L11n347

L11n346

L11n348

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Link Presentations

[edit Notes on L11n347's Link Presentations]

Planar diagram presentation	X6172 X3,15,4,14 X11,20,12,21 X7,18,8,19 X17,22,18,13 X16,9,17,10 X10,15,11,16 X19,12,20,5 X21,8,22,9 X2536 X13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -4, 9, 6, -7, -3, 8}, {-11, 2, 7, -6, -5, 4, -8, 3, -9, 5}

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{u v^2 w^3-u v^2 w^2+u v^2 w-u v^2-u v w+u v+u w^2+v^3 (-w)-v^2 w^3+v^2 w^2+v w^3-v w^2+v w-v}{\sqrt{u} v^{3/2} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 1-2 q^{-1} +4 q^{-2} -3 q^{-3} +4 q^{-4} -3 q^{-5} +3 q^{-6} - q^{-7} + q^{-9} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^2+a^8 z^{-2} +2 a^8-2 a^6 z^2-2 a^6 z^{-2} -5 a^6-a^4 z^6-4 a^4 z^4-3 a^4 z^2+a^4 z^{-2} +a^4+a^2 z^4+3 a^2 z^2+2 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^6-6 a^{10} z^4+8 a^{10} z^2-2 a^{10}-a^9 z^5+2 a^9 z^3+a^9 z-a^8 z^6+4 a^8 z^4-3 a^8 z^2-a^8 z^{-2} +3 a^8+a^7 z^7-5 a^7 z^5+10 a^7 z^3-8 a^7 z+2 a^7 z^{-1} +a^6 z^8-4 a^6 z^6+8 a^6 z^4-13 a^6 z^2-2 a^6 z^{-2} +9 a^6+3 a^5 z^7-11 a^5 z^5+12 a^5 z^3-8 a^5 z+2 a^5 z^{-1} +a^4 z^8-a^4 z^6-6 a^4 z^4+3 a^4 z^2-a^4 z^{-2} +3 a^4+2 a^3 z^7-7 a^3 z^5+4 a^3 z^3+a^3 z+a^2 z^6-4 a^2 z^4+5 a^2 z^2-2 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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                     1 1 -1                   1   -1 -3                 3 1   2 -5               2 3     1 -7           1 3 1       1 -9           1 2         1 -11       1 4 3           0 -13         2             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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/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}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/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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