L11n357

L11n356

L11n358

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Link Presentations

[edit Notes on L11n357's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,19,12,18 X7,16,8,17 X15,8,16,9 X17,15,18,22 X13,21,14,20 X19,13,20,12 X21,5,22,14 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -6, 3, -8, 7, -9, 6}, {10, -1, -4, 5, 11, -2, -3, 8, -7, 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{(w-1) \left(-u v^2 w-u v+u+v^2 w^3-v w^3-w^2\right)}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-5} -q^4- q^{-4} +2 q^3+3 q^{-3} -2 q^2-3 q^{-2} +4 q+4 q^{-1} -3 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 z^2+a^4 z^{-2} +3 a^4-2 a^2 z^4-z^4 a^{-2} -8 a^2 z^2-3 z^2 a^{-2} -2 a^2 z^{-2} -8 a^2- a^{-2} +z^6+5 z^4+8 z^2+ z^{-2} +6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^3 z^9+a z^9+a^4 z^8+4 a^2 z^8+3 z^8-5 a^3 z^7-2 a z^7+3 z^7 a^{-1} -7 a^4 z^6-24 a^2 z^6+2 z^6 a^{-2} -15 z^6+5 a^3 z^5-8 a z^5-12 z^5 a^{-1} +z^5 a^{-3} +17 a^4 z^4+47 a^2 z^4-6 z^4 a^{-2} +24 z^4+5 a^3 z^3+18 a z^3+11 z^3 a^{-1} -2 z^3 a^{-3} -17 a^4 z^2-38 a^2 z^2+3 z^2 a^{-2} +2 z^2 a^{-4} -20 z^2-7 a^3 z-10 a z-3 z a^{-1} +z a^{-3} +z a^{-5} +7 a^4+14 a^2- a^{-2} - a^{-4} +8+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 9                   1 -1 7                 1   1 5               2 2   0 3             2 1 1   2 1           2 3       1 -1         2 2 1       1 -3       2 3           1 -5     1 1             0 -7   1 3               2 -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}^{3} }[/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=-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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}_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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