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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,17,12,16 X14,7,15,8 X8,15,9,16 X17,5,18,20 X13,18,14,19 X19,12,20,13 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-15/2 + 3q-13/2 - 5q-11/2 + 6q-9/2 - 7q-7/2 + 5q-5/2 - 5q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- az - az3 - a3z-1 + 2a3z3 + a3z5 + a5z-1 - 2a5z - 2a5z3 + a7z

Kauffman Polynomial:

- az + 2az3 - az5 - 2a2z2 + 7a2z4 - 3a2z6 - a3z-1 + a3z - a3z3 + 6a3z5 - 3a3z7 + a4 - 3a4z2 + 7a4z4 - 2a4z6 - a4z8 - a5z-1 + 2a5z - 6a5z3 + 8a5z5 - 4a5z7 + a6z2 - 3a6z4 + a6z6 - a6z8 + a7z - 4a7z3 + a7z5 - a7z7 + 2a8z2 - 3a8z4 + a9z - a9z3

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                 1 j = 0               2   j = -2             3 1   j = -4           3 3     j = -6         4 2       j = -8       2 3         j = -10     3 4           j = -12   1 3             j = -14   2               j = -16 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, 27]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 27]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 17, 12, 16], X[14, 7, 15, 8], > X[8, 15, 9, 16], X[17, 5, 18, 20], X[13, 18, 14, 19], X[19, 12, 20, 13], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 4, -5, 10, -2, -3, 8, -7, -4, 5, 3, -6, 7, > -8, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 5 6 7 5 5 3 -q + ----- - ----- + ---- - ---- + ---- - ---- + ------- - Sqrt[q] 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 -18 2 3 -10 3 -6 -4 2 -1 + q - --- + q + --- + --- + q + -- - q + q + q 20 14 12 8 q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 27]][a, z]
Out[8]=	3 5 a a 5 7 3 3 3 5 3 3 5 -(--) + -- - a z - 2 a z + a z - a z + 2 a z - 2 a z + a z z z
In[9]:=	Kauffman[Link[10, NonAlternating, 27]][a, z]
Out[9]=	3 5 4 a a 3 5 7 9 2 2 4 2 6 2 a - -- - -- - a z + a z + 2 a z + a z + a z - 2 a z - 3 a z + a z + z z 8 2 3 3 3 5 3 7 3 9 3 2 4 4 4 > 2 a z + 2 a z - a z - 6 a z - 4 a z - a z + 7 a z + 7 a z - 6 4 8 4 5 3 5 5 5 7 5 2 6 4 6 > 3 a z - 3 a z - a z + 6 a z + 8 a z + a z - 3 a z - 2 a z + 6 6 3 7 5 7 7 7 4 8 6 8 > a z - 3 a z - 4 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 2 1 3 3 4 2 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 4 2 16 6 14 5 12 5 12 4 10 4 10 3 8 3 8 2 q q q t q t q t q t q t q t q t q t 4 2 3 t 2 2 > ----- + ---- + ---- + 2 t + -- + q t 6 2 6 4 2 q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n27

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