L11n7

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Link Presentations

[edit Notes on L11n7's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X3849 X5,11,6,10 X11,20,12,21 X19,22,20,5 X13,19,14,18 X21,12,22,13 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 10, -9, 4, -11, -2, 3, 9, -8, 7, -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{t(1) t(2)^5-4 t(1) t(2)^4+4 t(1) t(2)^3-2 t(2)^3-2 t(1) t(2)^2+4 t(2)^2-4 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{5}{q^{15/2}}-\frac{4}{q^{17/2}}+\frac{2}{q^{19/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{11} z^{-1} +z^3 a^9+4 z a^9+3 a^9 z^{-1} -2 z^5 a^7-8 z^3 a^7-8 z a^7-3 a^7 z^{-1} +z^7 a^5+5 z^5 a^5+8 z^3 a^5+6 z a^5+2 a^5 z^{-1} -z^5 a^3-4 z^3 a^3-4 z a^3-a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 z^2 a^{12}+a^{12}-z^5 a^{11}-3 z^3 a^{11}+3 z a^{11}-a^{11} z^{-1} -3 z^6 a^{10}+5 z^4 a^{10}-6 z^2 a^{10}+2 a^{10}-4 z^7 a^9+11 z^5 a^9-15 z^3 a^9+12 z a^9-3 a^9 z^{-1} -3 z^8 a^8+7 z^6 a^8-4 z^4 a^8+3 z^2 a^8-z^9 a^7-3 z^7 a^7+21 z^5 a^7-27 z^3 a^7+15 z a^7-3 a^7 z^{-1} -5 z^8 a^6+19 z^6 a^6-20 z^4 a^6+9 z^2 a^6-2 a^6-z^9 a^5+14 z^5 a^5-23 z^3 a^5+11 z a^5-2 a^5 z^{-1} -2 z^8 a^4+9 z^6 a^4-11 z^4 a^4+3 z^2 a^4-z^7 a^3+5 z^5 a^3-8 z^3 a^3+5 z a^3-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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 0                   1 1 -2                 1   -1 -4               3 1   2 -6             3 2     -1 -8           4 2       2 -10         3 3         0 -12       4 4           0 -14     2 4             2 -16   2 3               -1 -18   2                 2 -20 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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