L11n163

L11n162

L11n164

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Link Presentations

[edit Notes on L11n163's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X17,13,18,12 X14,5,15,6 X4,13,5,14 X11,19,12,18 X22,19,7,20 X20,15,21,16 X16,21,17,22 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, -6, 3, 5, -4, 8, -9, -3, 6, 7, -8, 9, -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{t(1) t(2)^4-t(2)^4+t(1)^2 t(2)^3-4 t(1) t(2)^3+2 t(2)^3-3 t(1)^2 t(2)^2+5 t(1) t(2)^2-3 t(2)^2+2 t(1)^2 t(2)-4 t(1) t(2)+t(2)-t(1)^2+t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{5}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^9-z a^9+z^5 a^7+2 z^3 a^7+2 z a^7+a^7 z^{-1} +z^5 a^5+z^3 a^5-z a^5-a^5 z^{-1} -2 z^3 a^3-3 z a^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{12}+3 z^4 a^{12}-2 z^2 a^{12}-3 z^7 a^{11}+10 z^5 a^{11}-9 z^3 a^{11}+2 z a^{11}-3 z^8 a^{10}+7 z^6 a^{10}-z^4 a^{10}-2 z^2 a^{10}-z^9 a^9-4 z^7 a^9+17 z^5 a^9-14 z^3 a^9+5 z a^9-5 z^8 a^8+9 z^6 a^8-z^4 a^8-z^2 a^8-z^9 a^7-3 z^7 a^7+7 z^5 a^7-2 z^3 a^7-2 z a^7+a^7 z^{-1} -2 z^8 a^6+z^4 a^6-a^6-2 z^7 a^5-2 z a^5+a^5 z^{-1} -z^6 a^4-2 z^4 a^4+z^2 a^4-3 z^3 a^3+3 z a^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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                   2 2 -4                 3 1 -2 -6               4 1   3 -8             5 3     -2 -10           5 4       1 -12         4 5         1 -14       4 5           -1 -16     2 5             3 -18   1 3               -2 -20   2                 2 -22 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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