L11n432

L11n431

L11n433

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Link Presentations

[edit Notes on L11n432's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X15,20,16,21 X5,17,6,16 X11,19,12,18 X17,11,18,10 X19,5,20,4
Gauss code	{1, -4, 2, 11, -8, -6}, {4, -1, 5, 10, -9, -3}, {3, -2, -7, 8, -10, 9, -11, 7, 6, -5}

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)) \left(-t(1) t(3)^2+t(1) t(2) t(3)^2-t(2) t(3)^2+t(1)+t(2)-1\right)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^9-q^8+q^7+q^5+q^4+q^2-q+1 }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^6 a^{-4} +z^4 a^{-2} -5 z^4 a^{-4} +4 z^2 a^{-2} -5 z^2 a^{-4} +z^2 a^{-8} +2 a^{-2} -3 a^{-6} + a^{-8} + a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-10} -5 z^4 a^{-10} +5 z^2 a^{-10} +z^7 a^{-9} -5 z^5 a^{-9} +5 z^3 a^{-9} +z^8 a^{-8} -7 z^6 a^{-8} +16 z^4 a^{-8} -17 z^2 a^{-8} - a^{-8} z^{-2} +7 a^{-8} +z^7 a^{-7} -7 z^5 a^{-7} +13 z^3 a^{-7} -9 z a^{-7} +2 a^{-7} z^{-1} +z^8 a^{-6} -8 z^6 a^{-6} +21 z^4 a^{-6} -25 z^2 a^{-6} -2 a^{-6} z^{-2} +11 a^{-6} +z^7 a^{-5} -7 z^5 a^{-5} +13 z^3 a^{-5} -9 z a^{-5} +2 a^{-5} z^{-1} +z^6 a^{-4} -5 z^4 a^{-4} +3 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +5 z^3 a^{-3} +z^6 a^{-2} -5 z^4 a^{-2} +6 z^2 a^{-2} -2 a^{-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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 19                     1 1 17                   1 1 0 15               1 1     0 13             1 1 1     1 11           1 3 1       1 9         2 1 1         2 7       1 4 2           1 5     1 1 1             1 3   1 2 1               0 1                       0 -1 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=3 }[/math] [math]\displaystyle{ i=5 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/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} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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}\oplus{\mathbb Z}_2 }[/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=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=8 }[/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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