L11n358

L11n357

L11n359

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Link Presentations

[edit Notes on L11n358'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 X21,13,22,12 X13,21,14,20 X19,5,20,14 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -6, 3, -9, 8, -7, 6}, {10, -1, -4, 5, 11, -2, -3, 7, -8, 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(2) t(3)^3-t(3)^3+t(2)^2 t(3)^2-2 t(2) t(3)^2+t(1) t(3)-2 t(1) t(2) t(3)-t(1) t(2)^2+t(1) t(2)\right)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^3+3 q^2-4 q+7-6 q^{-1} +7 q^{-2} -5 q^{-3} +4 q^{-4} -2 q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6-2 a^4 z^2+a^4 z^{-2} -a^4+a^2 z^4-2 a^2 z^{-2} -z^2 a^{-2} -2 a^2+z^4+z^2+ z^{-2} +2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^3 z^9+a z^9+2 a^4 z^8+5 a^2 z^8+3 z^8+2 a^5 z^7+2 z^7 a^{-1} +a^6 z^6-6 a^4 z^6-22 a^2 z^6-15 z^6-7 a^5 z^5-10 a^3 z^5-11 a z^5-8 z^5 a^{-1} -4 a^6 z^4+4 a^4 z^4+37 a^2 z^4+3 z^4 a^{-2} +32 z^4+5 a^5 z^3+15 a^3 z^3+24 a z^3+15 z^3 a^{-1} +z^3 a^{-3} +4 a^6 z^2-5 a^4 z^2-30 a^2 z^2-5 z^2 a^{-2} -26 z^2-a^5 z-10 a^3 z-14 a z-7 z a^{-1} -2 z a^{-3} -a^6+4 a^4+12 a^2+2 a^{-2} +10+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 7                   1 -1 5                 2   2 3               3 2   -1 1             4 1     3 -1           3 4       1 -3         4 3         1 -5       2 4           2 -7     2 3             -1 -9   1 3               2 -11   1                 -1 -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=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{2}\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} }[/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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