L11n174

L11n173

L11n175

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Link Presentations

[edit Notes on L11n174's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X14,5,15,6 X11,18,12,19 X3,10,4,11 X7,13,8,12 X16,13,17,14 X17,7,18,22 X6,15,1,16 X4,21,5,22 X19,2,20,3
Gauss code	{1, 11, -5, -10, 3, -9}, {-6, -1, 2, 5, -4, 6, 7, -3, 9, -7, -8, 4, -11, -2, 10, 8}

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{\left(t(2)^2-3 t(2)+1\right) \left(t(1)^2 t(2)^2+t(1) t(2)+1\right)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{2}{q^{17/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z+a^9 z^{-1} -a^7 z^5-4 a^7 z^3-4 a^7 z-2 a^7 z^{-1} +a^5 z^7+5 a^5 z^5+7 a^5 z^3+4 a^5 z+2 a^5 z^{-1} -a^3 z^5-4 a^3 z^3-4 a^3 z-a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 a^{11} z+a^{10} z^4+a^9 z^7-3 a^9 z^5+4 a^9 z^3-3 a^9 z+a^9 z^{-1} +2 a^8 z^8-8 a^8 z^6+10 a^8 z^4-6 a^8 z^2+a^7 z^9-a^7 z^7-9 a^7 z^5+15 a^7 z^3-9 a^7 z+2 a^7 z^{-1} +4 a^6 z^8-17 a^6 z^6+19 a^6 z^4-8 a^6 z^2+a^6+a^5 z^9-a^5 z^7-11 a^5 z^5+19 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +2 a^4 z^8-9 a^4 z^6+10 a^4 z^4-2 a^4 z^2+a^3 z^7-5 a^3 z^5+8 a^3 z^3-5 a^3 z+a^3 z^{-1} }[/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 χ 0                 1 1 -2               1   -1 -4             3 1   2 -6           2 2     0 -8         3 2       1 -10       2 2         0 -12     2 3           -1 -14   1 2             1 -16 1 2               -1 -18 2                 2

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/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}^{2}\oplus{\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}^{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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{2}\oplus{\mathbb Z}_2^{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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