L11n360

L11n359

L11n361

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Link Presentations

[edit Notes on L11n360's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X5,18,6,19 X8493 X9,21,10,20 X19,11,20,10 X17,14,18,15 X15,22,16,17 X21,16,22,5 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {-8, 4, -7, 6, -10, 9}, {-4, -1, 2, -5, -6, 7, 11, -2, 3, 8, -9, 10}

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)-1) (t(3)-1)^2}{\sqrt{t(1)} t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^3-2 q^2+3 q-3+4 q^{-1} -2 q^{-2} +3 q^{-3} + q^{-6} - q^{-7} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^6-2 a^6+z^4 a^4+5 z^2 a^4+a^4 z^{-2} +5 a^4-z^4 a^2-3 z^2 a^2-2 a^2 z^{-2} -4 a^2-z^4-2 z^2+ z^{-2} +z^2 a^{-2} + a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^7-6 a^7 z^5+9 a^7 z^3-3 a^7 z+a^6 z^8-7 a^6 z^6+14 a^6 z^4-12 a^6 z^2+5 a^6+a^5 z^7-9 a^5 z^5+18 a^5 z^3-10 a^5 z+2 a^4 z^8-16 a^4 z^6+39 a^4 z^4-38 a^4 z^2-a^4 z^{-2} +15 a^4+a^3 z^9-5 a^3 z^7+2 a^3 z^5+13 a^3 z^3-13 a^3 z+2 a^3 z^{-1} +3 a^2 z^8-18 a^2 z^6+z^6 a^{-2} +35 a^2 z^4-4 z^4 a^{-2} -31 a^2 z^2+3 z^2 a^{-2} -2 a^2 z^{-2} +13 a^2- a^{-2} +a z^9-3 a z^7+2 z^7 a^{-1} -3 a z^5-8 z^5 a^{-1} +10 a z^3+6 z^3 a^{-1} -7 a z-z a^{-1} +2 a z^{-1} +2 z^8-8 z^6+6 z^4-2 z^2- z^{-2} +3 }[/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 3 4 χ 7                       1 1 5                     1   -1 3                   2 1   1 1               1 2 1     0 -1             1 4 2       1 -3           1 2 3         2 -5         1 4 2           1 -7       1 1 3             3 -9       1 1               0 -11   1 1 1                 1 -13                         0 -15 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{ i=1 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/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=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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