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L9n27 L10a1

Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X4,13,1,14 X18,10,15,9 X8493 X5,17,6,16 X17,5,18,14 X10,16,11,15 X2,12,3,11

Gauss Code:

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

Jones Polynomial:

- q-2 + 3q-1 - 4 + 6q - 5q2 + 6q3 - 4q4 + 3q5

A2 (sl(3)) Invariant:

- q-6 + q-4 + 1 + 2q2 + 4q6 + 2q8 + 5q10 + 4q12 + 3q14 + 4q16 + q18 + q20

HOMFLY-PT Polynomial:

a-6z-2 + a-6 - 2a-4z-2 - 4a-4 - 3a-4z2 - a-4z4 + a-2z-2 + 3a-2 + 5a-2z2 + 4a-2z4 + a-2z6 - 2z2 - z4

Kauffman Polynomial:

a-6z-2 - 5a-6 + 6a-6z2 - 2a-5z-1 + 5a-5z - 3a-5z3 + 3a-5z5 + 2a-4z-2 - 10a-4 + 18a-4z2 - 13a-4z4 + 5a-4z6 - 2a-3z-1 + 7a-3z - 8a-3z3 + a-3z5 + 2a-3z7 + a-2z-2 - 7a-2 + 17a-2z2 - 21a-2z4 + 8a-2z6 + 3a-1z - 7a-1z3 - a-1z5 + 2a-1z7 - 1 + 5z2 - 8z4 + 3z6 + az - 2az3 + az5

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n28

L9n27 L10a1