L11n178

L11n177

L11n179

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Link Presentations

[edit Notes on L11n178's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X13,21,14,20 X3,10,4,11 X5,14,6,15 X7,16,8,17 X15,22,16,7 X11,4,12,5 X19,13,20,12 X21,1,22,6 X2,18,3,17
Gauss code	{1, -11, -4, 8, -5, 10}, {-6, -1, 2, 4, -8, 9, -3, 5, -7, 6, 11, -2, -9, 3, -10, 7}

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

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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