z - 4 a z + 3 a z + z z z 7 3 9 3 3 5 5 5 7 5 9 5 5 7 7 7 > 9 a z - 4 a z - a z + 4 a z + 5 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 259]][a, z]
Out[9]=	5 7 9 6 8 10 2 a 3 a a 3 5 7 9 -3 a - 3 a - a + ---- + ---- + -- + 4 a z - 6 a z - 15 a z - a z + z z z 11 4 2 6 2 8 2 10 2 12 2 14 2 > 4 a z + a z + 6 a z + 14 a z + 4 a z - 3 a z + 2 a z - 3 3 5 3 7 3 9 3 11 3 13 3 4 4 > 8 a z + 3 a z + 30 a z + 3 a z - 13 a z + 3 a z - 8 a z - 6 4 8 4 10 4 12 4 14 4 3 5 5 5 > 8 a z - 20 a z - 14 a z + 5 a z - a z + 5 a z - 3 a z - 7 5 9 5 11 5 13 5 4 6 6 6 8 6 > 29 a z - 9 a z + 10 a z - 2 a z + 8 a z + 8 a z + 14 a z + 10 6 12 6 3 7 5 7 7 7 9 7 11 7 > 11 a z - 3 a z - a z + 6 a z + 19 a z + 8 a z - 4 a z - 4 8 8 8 10 8 5 9 7 9 9 9 6 10 8 10 > 2 a z - 2 a z - 4 a z - 2 a z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 1 1 3 1 4 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q q q t q t q t q t q t q t q t 5 4 6 6 5 5 4 5 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 q t q t q t q t q t q t q t q t 3 4 t t 2 > ---- + ---- + -- + -- + t 8 6 4 2 q t q t q q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a259

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