L11n52

L11n51

L11n53

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Link Presentations

[edit Notes on L11n52's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X13,18,14,19 X16,9,17,10 X8,17,9,18 X19,22,20,5 X15,21,16,20 X21,15,22,14 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -2, 11, -3, -4, 9, -8, -5, 6, 4, -7, 8, -9, 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(1)-1) (t(2)-1)^3}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -2 q^{5/2}+3 q^{3/2}-5 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^5+a^5 z^{-1} -2 z^3 a^3-4 z a^3-2 a^3 z^{-1} +z^5 a+3 z^3 a+3 z a+a z^{-1} -z^3 a^{-1} + a^{-1} z^{-1} - a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^5 z^7-5 a^5 z^5+8 a^5 z^3-5 a^5 z+a^5 z^{-1} +2 a^4 z^8-9 a^4 z^6+11 a^4 z^4-5 a^4 z^2+a^4+a^3 z^9-16 a^3 z^5+27 a^3 z^3-15 a^3 z+z a^{-3} +2 a^3 z^{-1} - a^{-3} z^{-1} +5 a^2 z^8-22 a^2 z^6+26 a^2 z^4+z^4 a^{-2} -12 a^2 z^2-z^2 a^{-2} +3 a^2+ a^{-2} +a z^9+a z^7+2 z^7 a^{-1} -19 a z^5-8 z^5 a^{-1} +28 a z^3+9 z^3 a^{-1} -12 a z-z a^{-1} +a z^{-1} - a^{-1} z^{-1} +3 z^8-13 z^6+16 z^4-8 z^2+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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 6                 2 2 4               1   -1 2             4 2   2 0           3 3     0 -2         2 3 1     0 -4       3 3         0 -6     1 2           -1 -8   1 3             2 -10   1               -1 -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=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/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}_2^{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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