L11n419

L11n418

L11n420

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Link Presentations

[edit Notes on L11n419's Link Presentations]

Planar diagram presentation	X8192 X16,8,17,7 X3,10,4,11 X2,18,3,17 X9,19,10,18 X20,12,21,11 X5,14,6,15 X15,13,16,22 X13,6,14,1 X19,5,20,4 X12,22,7,21
Gauss code	{1, -4, -3, 10, -7, 9}, {2, -1, -5, 3, 6, -11}, {-9, 7, -8, -2, 4, 5, -10, -6, 11, 8}

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) t(3)^3-t(2) t(3)^3+t(1) t(2)^2 t(3)^2-2 t(1) t(2) t(3)^2+t(2) t(3)^2-t(1) t(3)-t(1)^2 t(2) t(3)+2 t(1) t(2) t(3)+t(1)^2 t(2)-t(1) t(2)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^6+3 q^5-3 q^4+5 q^3-4 q^2+4 q+2 q^{-1} -2 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -2 z^4 a^{-2} -7 z^2 a^{-2} +3 z^2 a^{-4} +2 z^2-8 a^{-2} +5 a^{-4} - a^{-6} +4-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{-7} -z a^{-7} +3 z^4 a^{-6} -4 z^2 a^{-6} + a^{-6} +z^7 a^{-5} -3 z^5 a^{-5} +7 z^3 a^{-5} -3 z a^{-5} +z^8 a^{-4} -5 z^6 a^{-4} +16 z^4 a^{-4} -18 z^2 a^{-4} - a^{-4} z^{-2} +8 a^{-4} +2 z^7 a^{-3} -6 z^5 a^{-3} +11 z^3 a^{-3} -8 z a^{-3} +2 a^{-3} z^{-1} +z^8 a^{-2} -5 z^6 a^{-2} +16 z^4 a^{-2} -23 z^2 a^{-2} -2 a^{-2} z^{-2} +12 a^{-2} +z^7 a^{-1} -3 z^5 a^{-1} +5 z^3 a^{-1} -6 z a^{-1} +2 a^{-1} z^{-1} +3 z^4-9 z^2- z^{-2} +6 }[/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   \ -2 -1 0 1 2 3 4 5 χ 13               1 -1 11             2   2 9           2 2   0 7         3 1     2 5       2 3       1 3     2 2         0 1   1 3           2 -1 1 1             0 -3 2               2

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=3 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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