L11n287

L11n286

L11n288

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Link Presentations

[edit Notes on L11n287's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X4,15,1,16 X21,18,22,19 X9,21,10,20 X19,5,20,10 X11,16,12,17 X17,22,18,11
Gauss code	{1, -4, 3, -6}, {-2, -1, 5, -3, -8, 9}, {-10, 2, 4, -5, 6, 10, -11, 7, -9, 8, -7, 11}

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