L10a98

L10a97

L10a99

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Link Presentations

[edit Notes on L10a98's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X18,7,19,8 X20,15,9,16 X16,19,17,20 X8,9,1,10 X4,13,5,14 X6,17,7,18
Gauss code	{1, -2, 3, -9, 4, -10, 5, -8}, {8, -1, 2, -3, 9, -4, 6, -7, 10, -5, 7, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{2 t(2)^3 t(1)^3-t(2)^2 t(1)^3-t(2)^3 t(1)^2+3 t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+3 t(2) t(1)-t(1)-t(2)+2}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}-\frac{1}{q^{27/2}}+\frac{2}{q^{25/2}}-\frac{3}{q^{23/2}}+\frac{4}{q^{21/2}}-\frac{5}{q^{19/2}}+\frac{5}{q^{17/2}}-\frac{4}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{3}{q^{11/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{11} z^5+4 a^{11} z^3+3 a^{11} z-a^9 z^7-5 a^9 z^5-6 a^9 z^3+a^9 z^{-1} -a^7 z^7-6 a^7 z^5-11 a^7 z^3-7 a^7 z-a^7 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^3 a^{17}+z a^{17}-2 z^4 a^{16}+2 z^2 a^{16}-2 z^5 a^{15}+z^3 a^{15}-2 z^6 a^{14}+2 z^4 a^{14}-z^2 a^{14}-2 z^7 a^{13}+4 z^5 a^{13}-3 z^3 a^{13}+z a^{13}-2 z^8 a^{12}+7 z^6 a^{12}-9 z^4 a^{12}+5 z^2 a^{12}-z^9 a^{11}+3 z^7 a^{11}-3 z^5 a^{11}+5 z^3 a^{11}-3 z a^{11}-3 z^8 a^{10}+13 z^6 a^{10}-15 z^4 a^{10}+5 z^2 a^{10}-z^9 a^9+4 z^7 a^9-3 z^5 a^9-z^3 a^9+2 z a^9-a^9 z^{-1} -z^8 a^8+4 z^6 a^8-2 z^4 a^8-3 z^2 a^8+a^8-z^7 a^7+6 z^5 a^7-11 z^3 a^7+7 z a^7-a^7 z^{-1} }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\   r    \    j   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                     1 1 -8                   1 1 0 -10                 2     2 -12               1 1     0 -14             3 2       1 -16           2 1         -1 -18         3 3           0 -20       1 2             1 -22     2 3               -1 -24   1 2                 1 -26   1                   -1 -28 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-8 }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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