L11n215

L11n214

L11n216

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Link Presentations

[edit Notes on L11n215's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X14,5,15,6 X9,18,10,19 X17,22,18,9 X21,1,22,8 X20,15,21,16 X7,16,8,17 X6,20,7,19 X4,11,5,12 X2,13,3,14
Gauss code	{1, -11, 2, -10, 3, -9, -8, 6}, {-4, -1, 10, -2, 11, -3, 7, 8, -5, 4, 9, -7, -6, 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{t(2)^2 t(1)^3-t(2) t(1)^3+t(2)^3 t(1)^2-4 t(2)^2 t(1)^2+4 t(2) t(1)^2-2 t(1)^2-2 t(2)^3 t(1)+4 t(2)^2 t(1)-4 t(2) t(1)+t(1)-t(2)^2+t(2)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^9-2 z a^9-a^9 z^{-1} +z^5 a^7+3 z^3 a^7+5 z a^7+3 a^7 z^{-1} +z^5 a^5+z^3 a^5-2 z a^5-2 a^5 z^{-1} -2 z^3 a^3-3 z a^3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{12}+4 z^4 a^{12}-4 z^2 a^{12}-2 z^7 a^{11}+7 z^5 a^{11}-7 z^3 a^{11}+2 z a^{11}-2 z^8 a^{10}+5 z^6 a^{10}-3 z^4 a^{10}+2 z^2 a^{10}-a^{10}-z^9 a^9+2 z^5 a^9+3 z^3 a^9-2 z a^9+a^9 z^{-1} -4 z^8 a^8+9 z^6 a^8-10 z^4 a^8+11 z^2 a^8-3 a^8-z^9 a^7-5 z^5 a^7+15 z^3 a^7-12 z a^7+3 a^7 z^{-1} -2 z^8 a^6+2 z^6 a^6-5 z^4 a^6+6 z^2 a^6-3 a^6-2 z^7 a^5+2 z^3 a^5-5 z a^5+2 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             4 3     -1 -10           5 4       1 -12         4 5         1 -14       3 4           -1 -16     1 4             3 -18   1 3               -2 -20   1                 1 -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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}^{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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