L10a15

L10a14

L10a16

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Link Presentations

[edit Notes on L10a15's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,8,17,7 X18,11,19,12 X20,13,5,14 X12,19,13,20 X14,17,15,18 X8,16,9,15 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 10, -2, 4, -6, 5, -7, 8, -3, 7, -4, 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)^5+2 t(1) t(2)^4-4 t(2)^4-5 t(1) t(2)^3+5 t(2)^3+5 t(1) t(2)^2-5 t(2)^2-4 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{11}{q^{9/2}}-\frac{11}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z+a^9 z^{-1} -3 a^7 z^3-6 a^7 z-3 a^7 z^{-1} +2 a^5 z^5+6 a^5 z^3+7 a^5 z+4 a^5 z^{-1} +a^3 z^5+a^3 z^3-2 a^3 z-2 a^3 z^{-1} -a z^3-2 a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+6 z^4 a^{10}-3 z^2 a^{10}+a^{10}-4 z^7 a^9+6 z^5 a^9-z^3 a^9+2 z a^9-a^9 z^{-1} -3 z^8 a^8+10 z^4 a^8-9 z^2 a^8+3 a^8-z^9 a^7-7 z^7 a^7+21 z^5 a^7-25 z^3 a^7+14 z a^7-3 a^7 z^{-1} -6 z^8 a^6+10 z^6 a^6-4 z^4 a^6-6 z^2 a^6+3 a^6-z^9 a^5-6 z^7 a^5+21 z^5 a^5-31 z^3 a^5+19 z a^5-4 a^5 z^{-1} -3 z^8 a^4+5 z^6 a^4-4 z^4 a^4-z^2 a^4+2 a^4-3 z^7 a^3+6 z^5 a^3-6 z^3 a^3+6 z a^3-2 a^3 z^{-1} -2 z^6 a^2+4 z^4 a^2-z^2 a^2-z^5 a+3 z^3 a-2 z a }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                     1 1 0                   1   -1 -2                 4 1   3 -4               5 2     -3 -6             6 3       3 -8           5 5         0 -10         6 6           0 -12       4 6             2 -14     2 5               -3 -16   1 4                 3 -18   2                   -2 -20 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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 }[/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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