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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X16,12,9,11 X12,16,13,15 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-7 - q-6 + 4q-5 - 4q-4 + 6q-3 - 4q-2 + 4q-1 - 3 + q

A2 (sl(3)) Invariant:

q-24 + 3q-22 + 3q-20 + 4q-18 + 6q-16 + 3q-14 + 3q-12 + 2q-10 + q-8 + 2q-6 - q-4 + q-2 - 1 - q2 + q4

HOMFLY-PT Polynomial:

z2 - a2z2 - a2z4 + a4z-2 + 3a4 + 3a4z2 - 2a6z-2 - 3a6 + a8z-2

Kauffman Polynomial:

- z2 + z4 - 5az3 + 3az5 + 2a2z2 - 5a2z4 + 3a2z6 - 3a3z3 + 2a3z5 + a3z7 - a4z-2 + 3a4 - 5a4z4 + 4a4z6 + 2a5z-1 - 3a5z + 2a5z3 + a5z7 - 2a6z-2 + 5a6 - 6a6z2 + 2a6z4 + a6z6 + 2a7z-1 - 3a7z + a7z5 - a8z-2 + 3a8 - 3a8z2 + a8z4

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                 1 j = 1               2   j = -1             2 1   j = -3           3 3     j = -5         3 1       j = -7       1 3         j = -9     3 3           j = -11   1 4             j = -13                   j = -15 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[8, Alternating, 15]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[8, Alternating, 15]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], > X[16, 12, 9, 11], X[12, 16, 13, 15], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, 2, -8}, {7, -1, 3, -4}, {8, -2, 5, -6, 4, -3, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-7 -6 4 4 6 4 4 -3 + q - q + -- - -- + -- - -- + - + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 3 3 4 6 3 3 2 -8 2 -4 -2 -1 + q + --- + --- + --- + --- + --- + --- + --- + q + -- - q + q - 22 20 18 16 14 12 10 6 q q q q q q q q 2 4 > q + q
In[8]:=	HOMFLYPT[Link[8, Alternating, 15]][a, z]
Out[8]=	4 6 8 4 6 a 2 a a 2 2 2 4 2 2 4 3 a - 3 a + -- - ---- + -- + z - a z + 3 a z - a z 2 2 2 z z z
In[9]:=	Kauffman[Link[8, Alternating, 15]][a, z]
Out[9]=	4 6 8 5 7 4 6 8 a 2 a a 2 a 2 a 5 7 2 3 a + 5 a + 3 a - -- - ---- - -- + ---- + ---- - 3 a z - 3 a z - z + 2 2 2 z z z z z 2 2 6 2 8 2 3 3 3 5 3 4 2 4 > 2 a z - 6 a z - 3 a z - 5 a z - 3 a z + 2 a z + z - 5 a z - 4 4 6 4 8 4 5 3 5 7 5 2 6 4 6 > 5 a z + 2 a z + a z + 3 a z + 2 a z + a z + 3 a z + 4 a z + 6 6 3 7 5 7 > a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 2 1 1 4 3 3 1 3 3 -- + - + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 3 q 15 6 11 5 11 4 9 4 9 3 7 3 7 2 5 2 q q t q t q t q t q t q t q t q t 1 3 t 3 2 > ---- + ---- + - + 2 q t + q t 5 3 q q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L8a15

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