L10n78

L10n77

L10n79

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Link Presentations

[edit Notes on L10n78's Link Presentations]

Planar diagram presentation	X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X9,18,10,19 X11,20,12,13 X13,12,14,5 X17,1,18,4 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}

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(3)^2 t(2)^3-t(1) t(3)^3 t(2)^2+t(1) t(3)^2 t(2)^2-t(3)^2 t(2)^2+t(3) t(2)^2-t(1) t(3)^2 t(2)+t(1) t(3) t(2)-t(3) t(2)+t(2)-t(1) t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{-2} - q^{-3} +3 q^{-4} -3 q^{-5} +4 q^{-6} -3 q^{-7} +2 q^{-8} - q^{-9} + q^{-10} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^4+4 a^8 z^2+a^8 z^{-2} +4 a^8-a^6 z^6-6 a^6 z^4-13 a^6 z^2-2 a^6 z^{-2} -11 a^6+2 a^4 z^4+8 a^4 z^2+a^4 z^{-2} +7 a^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{12}-3 z^2 a^{12}+a^{12}+z^5 a^{11}-2 z^3 a^{11}+z^6 a^{10}-2 z^4 a^{10}+z^2 a^{10}+z^7 a^9-3 z^5 a^9+4 z^3 a^9+z^8 a^8-5 z^6 a^8+11 z^4 a^8-8 z^2 a^8-a^8 z^{-2} +5 a^8+2 z^7 a^7-9 z^5 a^7+17 z^3 a^7-11 z a^7+2 a^7 z^{-1} +z^8 a^6-6 z^6 a^6+17 z^4 a^6-23 z^2 a^6-2 a^6 z^{-2} +13 a^6+z^7 a^5-5 z^5 a^5+11 z^3 a^5-11 z a^5+2 a^5 z^{-1} +3 z^4 a^4-11 z^2 a^4-a^4 z^{-2} +8 a^4 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                 2 2 -5               1 2 1 -7             2     2 -9           1 1     0 -11         3 2       1 -13       1 2         1 -15     1 2           -1 -17     1             1 -19 1 1               0 -21 1                 1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{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}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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