L11n426

L11n425

L11n427

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Link Presentations

[edit Notes on L11n426's Link Presentations]

Planar diagram presentation	X8192 X16,8,17,7 X14,6,15,5 X3,10,4,11 X4,14,5,13 X17,2,18,3 X9,19,10,18 X12,21,7,22 X22,11,13,12 X20,16,21,15 X6,19,1,20
Gauss code	{1, 6, -4, -5, 3, -11}, {2, -1, -7, 4, 9, -8}, {5, -3, 10, -2, -6, 7, 11, -10, 8, -9}

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)-t(2))^2 (t(3)-1)}{t(1) t(2) \sqrt{t(3)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-5} - q^{-4} -q^3+ q^{-3} +q^2+ q^{-1} +2 }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 z^2+a^4 z^{-2} +2 a^4-a^2 z^4-5 a^2 z^2-2 a^2 z^{-2} -z^2 a^{-2} -7 a^2-2 a^{-2} +z^4+5 z^2+ z^{-2} +7 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^4 z^8-7 a^4 z^6+15 a^4 z^4-12 a^4 z^2-a^4 z^{-2} +5 a^4+a^3 z^9-7 a^3 z^7+14 a^3 z^5-7 a^3 z^3+z^3 a^{-3} -3 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +2 a^2 z^8-15 a^2 z^6+35 a^2 z^4+z^4 a^{-2} -32 a^2 z^2-4 z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+4 a^{-2} +a z^9-7 a z^7+13 a z^5-z^5 a^{-1} -3 a z^3+5 z^3 a^{-1} -7 a z-6 z a^{-1} +2 a z^{-1} +z^8-8 z^6+21 z^4-24 z^2- z^{-2} +13 }[/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 χ 7                   1 -1 5                     0 3             1 1 1   1 1             2       2 -1         1 1 3       3 -3       1 2           1 -5       1 2           1 -7   1 1               0 -9                     0 -11 1                   1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/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} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_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=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.

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