L10n97

L10n96

L10n98

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Link Presentations

[edit Notes on L10n97's Link Presentations]

Planar diagram presentation	X6172 X2536 X20,13,15,14 X3,11,4,10 X9,1,10,4 X7,17,8,16 X15,5,16,8 X18,11,19,12 X12,19,13,20 X14,17,9,18
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 8, -9, 3, -10}, {-7, 6, 10, -8, 9, -3}

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(4)^2 t(3)^2+t(1) t(4) t(3)^2+t(2) t(4)^2 t(3)+t(1) t(3)-t(1) t(4) t(3)-t(2) t(4) t(3)-t(1) t(2)+t(2) t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{q^{9/2}}+\frac{1}{q^{7/2}}-q^{5/2}-\frac{3}{q^{5/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{11/2}}-\sqrt{q}-\frac{2}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^3+a^5 z^{-3} +3 a^5 z+3 a^5 z^{-1} -a^3 z^5-6 a^3 z^3-3 a^3 z^{-3} -13 a^3 z-10 a^3 z^{-1} +a z^5+7 a z^3+3 a z^{-3} -z^3 a^{-1} - a^{-1} z^{-3} +14 a z+11 a z^{-1} -4 z a^{-1} -4 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^7+4 z^3 a^7-3 z a^7+a^7 z^{-1} -z^6 a^6+3 z^4 a^6-a^6-z^7 a^5+4 z^5 a^5-4 z^3 a^5+3 z a^5-3 a^5 z^{-1} +a^5 z^{-3} -2 z^6 a^4+10 z^4 a^4-16 z^2 a^4-3 a^4 z^{-2} +11 a^4-z^7 a^3+7 z^5 a^3-18 z^3 a^3+21 z a^3-12 a^3 z^{-1} +3 a^3 z^{-3} -2 z^6 a^2+15 z^4 a^2-33 z^2 a^2-6 a^2 z^{-2} +24 a^2-z^7 a+9 z^5 a-25 z^3 a+28 z a-14 a z^{-1} +3 a z^{-3} -z^6+8 z^4-17 z^2-3 z^{-2} +13-z^7 a^{-1} +7 z^5 a^{-1} -15 z^3 a^{-1} +13 z a^{-1} -6 a^{-1} z^{-1} + a^{-1} z^{-3} }[/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 4 χ 6                     1 1 4                     1 1 2                 1     1 0             3         3 -2           1 4 1       2 -4         3             3 -6       1 4 1           2 -8     1 1 1             1 -10     1                 1 -12 1 1                   0 -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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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} }[/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} }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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