L11n356

L11n355

L11n357

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Link Presentations

[edit Notes on L11n356's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,18,12,19 X16,8,17,7 X8,16,9,15 X13,21,14,20 X19,22,20,15 X21,13,22,12 X17,14,18,5 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {5, -4, -9, 3, -7, 6, -8, 7}, {10, -1, 4, -5, 11, -2, -3, 8, -6, 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{(t(3)-1) \left(-t(1) t(2)^2+t(1) t(3) t(2)^2-t(3) t(2)^2+t(2)^2+t(1) t(2)-2 t(1) t(3) t(2)+t(3) t(2)-2 t(2)-t(1)+t(1) t(3)-t(3)+1\right)}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-6} +4 q^{-5} -6 q^{-4} +q^3+9 q^{-3} -3 q^2-9 q^{-2} +6 q+10 q^{-1} -7 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^2 z^6-a^4 z^4+3 a^2 z^4-2 z^4-a^4 z^2+2 a^2 z^2+z^2 a^{-2} -4 z^2+a^4-2 a^2+ a^{-2} +a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^3+4 a^6 z^4-2 a^6 z^2+a^5 z^7+2 a^5 z^5+2 a^4 z^8-3 a^4 z^6+8 a^4 z^4-8 a^4 z^2-a^4 z^{-2} +4 a^4+a^3 z^9+2 a^3 z^7-5 a^3 z^5+4 a^3 z^3-5 a^3 z+2 a^3 z^{-1} +5 a^2 z^8-10 a^2 z^6+z^6 a^{-2} +5 a^2 z^4-3 z^4 a^{-2} -6 a^2 z^2+3 z^2 a^{-2} -2 a^2 z^{-2} +6 a^2- a^{-2} +a z^9+4 a z^7+3 z^7 a^{-1} -16 a z^5-9 z^5 a^{-1} +11 a z^3+6 z^3 a^{-1} -5 a z+2 a z^{-1} +3 z^8-6 z^6-2 z^4+3 z^2- z^{-2} +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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                 2   -2 3               4 1   3 1             4 3     -1 -1           6 3       3 -3         4 5         1 -5       5 5           0 -7     2 5             3 -9   2 4               -2 -11   3                 3 -13 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{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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 }[/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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