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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X8493 X5,11,6,10 X11,5,12,20 X13,19,14,18 X19,13,20,12 X2,16,3,15

Gauss Code:

{{1, -10, 5, -3}, {-6, -1, 2, -5, -4, 6, -7, 9, -8, 4, 10, -2, 3, 8, -9, 7}}

Jones Polynomial:

- q-5/2 - q-1/2 + q1/2 - q3/2 + q5/2 - q7/2 + q9/2 - q11/2

A2 (sl(3)) Invariant:

q-10 + 2q-8 + 2q-6 + 2q-4 + q-2 - q2 - q4 - q6 + q12 + q14 + q16 + q18

HOMFLY-PT Polynomial:

- a-5z-1 - a-5z + 2a-3z-1 + 3a-3z + a-3z3 - a-1z-1 - a-1z - az-1 - az + a3z-1

Kauffman Polynomial:

- a-5z-1 + 5a-5z - 10a-5z3 + 6a-5z5 - a-5z7 - a-4 + 5a-4z2 - 10a-4z4 + 6a-4z6 - a-4z8 - 2a-3z-1 + 13a-3z - 21a-3z3 + 12a-3z5 - 2a-3z7 - 3a-2 + 9a-2z2 - 11a-2z4 + 6a-2z6 - a-2z8 - a-1z-1 + 8a-1z - 11a-1z3 + 6a-1z5 - a-1z7 - 2 + 4z2 - z4 + az-1 - az - a2 + a3z-1 - a3z

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 12                 1 j = 10                   j = 8             1 1   j = 6         1 1       j = 4         1 1       j = 2     1 2 1         j = 0   1 2 1           j = -2   1 2             j = -4 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[10, NonAlternating, 9]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 9]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[9, 14, 10, 15], > X[8, 4, 9, 3], X[5, 11, 6, 10], X[11, 5, 12, 20], X[13, 19, 14, 18], > X[19, 13, 20, 12], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {-6, -1, 2, -5, -4, 6, -7, 9, -8, 4, 10, -2, 3, 8, > -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 1 3/2 5/2 7/2 9/2 11/2 -q - ------- + Sqrt[q] - q + q - q + q - q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 2 2 -2 2 4 6 12 14 16 18 q + -- + -- + -- + q - q - q - q + q + q + q + q 8 6 4 q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 9]][a, z]
Out[8]=	3 3 1 2 1 a a z 3 z z z -(----) + ---- - --- - - + -- - -- + --- - - - a z + -- 5 3 a z z z 5 3 a 3 a z a z a a a
In[9]:=	Kauffman[Link[10, NonAlternating, 9]][a, z]
Out[9]=	3 -4 3 2 1 2 1 a a 5 z 13 z 8 z -2 - a - -- - a - ---- - ---- - --- + - + -- + --- + ---- + --- - a z - 2 5 3 a z z z 5 3 a a a z a z a a 2 2 3 3 3 4 4 3 2 5 z 9 z 10 z 21 z 11 z 4 10 z 11 z > a z + 4 z + ---- + ---- - ----- - ----- - ----- - z - ----- - ----- + 4 2 5 3 a 4 2 a a a a a a 5 5 5 6 6 7 7 7 8 8 6 z 12 z 6 z 6 z 6 z z 2 z z z z > ---- + ----- + ---- + ---- + ---- - -- - ---- - -- - -- - -- 5 3 a 4 2 5 3 a 4 2 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 1 2 2 2 4 2 6 2 2 + -- + q + ----- + ----- + - + ---- + t + 2 q t + q t + q t + q t + 2 6 2 4 2 t 2 q q t q t q t 4 3 6 3 8 4 8 5 12 6 > q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n9

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