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L9n13 L9n15

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X11,17,12,16 X3,10,4,11 X15,3,16,2 X5,13,6,12 X6718 X14,10,15,9 X18,14,7,13 X17,4,18,5

Gauss Code:

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

Jones Polynomial:

- q-5/2 + q-3/2 - 2q-1/2 + q1/2 - 2q3/2 + q5/2 - q7/2 + q9/2

A2 (sl(3)) Invariant:

q-8 + q-6 + 2q-4 + 2q-2 + 2 + 2q2 + q4 + q6 - q10 - q12 - q14

HOMFLY-PT Polynomial:

a-3z-1 + 3a-3z + a-3z3 - 3a-1z-1 - 7a-1z - 5a-1z3 - a-1z5 + 2az-1 + 3az + az3

Kauffman Polynomial:

a-4 - 6a-4z2 + 5a-4z4 - a-4z6 - a-3z-1 + 4a-3z - 7a-3z3 + 5a-3z5 - a-3z7 + 3a-2 - 10a-2z2 + 9a-2z4 - 2a-2z6 - 3a-1z-1 + 9a-1z - 9a-1z3 + 5a-1z5 - a-1z7 + 3 - 5z2 + 4z4 - z6 - 2az-1 + 4az - 2az3 - a2z2 - a3z

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n14

L9n13 L9n15