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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X8493 X14,10,15,9 X10,14,11,13 X18,12,5,11 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:

q-5/2 - 3q-3/2 + 5q-1/2 - 9q1/2 + 9q3/2 - 10q5/2 + 8q7/2 - 6q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-8 + q-6 - q-2 + 4 + q2 + 4q4 + 3q6 + q8 + 2q10 - 3q12 - q16 - 2q18 + q20

HOMFLY-PT Polynomial:

a-5z-1 - a-5z3 - 3a-3z-1 - 2a-3z + a-3z3 + a-3z5 + 2a-1z-1 + 3a-1z + 2a-1z3 + a-1z5 - az - az3

Kauffman Polynomial:

a-7z3 - a-7z5 - a-6 - 2a-6z2 + 8a-6z4 - 4a-6z6 + a-5z-1 - 4a-5z3 + 10a-5z5 - 5a-5z7 - 3a-4 - 3a-4z2 + 13a-4z4 - 4a-4z6 - 2a-4z8 + 3a-3z-1 - 3a-3z - 6a-3z3 + 16a-3z5 - 9a-3z7 - 3a-2 - 2a-2z2 + 9a-2z4 - 4a-2z6 - 2a-2z8 + 2a-1z-1 - 5a-1z + 3a-1z3 + 2a-1z5 - 4a-1z7 + 3z4 - 4z6 - 2az + 4az3 - 3az5 + a2z2 - a2z4

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9a3

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