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L9n12 L9n14

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 X9,14,10,15 X13,18,14,7 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-13/2 + 2q-11/2 - 2q-9/2 + 3q-7/2 - 4q-5/2 + 2q-3/2 - 3q-1/2 + q1/2

A2 (sl(3)) Invariant:

q-20 - q-14 - 2q-12 + q-10 + 2q-8 + 4q-6 + 3q-4 + q-2 + 1 - q2

HOMFLY-PT Polynomial:

- az-1 + az + az3 + a3z-1 - 4a3z - 4a3z3 - a3z5 + 2a5z + a5z3

Kauffman Polynomial:

- z2 + az-1 + 2az - 3az3 - a2 - 3a2z2 + 2a2z4 - a2z6 + a3z-1 + 4a3z - 6a3z3 + 3a3z5 - a3z7 - 7a4z2 + 9a4z4 - 3a4z6 + a5z + 2a5z5 - a5z7 - 5a6z2 + 7a6z4 - 2a6z6 - a7z + 3a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 2               1 j = 0             2   j = -2           1 2   j = -4         3 1     j = -6       1 2       j = -8     1 2         j = -10   1 1           j = -12   1             j = -14 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, 13]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 13]]
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[9, 14, 10, 15], X[13, 18, 14, 7], > 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]=	-(13/2) 2 2 3 4 2 3 -q + ----- - ---- + ---- - ---- + ---- - ------- + Sqrt[q] 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -14 2 -10 2 4 3 -2 2 1 + q - q - --- + q + -- + -- + -- + q - q 12 8 6 4 q q q q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 13]][a, z]
Out[8]=	3 a a 3 5 3 3 3 5 3 3 5 -(-) + -- + a z - 4 a z + 2 a z + a z - 4 a z + a z - a z z z
In[9]:=	Kauffman[Link[9, NonAlternating, 13]][a, z]
Out[9]=	3 2 a a 3 5 7 2 2 2 4 2 -a + - + -- + 2 a z + 4 a z + a z - a z - z - 3 a z - 7 a z - z z 6 2 3 3 3 7 3 2 4 4 4 6 4 > 5 a z - 3 a z - 6 a z + 3 a z + 2 a z + 9 a z + 7 a z + 3 5 5 5 7 5 2 6 4 6 6 6 3 7 5 7 > 3 a z + 2 a z - a z - a z - 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 1 1 2 1 2 2 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 3 1 1 2 > ----- + ---- + ---- + q t 4 2 4 2 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n13

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