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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X15,1,16,4 X5,10,6,11 X3849 X11,18,12,5 X17,12,18,13 X9,16,10,17 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

q-15/2 - q-13/2 + 2q-11/2 - 3q-9/2 + 2q-7/2 - 3q-5/2 + 2q-3/2 - 2q-1/2

A2 (sl(3)) Invariant:

- q-24 - q-22 - q-20 - 2q-18 + q-14 + 2q-12 + 3q-10 + 2q-8 + 2q-6 + q-4 + q-2 + 2

HOMFLY-PT Polynomial:

- az-1 - 2az + a3z + a3z3 + 2a5z-1 + 2a5z + a5z3 - a7z-1 - a7z

Kauffman Polynomial:

az-1 - 3az - a2 - a2z4 - 2a3z + 4a3z3 - 2a3z5 + 3a4 - 6a4z2 + 6a4z4 - 2a4z6 - 2a5z-1 + 3a5z + 2a5z5 - a5z7 + 5a6 - 13a6z2 + 12a6z4 - 3a6z6 - a7z-1 + 2a7z - 4a7z3 + 4a7z5 - a7z7 + 2a8 - 7a8z2 + 5a8z4 - a8z6

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n2

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