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L11a343 L11a345

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DrawMorseLink

PD Presentation:

X12,1,13,2 X18,7,19,8 X10,5,1,6 X6374 X4,9,5,10 X20,14,21,13 X22,16,11,15 X14,22,15,21 X16,20,17,19 X2,11,3,12 X8,17,9,18

Gauss Code:

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

Jones Polynomial:

q-15/2 - 2q-13/2 + 5q-11/2 - 10q-9/2 + 13q-7/2 - 18q-5/2 + 17q-3/2 - 16q-1/2 + 13q1/2 - 8q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 - 2q-22 - q-20 - 2q-18 + q-16 + 6q-14 + 2q-12 + 7q-10 + 3q-8 + 2q-4 - 5q-2 + 2 - 3q2 - q4 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z - 2a-1z3 - a-1z5 + 2az-1 + 6az + 7az3 + 4az5 + az7 - 7a3z-1 - 15a3z - 11a3z3 - 3a3z5 + 7a5z-1 + 9a5z + 3a5z3 - 2a7z-1 - a7z

Kauffman Polynomial:

a-3z3 - a-3z5 - 2a-2z2 + 6a-2z4 - 4a-2z6 + a-1z - 4a-1z3 + 11a-1z5 - 7a-1z7 + 2 - 4z2 + 2z4 + 8z6 - 7z8 - 2az-1 + 4az - 9az3 + 12az5 - 2az7 - 4az9 + 8a2 - 13a2z2 - 2a2z4 + 16a2z6 - 9a2z8 - a2z10 - 7a3z-1 + 15a3z - 15a3z3 + 5a3z5 + 6a3z7 - 6a3z9 + 13a4 - 24a4z2 + 11a4z4 + 5a4z6 - 4a4z8 - a4z10 - 7a5z-1 + 16a5z - 17a5z3 + 11a5z5 - a5z7 - 2a5z9 + 8a6 - 18a6z2 + 13a6z4 - 2a6z8 - 2a7z-1 + 4a7z - 6a7z3 + 6a7z5 - 2a7z7 + 2a8 - 5a8z2 + 4a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     3   j = 4                   5 1   j = 2                 8 3     j = 0               8 5       j = -2             10 9         j = -4           8 7           j = -6         5 10             j = -8       5 8               j = -10     1 6                 j = -12   1 4                   j = -14   1                     j = -16 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, 344]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 344]]
Out[4]=	PD[X[12, 1, 13, 2], X[18, 7, 19, 8], X[10, 5, 1, 6], X[6, 3, 7, 4], > X[4, 9, 5, 10], X[20, 14, 21, 13], X[22, 16, 11, 15], X[14, 22, 15, 21], > X[16, 20, 17, 19], X[2, 11, 3, 12], X[8, 17, 9, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, > {10, -1, 6, -8, 7, -9, 11, -2, 9, -6, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 2 5 10 13 18 17 16 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 13 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 8 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 -20 2 -16 6 2 7 3 2 5 2 2 - q - --- - q - --- + q + --- + --- + --- + -- + -- - -- - 3 q - 22 18 14 12 10 8 4 2 q q q q q q q q 4 6 8 10 > q + 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 344]][a, z]
Out[8]=	3 5 7 3 2 a 7 a 7 a 2 a z 3 5 7 2 z --- - ---- + ---- - ---- - - + 6 a z - 15 a z + 9 a z - a z - ---- + z z z z a a 5 3 3 3 5 3 z 5 3 5 7 > 7 a z - 11 a z + 3 a z - -- + 4 a z - 3 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 344]][a, z]
Out[9]=	3 5 7 2 4 6 8 2 a 7 a 7 a 2 a z 2 + 8 a + 13 a + 8 a + 2 a - --- - ---- - ---- - ---- + - + 4 a z + z z z z a 2 3 5 7 2 2 z 2 2 4 2 6 2 > 15 a z + 16 a z + 4 a z - 4 z - ---- - 13 a z - 24 a z - 18 a z - 2 a 3 3 8 2 z 4 z 3 3 3 5 3 7 3 4 > 5 a z + -- - ---- - 9 a z - 15 a z - 17 a z - 6 a z + 2 z + 3 a a 4 5 5 6 z 2 4 4 4 6 4 8 4 z 11 z 5 > ---- - 2 a z + 11 a z + 13 a z + 4 a z - -- + ----- + 12 a z + 2 3 a a a 6 3 5 5 5 7 5 6 4 z 2 6 4 6 8 6 > 5 a z + 11 a z + 6 a z + 8 z - ---- + 16 a z + 5 a z - a z - 2 a 7 7 z 7 3 7 5 7 7 7 8 2 8 4 8 > ---- - 2 a z + 6 a z - a z - 2 a z - 7 z - 9 a z - 4 a z - a 6 8 9 3 9 5 9 2 10 4 10 > 2 a z - 4 a z - 6 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 1 1 4 1 6 5 8 8 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 5 10 8 7 10 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 5 t + 8 q t + 3 q t + 5 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a344

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