L10n1

L10a174

L10n2

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Link Presentations

[edit Notes on L10n1's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X3849 X11,19,12,18 X17,5,18,20 X19,13,20,12 X9,16,10,17 X13,2,14,3
Gauss code	{1, 10, -5, -3}, {-4, -1, 2, 5, -9, 4, -6, 8, -10, -2, 3, 9, -7, 6, -8, 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{-u v^5+2 u v^4-2 u v^3-2 v^2+2 v-1}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{2}{q^{9/2}}-\frac{4}{q^{7/2}}+\frac{3}{q^{5/2}}+q^{3/2}-\frac{3}{q^{3/2}}+\frac{1}{q^{13/2}}-\frac{2}{q^{11/2}}-2 \sqrt{q}+\frac{2}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7-a^7 z^{-1} +z^5 a^5+5 z^3 a^5+7 z a^5+4 a^5 z^{-1} -z^7 a^3-6 z^5 a^3-12 z^3 a^3-11 z a^3-4 a^3 z^{-1} +z^5 a+4 z^3 a+3 z a+a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^8 z^2-a^8+2 a^7 z^3-3 a^7 z+a^7 z^{-1} +a^6 z^6-4 a^6 z^4+8 a^6 z^2-4 a^6+2 a^5 z^7-10 a^5 z^5+19 a^5 z^3-15 a^5 z+4 a^5 z^{-1} +a^4 z^8-2 a^4 z^6-6 a^4 z^4+15 a^4 z^2-7 a^4+4 a^3 z^7-19 a^3 z^5+27 a^3 z^3-16 a^3 z+4 a^3 z^{-1} +a^2 z^8-2 a^2 z^6-6 a^2 z^4+11 a^2 z^2-4 a^2+2 a z^7-9 a z^5+10 a z^3-4 a z+a z^{-1} +z^6-4 z^4+3 z^2-1 }[/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 χ 4                 1 -1 2               1   1 0             1 1   0 -2         1 3 1     1 -4         2 2       0 -6       3 2         1 -8   1 1 2           2 -10   2 2             0 -12   1               1 -14 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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/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.

L10a174 L10n2