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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X12,8,5,7 X8,12,9,11 X2536 X4,9,1,10

Gauss Code:

{{1, -5, 2, -6}, {5, -1, 3, -4, 6, -2, 4, -3}}

Jones Polynomial:

- q-9/2 + q-7/2 - 3q-5/2 + 2q-3/2 - 2q-1/2 + 2q1/2 - q3/2

A2 (sl(3)) Invariant:

q-16 + 2q-14 + q-12 + 2q-10 + 2q-8 + q-6 + q-4 - q-2 - q2 + q6

HOMFLY-PT Polynomial:

- a-1z + az + az3 - a3z-1 - 2a3z + a5z-1

Kauffman Polynomial:

a-1z - a-1z3 + 3z2 - 2z4 - az5 + 3a2z2 - 3a2z4 - a3z-1 + a3z - a3z5 + a4 - a4z4 - a5z-1 + 2a5z - a5z3

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L6a1

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