L11n390

L11n389

L11n391

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Link Presentations

[edit Notes on L11n390's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X3849 X22,14,19,13 X20,10,21,9 X10,20,11,19 X14,22,15,21 X11,18,12,5 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {8, -7, 9, -6}, {-4, -1, 2, 5, 7, -8, -10, 4, 6, -9, -11, -2, 3, 10}

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)-1) \left(t(1) t(3)^4+t(1) t(3)^3-t(1) t(3)^2+t(1) t(2) t(3)^2-t(2) t(3)^2+t(3)^2+t(2) t(3)+t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-7} -3 q^{-6} +3 q^{-5} -3 q^{-4} +4 q^{-3} +q^2-2 q^{-2} +2 q^{-1} +1 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 z^2-a^6 z^{-2} -a^6+4 a^4 z^2+4 a^4 z^{-2} +8 a^4-a^2 z^6-7 a^2 z^4-14 a^2 z^2-5 a^2 z^{-2} -13 a^2+z^4+5 z^2+2 z^{-2} +6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^8-z^2 a^8+3 z^5 a^7-6 z^3 a^7+a^7 z^{-1} +2 z^6 a^6-3 z^4 a^6-a^6 z^{-2} +a^6+3 z^5 a^5-9 z a^5+5 a^5 z^{-1} +6 z^4 a^4-13 z^2 a^4-4 a^4 z^{-2} +12 a^4+z^7 a^3-9 z^5 a^3+28 z^3 a^3-27 z a^3+9 a^3 z^{-1} +z^8 a^2-10 z^6 a^2+31 z^4 a^2-37 z^2 a^2-5 a^2 z^{-2} +21 a^2+z^7 a-9 z^5 a+22 z^3 a-18 z a+5 a z^{-1} +z^8-8 z^6+21 z^4-23 z^2-2 z^{-2} +11 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 5                     1 1 3                     1 1 1                 1     1 -1             3         3 -3           2 3 1       0 -5         3 1           2 -7         2 1           1 -9     3 3               0 -11   1 1                 0 -13   2                   -2 -15 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{ i=-1 }[/math] [math]\displaystyle{ r=-6 }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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} }[/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}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/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{ r=4 }[/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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