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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X9,18,10,19 X11,20,12,13 X13,12,14,5 X4,17,1,18 X19,10,20,11

Gauss Code:

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

Jones Polynomial:

q-10 - 2q-9 + 4q-8 - 5q-7 + 6q-6 - 5q-5 + 5q-4 - 2q-3 + 2q-2

A2 (sl(3)) Invariant:

q-30 + 2q-26 + 2q-24 + q-22 + 3q-20 + q-18 + 4q-16 + 3q-14 + 3q-12 + 4q-10 + q-8 + 2q-6

HOMFLY-PT Polynomial:

a4z-2 + 6a4 + 7a4z2 + 2a4z4 - 2a6z-2 - 9a6 - 10a6z2 - 5a6z4 - a6z6 + a8z-2 + 3a8 + 3a8z2 + a8z4

Kauffman Polynomial:

- a4z-2 + 6a4 - 9a4z2 + 3a4z4 + 2a5z-1 - 9a5z + 7a5z3 - 3a5z5 + a5z7 - 2a6z-2 + 9a6 - 12a6z2 + 8a6z4 - 3a6z6 + a6z8 + 2a7z-1 - 9a7z + 13a7z3 - 8a7z5 + 3a7z7 - a8z-2 + 3a8 - 3a8z2 + 3a8z4 - a8z6 + a8z8 + 3a9z3 - 3a9z5 + 2a9z7 - 2a10z2 - a10z4 + 2a10z6 - 3a11z3 + 2a11z5 + a12 - 2a12z2 + a12z4

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3                 2 j = -5               2 2 j = -7             3     j = -9           2 2     j = -11         4 3       j = -13       1 2         j = -15     3 4           j = -17   1 3             j = -19   1               j = -21 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, 79]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 79]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[8, 4, 9, 3], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[9, 18, 10, 19], X[11, 20, 12, 13], X[13, 12, 14, 5], > X[4, 17, 1, 18], X[19, 10, 20, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -9}, {-2, -1, 5, -3, -6, 10, -7, 8}, > {-8, 2, 4, -5, 9, 6, -10, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-10 2 4 5 6 5 5 2 2 q - -- + -- - -- + -- - -- + -- - -- + -- 9 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 2 2 -22 3 -18 4 3 3 4 -8 2 q + --- + --- + q + --- + q + --- + --- + --- + --- + q + -- 26 24 20 16 14 12 10 6 q q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 79]][a, z]
Out[8]=	4 6 8 4 6 8 a 2 a a 4 2 6 2 8 2 4 4 6 a - 9 a + 3 a + -- - ---- + -- + 7 a z - 10 a z + 3 a z + 2 a z - 2 2 2 z z z 6 4 8 4 6 6 > 5 a z + a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 79]][a, z]
Out[9]=	4 6 8 5 7 4 6 8 12 a 2 a a 2 a 2 a 5 7 6 a + 9 a + 3 a + a - -- - ---- - -- + ---- + ---- - 9 a z - 9 a z - 2 2 2 z z z z z 4 2 6 2 8 2 10 2 12 2 5 3 7 3 > 9 a z - 12 a z - 3 a z - 2 a z - 2 a z + 7 a z + 13 a z + 9 3 11 3 4 4 6 4 8 4 10 4 12 4 > 3 a z - 3 a z + 3 a z + 8 a z + 3 a z - a z + a z - 5 5 7 5 9 5 11 5 6 6 8 6 10 6 > 3 a z - 8 a z - 3 a z + 2 a z - 3 a z - a z + 2 a z + 5 7 7 7 9 7 6 8 8 8 > a z + 3 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 3 3 4 1 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 2 4 3 2 2 3 2 > ------ + ------ + ------ + ----- + ----- + ----- + ---- 13 4 11 4 11 3 9 3 9 2 7 2 5 q t q t q t q t q t q t q t

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