L11n285

L11n284

L11n286

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Link Presentations

[edit Notes on L11n285's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X4,15,1,16 X21,18,22,19 X9,21,10,20 X19,5,20,10 X11,16,12,17 X17,22,18,11
Gauss code	{1, 4, -3, -6}, {-2, -1, 5, 3, -8, 9}, {-10, 2, -4, -5, 6, 10, -11, 7, -9, 8, -7, 11}

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{2 t(1) t(3)^2 t(2)^2-2 t(1) t(3) t(2)^2-2 t(1) t(3)^2 t(2)+t(1) t(3) t(2)-t(3) t(2)+2 t(2)+2 t(3)-2}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 1- q^{-1} +3 q^{-2} -3 q^{-3} +5 q^{-4} -4 q^{-5} +5 q^{-6} -3 q^{-7} +2 q^{-8} - q^{-9} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10}+z^4 a^8+4 z^2 a^8+2 a^8-z^6 a^6-4 z^4 a^6-3 z^2 a^6+a^6 z^{-2} -z^6 a^4-4 z^4 a^4-4 z^2 a^4-2 a^4 z^{-2} -4 a^4+z^4 a^2+4 z^2 a^2+a^2 z^{-2} +3 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^3-2 a^{11} z+2 a^{10} z^4-4 a^{10} z^2+2 a^{10}+a^9 z^7-4 a^9 z^5+8 a^9 z^3-4 a^9 z+2 a^8 z^8-11 a^8 z^6+23 a^8 z^4-16 a^8 z^2+4 a^8+a^7 z^9-4 a^7 z^7+5 a^7 z^5+3 a^6 z^8-14 a^6 z^6+21 a^6 z^4-9 a^6 z^2+a^6 z^{-2} -2 a^6+a^5 z^9-4 a^5 z^7+6 a^5 z^5-8 a^5 z^3+6 a^5 z-2 a^5 z^{-1} +a^4 z^8-2 a^4 z^6-5 a^4 z^4+10 a^4 z^2+2 a^4 z^{-2} -7 a^4+a^3 z^7-3 a^3 z^5-a^3 z^3+4 a^3 z-2 a^3 z^{-1} +a^2 z^6-5 a^2 z^4+7 a^2 z^2+a^2 z^{-2} -4 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                   1 1 -1                     0 -3               3 1   2 -5           1 2 1     0 -7           4 2       2 -9       1 3 3         1 -11       3 2           1 -13     1 3             2 -15   1 2               -1 -17   1                 1 -19 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-7 }[/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=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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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