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L11n339 L11n341

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X18,12,19,11 X7,14,8,15 X13,8,14,9 X22,20,13,19 X20,16,21,15 X16,22,17,21 X12,18,5,17 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

3q-4 - 5q-3 + 11q-2 - 12q-1 + 14 - 13q + 11q2 - 7q3 + 3q4 - q5

A2 (sl(3)) Invariant:

2q-14 + 5q-12 + 3q-10 + 9q-8 + 7q-6 + 3q-4 + 5q-2 - 3 + q2 - 3q4 - q6 + 2q8 - 3q10 + q12 - q16

HOMFLY-PT Polynomial:

- a-4 - a-4z2 - a-2z-2 + a-2 + 3a-2z2 + 2a-2z4 + 4z-2 + 5 - 2z4 - z6 - 5a2z-2 - 7a2 - 2a2z2 + a2z4 + 2a4z-2 + 2a4

Kauffman Polynomial:

a-5z - 2a-5z3 + a-5z5 - a-4 + 3a-4z2 - 5a-4z4 + 3a-4z6 + a-3z-1 - 3a-3z + 5a-3z3 - 8a-3z5 + 5a-3z7 - a-2z-2 + 6a-2z2 - 10a-2z4 + 4a-2z8 + 5a-1z-1 - 20a-1z + 36a-1z3 - 35a-1z5 + 13a-1z7 + a-1z9 - 4z-2 + 11 - 10z2 + 7z4 - 11z6 + 8z8 + 9az-1 - 32az + 43az3 - 32az5 + 11az7 + az9 - 5a2z-2 + 18a2 - 26a2z2 + 18a2z4 - 8a2z6 + 4a2z8 + 5a3z-1 - 16a3z + 14a3z3 - 6a3z5 + 3a3z7 - 2a4z-2 + 9a4 - 13a4z2 + 6a4z4

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                   1 j = 9                 2   j = 7               5 1   j = 5             6 2     j = 3           7 5       j = 1         7 6         j = -1       7 9           j = -3     4 5             j = -5   1 7               j = -7 2 4                 j = -9 3

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, 340]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 340]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 12, 19, 11], X[7, 14, 8, 15], > X[13, 8, 14, 9], X[22, 20, 13, 19], X[20, 16, 21, 15], X[16, 22, 17, 21], > X[12, 18, 5, 17], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, 3, -9}, > {-5, 4, 7, -8, 9, -3, 6, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	3 5 11 12 2 3 4 5 14 + -- - -- + -- - -- - 13 q + 11 q - 7 q + 3 q - q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 5 3 9 7 3 5 2 4 6 8 10 -3 + --- + --- + --- + -- + -- + -- + -- + q - 3 q - q + 2 q - 3 q + 14 12 10 8 6 4 2 q q q q q q q 12 16 > q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 340]][a, z]
Out[8]=	2 4 2 2 -4 -2 2 4 4 1 5 a 2 a z 3 z 2 2 5 - a + a - 7 a + 2 a + -- - ----- - ---- + ---- - -- + ---- - 2 a z - 2 2 2 2 2 4 2 z a z z z a a 4 4 2 z 2 4 6 > 2 z + ---- + a z - z 2 a
In[9]:=	Kauffman[Link[11, NonAlternating, 340]][a, z]
Out[9]=	2 4 3 -4 2 4 4 1 5 a 2 a 1 5 9 a 5 a 11 - a + 18 a + 9 a - -- - ----- - ---- - ---- + ---- + --- + --- + ---- + 2 2 2 2 2 3 a z z z z a z z z a z 2 2 z 3 z 20 z 3 2 3 z 6 z 2 2 > -- - --- - ---- - 32 a z - 16 a z - 10 z + ---- + ---- - 26 a z - 5 3 a 4 2 a a a a 3 3 3 4 4 4 2 2 z 5 z 36 z 3 3 3 4 5 z 10 z > 13 a z - ---- + ---- + ----- + 43 a z + 14 a z + 7 z - ---- - ----- + 5 3 a 4 2 a a a a 5 5 5 6 2 4 4 4 z 8 z 35 z 5 3 5 6 3 z > 18 a z + 6 a z + -- - ---- - ----- - 32 a z - 6 a z - 11 z + ---- - 5 3 a 4 a a a 7 7 8 9 2 6 5 z 13 z 7 3 7 8 4 z 2 8 z > 8 a z + ---- + ----- + 11 a z + 3 a z + 8 z + ---- + 4 a z + -- + 3 a 2 a a a 9 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 3 2 4 1 7 4 5 7 - + 7 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 6 q t + q 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 11 5 > 7 q t + 5 q t + 6 q t + 2 q t + 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n340

L11n339 L11n341