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L9n15 L9n17

Knotscape

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DrawMorseLink

PD Presentation:

X8192 X11,17,12,16 X3,10,4,11 X2,15,3,16 X12,5,13,6 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-13/2 + q-11/2 - 2q-9/2 + 2q-7/2 - 3q-5/2 + 2q-3/2 - 2q-1/2 + q1/2

A2 (sl(3)) Invariant:

q-20 + q-18 + 2q-16 + 2q-14 + q-12 + 2q-10 + q-8 + q-6 - q-2 - q2

HOMFLY-PT Polynomial:

az-1 + 2az + az3 - 3a3z-1 - 8a3z - 5a3z3 - a3z5 + 2a5z-1 + 3a5z + a5z3

Kauffman Polynomial:

1 - z2 - az-1 + 2az - 2az3 + 3a2 - 8a2z2 + 4a2z4 - a2z6 - 3a3z-1 + 11a3z - 12a3z3 + 5a3z5 - a3z7 + 3a4 - 8a4z2 + 7a4z4 - 2a4z6 - 2a5z-1 + 6a5z - 6a5z3 + 4a5z5 - a5z7 - a6z2 + 3a6z4 - a6z6 - 3a7z + 4a7z3 - 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             1   j = -2           2 2   j = -4         1       j = -6       1 2       j = -8     1 1         j = -10     1           j = -12 1 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, 16]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 16]]
Out[4]=	PD[X[8, 1, 9, 2], X[11, 17, 12, 16], X[3, 10, 4, 11], X[2, 15, 3, 16], > X[12, 5, 13, 6], 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]=	-(13/2) -(11/2) 2 2 3 2 2 -q + q - ---- + ---- - ---- + ---- - ------- + Sqrt[q] 9/2 7/2 5/2 3/2 Sqrt[q] q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 2 2 -12 2 -8 -6 -2 2 q + q + --- + --- + q + --- + q + q - q - q 16 14 10 q q q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 16]][a, z]
Out[8]=	3 5 a 3 a 2 a 3 5 3 3 3 5 3 3 5 - - ---- + ---- + 2 a z - 8 a z + 3 a z + a z - 5 a z + a z - a z z z z
In[9]:=	Kauffman[Link[9, NonAlternating, 16]][a, z]
Out[9]=	3 5 2 4 a 3 a 2 a 3 5 7 2 1 + 3 a + 3 a - - - ---- - ---- + 2 a z + 11 a z + 6 a z - 3 a z - z - z z z 2 2 4 2 6 2 3 3 3 5 3 7 3 > 8 a z - 8 a z - a z - 2 a z - 12 a z - 6 a z + 4 a z + 2 4 4 4 6 4 3 5 5 5 7 5 2 6 4 6 > 4 a z + 7 a z + 3 a z + 5 a z + 4 a z - a z - a z - 2 a z - 6 6 3 7 5 7 > a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 1 1 1 1 2 1 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 6 12 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 1 2 2 > ----- + ---- + q t 4 2 2 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n16

L9n15 L9n17