L11n428

L11n427

L11n429

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Link Presentations

[edit Notes on L11n428's Link Presentations]

Planar diagram presentation	X8192 X9,20,10,21 X5,15,6,14 X12,14,7,13 X16,8,17,7 X22,18,13,17 X3,10,4,11 X18,11,19,12 X15,1,16,6 X19,4,20,5 X2,21,3,22
Gauss code	{1, -11, -7, 10, -3, 9}, {5, -1, -2, 7, 8, -4}, {4, 3, -9, -5, 6, -8, -10, 2, 11, -6}

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{(w-1) (u v-1)^2}{u v \sqrt{w}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^4+q^3+q+1+ q^{-2} + q^{-4} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 z^{-2} +2 a^4-a^2 z^4-z^4 a^{-2} -6 a^2 z^2-4 z^2 a^{-2} -2 a^2 z^{-2} -7 a^2-2 a^{-2} +z^6+6 z^4+10 z^2+ z^{-2} +7 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^4 z^4-4 a^4 z^2-a^4 z^{-2} +5 a^4+z^7 a^{-3} -6 z^5 a^{-3} +a^3 z^3+9 z^3 a^{-3} -3 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +z^8 a^{-2} -2 a^2 z^6-7 z^6 a^{-2} +13 a^2 z^4+15 z^4 a^{-2} -24 a^2 z^2-12 z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+4 a^{-2} +z^7 a^{-1} -a z^5-7 z^5 a^{-1} +5 a z^3+13 z^3 a^{-1} -7 a z-6 z a^{-1} +2 a z^{-1} +z^8-9 z^6+27 z^4-32 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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 9                   1 -1 7                     0 5             1 1 1   1 3             1       1 1         2 1 1       2 -1       1 3 1         1 -3     1 1 1           1 -5   1 3 1             1 -7     1               1 -9 1                   1

Integral Khovanov Homology (db, data source)

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