L11n10

L11n9

L11n11

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Link Presentations

[edit Notes on L11n10's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,10,6,11 X3849 X13,22,14,5 X11,20,12,21 X21,12,22,13 X19,14,20,15 X9,18,10,19 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -10, 4, -7, 8, -6, 9, -11, -2, 3, 10, -9, 7, -8, 6}

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{2 t(1) t(2)^5-2 t(1) t(2)^4+2 t(1) t(2)^3-t(2)^3-t(1) t(2)^2+2 t(2)^2-2 t(2)+2}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{4}{q^{21/2}}-\frac{2}{q^{23/2}}+\frac{1}{q^{25/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{13} (-z)-a^{13} z^{-1} +a^{11} z^5+5 a^{11} z^3+6 a^{11} z+a^{11} z^{-1} -a^9 z^7-5 a^9 z^5-6 a^9 z^3+2 a^9 z^{-1} -a^7 z^7-6 a^7 z^5-11 a^7 z^3-7 a^7 z-2 a^7 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{16} z^2-a^{16}+2 a^{15} z^3-a^{15} z+a^{14} z^6-2 a^{14} z^4+3 a^{14} z^2+2 a^{13} z^7-7 a^{13} z^5+8 a^{13} z^3-a^{13} z-a^{13} z^{-1} +2 a^{12} z^8-8 a^{12} z^6+11 a^{12} z^4-8 a^{12} z^2+3 a^{12}+a^{11} z^9-3 a^{11} z^7+2 a^{11} z^5-3 a^{11} z^3+2 a^{11} z-a^{11} z^{-1} +3 a^{10} z^8-13 a^{10} z^6+15 a^{10} z^4-6 a^{10} z^2+a^9 z^9-4 a^9 z^7+3 a^9 z^5+2 a^9 z^3-5 a^9 z+2 a^9 z^{-1} +a^8 z^8-4 a^8 z^6+2 a^8 z^4+4 a^8 z^2-3 a^8+a^7 z^7-6 a^7 z^5+11 a^7 z^3-7 a^7 z+2 a^7 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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                   1 1 -8                 1 1 0 -10               2     2 -12           1 2 1     0 -14           4 2       2 -16       1 2 2         1 -18       3 3           0 -20     2 2             0 -22   1 3               -2 -24   1                 1 -26 1                   -1

Integral Khovanov Homology (db, data source)

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