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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5 + 2q-4 - 2q-3 + 4q-2 - 3q-1 + 4 - 2q + 2q2

A2 (sl(3)) Invariant:

- q-16 + 3q-8 + 3q-6 + 4q-4 + 4q-2 + 3 + 4q2 + 2q4 + 3q6 + 2q8

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 - 2z-2 - 5 - 3z2 + a2z-2 + 4a2 + 3a2z2 + a2z4 - a4 - a4z2

Kauffman Polynomial:

a-2z-2 - 4a-2 + 3a-2z2 - 2a-1z-1 + 3a-1z - a-1z3 + a-1z5 + 2z-2 - 9 + 14z2 - 7z4 + 2z6 - 2az-1 + 5az - 2az3 - az5 + az7 + a2z-2 - 8a2 + 17a2z2 - 14a2z4 + 4a2z6 + 3a3z - 4a3z3 - a3z5 + a3z7 - 2a4 + 6a4z2 - 7a4z4 + 2a4z6 + a5z - 3a5z3 + a5z5

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 5               2 j = 3             1 1 j = 1           3 1   j = -1         2 3     j = -3       2 1       j = -5     1 3         j = -7   1 1           j = -9   1             j = -11 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]=	3
In[3]:=	Show[DrawMorseLink[Link[9, NonAlternating, 24]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 24]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[11, 16, 12, 17], X[9, 11, 10, 18], X[17, 5, 18, 10], > X[15, 1, 16, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 9}, {-2, -1, 5, 3, -7, 8}, {-6, 2, 4, -5, -9, 6, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-5 2 2 4 3 2 4 - q + -- - -- + -- - - - 2 q + 2 q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 3 3 4 4 2 4 6 8 3 - q + -- + -- + -- + -- + 4 q + 2 q + 3 q + 2 q 8 6 4 2 q q q q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 24]][a, z]
Out[8]=	2 2 2 4 2 1 a 2 2 2 4 2 2 4 -5 + -- + 4 a - a - -- + ----- + -- - 3 z + 3 a z - a z + a z 2 2 2 2 2 a z a z z
In[9]:=	Kauffman[Link[9, NonAlternating, 24]][a, z]
Out[9]=	2 4 2 4 2 1 a 2 2 a 3 z 3 -9 - -- - 8 a - 2 a + -- + ----- + -- - --- - --- + --- + 5 a z + 3 a z + 2 2 2 2 2 a z z a a z a z z 2 3 5 2 3 z 2 2 4 2 z 3 3 3 > a z + 14 z + ---- + 17 a z + 6 a z - -- - 2 a z - 4 a z - 2 a a 5 5 3 4 2 4 4 4 z 5 3 5 5 5 6 > 3 a z - 7 z - 14 a z - 7 a z + -- - a z - a z + a z + 2 z + a 2 6 4 6 7 3 7 > 4 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 1 1 3 2 1 2 - + 3 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 > q t + q t + q t + 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n24

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