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L11n365 L11n367

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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X15,17,16,22 X18,10,19,9 X8,18,9,17 X13,21,14,20 X21,15,22,14 X19,5,20,16 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

q - q2 + 3q3 - 2q4 + 3q5 - 2q6 + 2q7 - q8 + q9

A2 (sl(3)) Invariant:

q4 + q6 + 2q8 + 3q10 + 3q12 + 3q14 + 2q16 + 3q18 + q20 + 3q22 + 2q24 + 2q26 + q28 + q32 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + a-8z-2 + 6a-8 + 10a-8z2 + 6a-8z4 + a-8z6 - 2a-6z-2 - 12a-6 - 20a-6z2 - 17a-6z4 - 7a-6z6 - a-6z8 + a-4z-2 + 7a-4 + 11a-4z2 + 6a-4z4 + a-4z6

Kauffman Polynomial:

a-12 - a-11z + 2a-10 - 5a-10z2 + a-10z4 - 3a-9z + 11a-9z3 - 10a-9z5 + 2a-9z7 - a-8z-2 + 9a-8 - 26a-8z2 + 33a-8z4 - 18a-8z6 + 3a-8z8 + 2a-7z-1 - 12a-7z + 19a-7z3 - 6a-7z5 - 3a-7z7 + a-7z9 - 2a-6z-2 + 15a-6 - 39a-6z2 + 49a-6z4 - 25a-6z6 + 4a-6z8 + 2a-5z-1 - 10a-5z + 8a-5z3 + 4a-5z5 - 5a-5z7 + a-5z9 - a-4z-2 + 8a-4 - 18a-4z2 + 17a-4z4 - 7a-4z6 + a-4z8

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 19                 1 j = 17               2 2 j = 15             1 1 1 j = 13           2 2     j = 11         1 1 1     j = 9       1 2         j = 7     2 1           j = 5   1 3             j = 3                   j = 1 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, 366]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 366]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[15, 17, 16, 22], > X[18, 10, 19, 9], X[8, 18, 9, 17], X[13, 21, 14, 20], X[21, 15, 22, 14], > X[19, 5, 20, 16], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {6, -5, -9, 7, -8, 4}, > {10, -1, 3, -6, 5, -2, 11, -3, -7, 8, -4, 9}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 5 6 7 8 9 q - q + 3 q - 2 q + 3 q - 2 q + 2 q - q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	4 6 8 10 12 14 16 18 20 22 24 q + q + 2 q + 3 q + 3 q + 3 q + 2 q + 3 q + q + 3 q + 2 q + 26 28 32 34 > 2 q + q + q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 366]][a, z]
Out[8]=	2 2 2 2 -10 6 12 7 1 2 1 z 10 z 20 z 11 z -a + -- - -- + -- + ----- - ----- + ----- - --- + ----- - ----- + ----- + 8 6 4 8 2 6 2 4 2 10 8 6 4 a a a a z a z a z a a a a 4 4 4 6 6 6 8 6 z 17 z 6 z z 7 z z z > ---- - ----- + ---- + -- - ---- + -- - -- 8 6 4 8 6 4 6 a a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 366]][a, z]
Out[9]=	-12 2 9 15 8 1 2 1 2 2 z 3 z a + --- + -- + -- + -- - ----- - ----- - ----- + ---- + ---- - --- - --- - 10 8 6 4 8 2 6 2 4 2 7 5 11 9 a a a a a z a z a z a z a z a a 2 2 2 2 3 3 3 4 12 z 10 z 5 z 26 z 39 z 18 z 11 z 19 z 8 z z > ---- - ---- - ---- - ----- - ----- - ----- + ----- + ----- + ---- + --- + 7 5 10 8 6 4 9 7 5 10 a a a a a a a a a a 4 4 4 5 5 5 6 6 6 7 33 z 49 z 17 z 10 z 6 z 4 z 18 z 25 z 7 z 2 z > ----- + ----- + ----- - ----- - ---- + ---- - ----- - ----- - ---- + ---- - 8 6 4 9 7 5 8 6 4 9 a a a a a a a a a a 7 7 8 8 8 9 9 3 z 5 z 3 z 4 z z z z > ---- - ---- + ---- + ---- + -- + -- + -- 7 5 8 6 4 7 5 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	5 5 7 q q 7 9 9 2 11 2 11 3 13 3 3 q + 2 q + -- + -- + q t + q t + 2 q t + q t + q t + 2 q t + 2 t t 11 4 13 4 15 4 15 5 17 5 15 6 17 6 19 6 > q t + 2 q t + q t + q t + 2 q t + q t + 2 q t + q t

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