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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X8493 X9,14,10,15 X13,10,14,11 X11,18,12,5 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

2q-13/2 - 3q-11/2 + 4q-9/2 - 5q-7/2 + 3q-5/2 - 4q-3/2 + 2q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-22 - 3q-20 - q-18 - q-16 + q-14 + 3q-12 + 2q-10 + 4q-8 + q-6 + 2q-4 + q-2 + q2

HOMFLY-PT Polynomial:

- az-1 - 2az - az3 + 2a3z + 3a3z3 + a3z5 + 2a5z-1 - a5z3 - a7z-1

Kauffman Polynomial:

az-1 - 3az + 3az3 - az5 - a2 - a2z2 + 5a2z4 - 2a2z6 - 4a3z + 8a3z3 - a3z5 - a3z7 + 3a4 - 6a4z2 + 9a4z4 - 4a4z6 - 2a5z-1 - a5z + 4a5z3 - a5z5 - a5z7 + 5a6 - 8a6z2 + 4a6z4 - 2a6z6 - a7z-1 - a7z3 - a7z5 + 2a8 - 3a8z2

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2               1 j = 0             1   j = -2           3 1   j = -4         2 3     j = -6       3 1       j = -8     1 2         j = -10   2 3           j = -12   1             j = -14 2

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, 8]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 8]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[5, 12, 6, 13], > X[8, 4, 9, 3], X[9, 14, 10, 15], X[13, 10, 14, 11], X[11, 18, 12, 5], > X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 5, -3}, {-4, -1, 2, -5, -6, 7, -8, 4, -7, 6, 9, -2, 3, 8}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 5 3 4 2 ----- - ----- + ---- - ---- + ---- - ---- + ------- - Sqrt[q] 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 3 -18 -16 -14 3 2 4 -6 2 -2 2 -q - --- - q - q + q + --- + --- + -- + q + -- + q + q 20 12 10 8 4 q q q q q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 8]][a, z]
Out[8]=	5 7 a 2 a a 3 3 3 3 5 3 3 5 -(-) + ---- - -- - 2 a z + 2 a z - a z + 3 a z - a z + a z z z z
In[9]:=	Kauffman[Link[9, NonAlternating, 8]][a, z]
Out[9]=	5 7 2 4 6 8 a 2 a a 3 5 2 2 -a + 3 a + 5 a + 2 a + - - ---- - -- - 3 a z - 4 a z - a z - a z - z z z 4 2 6 2 8 2 3 3 3 5 3 7 3 > 6 a z - 8 a z - 3 a z + 3 a z + 8 a z + 4 a z - a z + 2 4 4 4 6 4 5 3 5 5 5 7 5 2 6 > 5 a z + 9 a z + 4 a z - a z - a z - a z - a z - 2 a z - 4 6 6 6 3 7 5 7 > 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 2 1 2 3 1 2 3 1 -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 4 2 14 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q q q t q t q t q t q t q t q t q t 2 t 2 2 > ---- + t + -- + q t 4 2 q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n8

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