L11n270

L11n269

L11n271

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Link Presentations

[edit Notes on L11n270's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X11,19,12,18 X17,9,18,22 X13,21,14,20 X19,13,20,12 X21,15,22,14 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, -5, 8, -7, 9, 4, -3, -6, 5, -8, 7, -9, 6}

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(3)^5+t(2) t(3)^5-t(3)^5-t(1) t(3)^4-t(2) t(3)^4+t(1) t(3)^3+t(2) t(3)^3-t(1) t(3)^2-t(2) t(3)^2+t(1) t(3)+t(2) t(3)-t(1)+t(1) t(2)-t(2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-5} -2 q^4- q^{-4} +2 q^3+4 q^{-3} -3 q^2-2 q^{-2} +4 q+5 q^{-1} -4 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ 2 z^6-3 a^2 z^4-2 z^4 a^{-2} +12 z^4+a^4 z^2-14 a^2 z^2-8 z^2 a^{-2} +24 z^2+4 a^4-18 a^2-7 a^{-2} +21+3 a^4 z^{-2} -8 a^2 z^{-2} -2 a^{-2} z^{-2} +7 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^3 z^9+a z^9+a^4 z^8+5 a^2 z^8+4 z^8-4 a^3 z^7+4 z^7 a^{-1} -7 a^4 z^6-30 a^2 z^6+2 z^6 a^{-2} -21 z^6-2 a^3 z^5-21 a z^5-17 z^5 a^{-1} +2 z^5 a^{-3} +18 a^4 z^4+59 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} +38 z^4+22 a^3 z^3+50 a z^3+24 z^3 a^{-1} -4 z^3 a^{-3} -22 a^4 z^2-51 a^2 z^2-8 z^2 a^{-2} -37 z^2-24 a^3 z-45 a z-21 z a^{-1} +3 z a^{-3} +3 z a^{-5} +13 a^4+28 a^2+7 a^{-2} + a^{-4} +22+8 a^3 z^{-1} +15 a z^{-1} +7 a^{-1} z^{-1} - a^{-3} z^{-1} - a^{-5} z^{-1} -3 a^4 z^{-2} -8 a^2 z^{-2} -2 a^{-2} z^{-2} -7 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                   2 -2 7                 1 1 0 5               3 2   -1 3             1 1 1   1 1           4 4       0 -1         1 1 1       1 -3       1 4           3 -5     3 1             2 -7   1 4               3 -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}^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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