L10n22

L10n21

L10n23

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Link Presentations

[edit Notes on L10n22's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X9,16,10,17 X20,17,5,18 X18,13,19,14 X14,19,15,20 X15,8,16,9 X2536 X4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 3, 8, -4, -2, 10, -3, 6, -7, -8, 4, 5, -6, 7, -5}

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

Integral Khovanov Homology (db, data source)

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