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L10a145 L10a147

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DrawMorseLink

PD Presentation:

X6172 X14,4,15,3 X20,12,13,11 X18,8,19,7 X16,10,17,9 X8,18,9,17 X10,14,11,13 X12,20,5,19 X2536 X4,16,1,15

Gauss Code:

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

Jones Polynomial:

q-1 - 2 + 5q - 8q2 + 10q3 - 9q4 + 10q5 - 7q6 + 5q7 - 2q8 + q9

A2 (sl(3)) Invariant:

q-4 + 3q2 - 2q4 + q8 + 4q12 + 2q14 + 5q16 + 4q18 + 2q20 + 5q22 + q24 + q28

HOMFLY-PT Polynomial:

a-8z-2 + a-8 + a-8z2 - 2a-6z-2 - 3a-6 - a-6z2 - a-6z4 + a-4z-2 + a-4 - 2a-4z2 - 2a-4z4 - a-2z4 + 1 + z2

Kauffman Polynomial:

4a-10z2 - 4a-10z4 + a-10z6 + 4a-9z3 - 6a-9z5 + 2a-9z7 - a-8z-2 + 8a-8 - 19a-8z2 + 18a-8z4 - 11a-8z6 + 3a-8z8 + 2a-7z-1 - 11a-7z + 17a-7z3 - 15a-7z5 + 3a-7z7 + a-7z9 - 2a-6z-2 + 13a-6 - 30a-6z2 + 28a-6z4 - 18a-6z6 + 6a-6z8 + 2a-5z-1 - 11a-5z + 21a-5z3 - 17a-5z5 + 5a-5z7 + a-5z9 - a-4z-2 + 5a-4 - 5a-4z2 + 3a-4z4 - 3a-4z6 + 3a-4z8 + 6a-3z3 - 6a-3z5 + 4a-3z7 - 2a-2z4 + 3a-2z6 - 2a-1z3 + 2a-1z5 + 1 - 2z2 + z4

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                     1 j = 17                   2 1 j = 15                 3     j = 13               4 2     j = 11             6 3       j = 9           4 5         j = 7         6 5           j = 5       2 4             j = 3     3 6               j = 1   1 4                 j = -1   1                   j = -3 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[10, Alternating, 146]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 146]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[20, 12, 13, 11], X[18, 8, 19, 7], > X[16, 10, 17, 9], X[8, 18, 9, 17], X[10, 14, 11, 13], X[12, 20, 5, 19], > X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 4, -6, 5, -7, 3, -8}, > {7, -2, 10, -5, 6, -4, 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	1 2 3 4 5 6 7 8 9 -2 + - + 5 q - 8 q + 10 q - 9 q + 10 q - 7 q + 5 q - 2 q + q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 2 4 8 12 14 16 18 20 22 24 q + 3 q - 2 q + q + 4 q + 2 q + 5 q + 4 q + 2 q + 5 q + q + 28 > q
In[8]:=	HOMFLYPT[Link[10, Alternating, 146]][a, z]
Out[8]=	2 2 2 4 4 -8 3 -4 1 2 1 2 z z 2 z z 2 z 1 + a - -- + a + ----- - ----- + ----- + z + -- - -- - ---- - -- - ---- - 6 8 2 6 2 4 2 8 6 4 6 4 a a z a z a z a a a a a 4 z > -- 2 a
In[9]:=	Kauffman[Link[10, Alternating, 146]][a, z]
Out[9]=	8 13 5 1 2 1 2 2 11 z 11 z 2 1 + -- + -- + -- - ----- - ----- - ----- + ---- + ---- - ---- - ---- - 2 z + 8 6 4 8 2 6 2 4 2 7 5 7 5 a a a a z a z a z a z a z a a 2 2 2 2 3 3 3 3 3 4 z 19 z 30 z 5 z 4 z 17 z 21 z 6 z 2 z 4 > ---- - ----- - ----- - ---- + ---- + ----- + ----- + ---- - ---- + z - 10 8 6 4 9 7 5 3 a a a a a a a a a 4 4 4 4 4 5 5 5 5 5 4 z 18 z 28 z 3 z 2 z 6 z 15 z 17 z 6 z 2 z > ---- + ----- + ----- + ---- - ---- - ---- - ----- - ----- - ---- + ---- + 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 8 z 11 z 18 z 3 z 3 z 2 z 3 z 5 z 4 z 3 z > --- - ----- - ----- - ---- + ---- + ---- + ---- + ---- + ---- + ---- + 10 8 6 4 2 9 7 5 3 8 a a a a a a a a a a 8 8 9 9 6 z 3 z z z > ---- + ---- + -- + -- 6 4 7 5 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 q 3 5 5 2 7 2 7 3 4 q + 3 q + ----- + --- + - + 6 q t + 2 q t + 4 q t + 6 q t + 5 q t + 3 2 q t t q t 9 3 9 4 11 4 11 5 13 5 13 6 15 6 > 4 q t + 5 q t + 6 q t + 3 q t + 4 q t + 2 q t + 3 q t + 17 7 17 8 19 8 > 2 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a146

L10a145 L10a147