L11n70

L11n69

L11n71

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Link Presentations

[edit Notes on L11n70's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,8,17,7 X17,22,18,5 X11,18,12,19 X13,20,14,21 X19,12,20,13 X21,14,22,15 X8,16,9,15 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, -5, 7, -6, 8, 9, -3, -4, 5, -7, 6, -8, 4}

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{t(1) t(2)^5-t(2)^5-2 t(1) t(2)^4+t(2)^4+t(1) t(2)^3-t(2)^3-t(1) t(2)^2+t(2)^2+t(1) t(2)-2 t(2)-t(1)+1}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{2}{q^{17/2}}+\frac{1}{q^{19/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^9 z^{-1} -z^5 a^7-3 z^3 a^7+z a^7+3 a^7 z^{-1} +z^7 a^5+5 z^5 a^5+6 z^3 a^5-2 a^5 z^{-1} -z^5 a^3-4 z^3 a^3-3 z a^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^2 a^{12}-2 z^3 a^{11}-3 z^4 a^{10}+3 z^2 a^{10}-a^{10}-4 z^5 a^9+8 z^3 a^9-4 z a^9+a^9 z^{-1} -z^8 a^8+3 z^6 a^8-6 z^4 a^8+10 z^2 a^8-3 a^8-z^9 a^7+4 z^7 a^7-8 z^5 a^7+14 z^3 a^7-10 z a^7+3 a^7 z^{-1} -3 z^8 a^6+13 z^6 a^6-16 z^4 a^6+9 z^2 a^6-3 a^6-z^9 a^5+3 z^7 a^5+z^5 a^5-3 z^3 a^5-3 z a^5+2 a^5 z^{-1} -2 z^8 a^4+10 z^6 a^4-13 z^4 a^4+3 z^2 a^4-z^7 a^3+5 z^5 a^3-7 z^3 a^3+3 z a^3 }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                   1 1 -2                 1   -1 -4               2 1   1 -6             2 2     0 -8           3 1       2 -10         1 2         1 -12       3 3           0 -14     1 2             1 -16   1 2               -1 -18   1                 1 -20 1                   -1

Integral Khovanov Homology (db, data source)

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