L10n56

L10n55

L10n57

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Link Presentations

[edit Notes on L10n56's Link Presentations]

Planar diagram presentation	X10,1,11,2 X18,11,19,12 X20,5,9,6 X7,15,8,14 X12,4,13,3 X13,16,14,17 X15,7,16,6 X8,9,1,10 X4,19,5,20 X2,18,3,17
Gauss code	{1, -10, 5, -9, 3, 7, -4, -8}, {8, -1, 2, -5, -6, 4, -7, 6, 10, -2, 9, -3}

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{(u-1) (u+1)^2 (v-1)}{u^{3/2} \sqrt{v}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{5/2}+q^{3/2}-\sqrt{q}-\frac{1}{q^{5/2}}-\frac{1}{q^{9/2}}+\frac{1}{q^{11/2}} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 (-z)-a^3 z+a z^5+5 a z^3-z^3 a^{-1} +5 a z+a z^{-1} -3 z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^8-z^8-a^3 z^7-2 a z^7-z^7 a^{-1} +6 a^2 z^6+6 z^6-a^5 z^5+6 a^3 z^5+13 a z^5+6 z^5 a^{-1} -a^6 z^4-8 a^2 z^4-9 z^4+4 a^5 z^3-8 a^3 z^3-22 a z^3-10 z^3 a^{-1} +3 a^6 z^2+a^4 z^2+a^2 z^2+3 z^2-2 a^5 z+2 a^3 z+10 a z+6 z a^{-1} +1-a z^{-1} - a^{-1} 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                   1 1 4                     0 2               1 1   0 0           2 1       1 -2         1 2 1       0 -4       1 1 1         1 -6       1             1 -8   1 1 1             1 -10                     0 -12 1                   -1

Integral Khovanov Homology (db, data source)

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