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L10n66 L10n68

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-6 - 2q-5 + 5q-4 - 5q-3 + 6q-2 - 6q-1 + 6 - 3q + 2q2

A2 (sl(3)) Invariant:

q-20 + 2q-18 + q-16 + 4q-14 + 3q-12 + q-10 + 2q-8 - q-6 + q-4 + q-2 + 2 + 4q2 + q4 + 3q6 + 2q8

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 - 3z-2 - 6 - 4z2 + 4a2z-2 + 8a2 + 6a2z2 + 2a2z4 - 3a4z-2 - 5a4 - 3a4z2 + a6z-2 + a6

Kauffman Polynomial:

a-2z-2 - 4a-2 + 3a-2z2 - a-1z-1 + a-1z + a-1z3 + a-1z5 + 3z-2 - 14 + 23z2 - 13z4 + 4z6 - az-1 + az + 9az3 - 11az5 + 4az7 + 4a2z-2 - 21a2 + 39a2z2 - 30a2z4 + 6a2z6 + a2z8 - a3z-1 + a3z + 11a3z3 - 18a3z5 + 6a3z7 + 3a4z-2 - 14a4 + 25a4z2 - 21a4z4 + 3a4z6 + a4z8 - a5z-1 + a5z + 3a5z3 - 6a5z5 + 2a5z7 + a6z-2 - 4a6 + 6a6z2 - 4a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 5                 2 j = 3               1   j = 1             5 2   j = -1           4 4     j = -3         2 3 1     j = -5       3 4         j = -7     2 2           j = -9   1 4             j = -11   1               j = -13 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[10, NonAlternating, 67]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 67]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 16, 12, 17], X[13, 19, 14, 18], > X[17, 20, 18, 9], X[19, 13, 20, 12], X[8, 16, 5, 15], X[14, 8, 15, 7], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 8, -7}, > {10, -2, -3, 6, -4, -8, 7, 3, -5, 4, -6, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-6 2 5 5 6 6 2 6 + q - -- + -- - -- + -- - - - 3 q + 2 q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 -16 4 3 -10 2 -6 -4 -2 2 4 2 + q + --- + q + --- + --- + q + -- - q + q + q + 4 q + q + 18 14 12 8 q q q q 6 8 > 3 q + 2 q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 67]][a, z]
Out[8]=	2 4 6 2 2 4 6 3 1 4 a 3 a a 2 2 2 -6 + -- + 8 a - 5 a + a - -- + ----- + ---- - ---- + -- - 4 z + 6 a z - 2 2 2 2 2 2 2 a z a z z z z 4 2 2 4 > 3 a z + 2 a z
In[9]:=	Kauffman[Link[10, NonAlternating, 67]][a, z]
Out[9]=	2 4 6 4 2 4 6 3 1 4 a 3 a a 1 a -14 - -- - 21 a - 14 a - 4 a + -- + ----- + ---- + ---- + -- - --- - - - 2 2 2 2 2 2 2 a z z a z a z z z z 3 5 2 a a z 3 5 2 3 z 2 2 4 2 > -- - -- + - + a z + a z + a z + 23 z + ---- + 39 a z + 25 a z + z z a 2 a 3 6 2 z 3 3 3 5 3 4 2 4 4 4 > 6 a z + -- + 9 a z + 11 a z + 3 a z - 13 z - 30 a z - 21 a z - a 5 6 4 z 5 3 5 5 5 6 2 6 4 6 > 4 a z + -- - 11 a z - 18 a z - 6 a z + 4 z + 6 a z + 3 a z + a 6 6 7 3 7 5 7 2 8 4 8 > a z + 4 a z + 6 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-3 4 1 1 1 4 2 2 3 q + - + 5 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 q t q t q t q t q t q t q t 4 2 3 4 3 5 2 > ----- + ----- + ---- + --- + 2 q t + q t + 2 q t 5 2 3 2 3 q t q t q t q t

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