L10n23

L10n22

L10n24

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Link Presentations

[edit Notes on L10n23's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X16,10,17,9 X17,5,18,20 X13,19,14,18 X19,15,20,14 X8,16,9,15 X2536 X4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 4, -2, 10, -3, -6, 7, 8, -4, -5, 6, -7, 5}

A Braid Representative	{{{braid_table}}}
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) \left(t(2)^4+1\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{15/2}-2 q^{13/2}+2 q^{11/2}-3 q^{9/2}+2 q^{7/2}-3 q^{5/2}+q^{3/2}-\sqrt{q} }[/math] (db)
Signature	5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{-9} +z^5 a^{-7} +5 z^3 a^{-7} +6 z a^{-7} +2 a^{-7} z^{-1} -z^7 a^{-5} -6 z^5 a^{-5} -12 z^3 a^{-5} -12 z a^{-5} -5 a^{-5} z^{-1} +z^5 a^{-3} +5 z^3 a^{-3} +7 z a^{-3} +3 a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{-10} +z^3 a^{-9} -2 z a^{-9} +2 z^6 a^{-8} -8 z^4 a^{-8} +6 z^2 a^{-8} +3 z^7 a^{-7} -15 z^5 a^{-7} +21 z^3 a^{-7} -11 z a^{-7} +2 a^{-7} z^{-1} +z^8 a^{-6} -2 z^6 a^{-6} -7 z^4 a^{-6} +13 z^2 a^{-6} -5 a^{-6} +4 z^7 a^{-5} -21 z^5 a^{-5} +32 z^3 a^{-5} -19 z a^{-5} +5 a^{-5} z^{-1} +z^8 a^{-4} -4 z^6 a^{-4} +z^4 a^{-4} +7 z^2 a^{-4} -5 a^{-4} +z^7 a^{-3} -6 z^5 a^{-3} +12 z^3 a^{-3} -10 z a^{-3} +3 a^{-3} 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   \ -2 -1 0 1 2 3 4 5 χ 16               2 -2 14             1 1 0 12           2 2   0 10         1 1 1   1 8       1 2       1 6     2 1         1 4   1 3           2 2                 0 0 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=4 }[/math] [math]\displaystyle{ i=6 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{3} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\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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