L9n15

L9n14

L9n16

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n15 at Knotilus!
	This is a Seifert-fibered link: it is the trefoil plus the core of one solid torus. L9n15 is [math]\displaystyle{ 9^2_{49} }[/math] in the Rolfsen table of links.

The Triune sculpture in Philadelpdia by Robert Engman, showing a Seifert surface for L9n15. Photo by sameold2008

Link Presentations

[edit Notes on L9n15's Link Presentations]

Planar diagram presentation	X8192 X16,11,17,12 X3,10,4,11 X2,15,3,16 X12,5,13,6 X6718 X9,14,10,15 X13,18,14,7 X17,4,18,5
Gauss code	{1, -4, -3, 9, 5, -6}, {6, -1, -7, 3, 2, -5, -8, 7, 4, -2, -9, 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)^2 t(2)^4+t(1) t(2)^2+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{7/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{11} (-z)-a^{11} z^{-1} +a^9 z^5+6 a^9 z^3+9 a^9 z+3 a^9 z^{-1} -a^7 z^7-7 a^7 z^5-15 a^7 z^3-11 a^7 z-2 a^7 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12}+z a^{11}-a^{11} z^{-1} -z^6 a^{10}+6 z^4 a^{10}-9 z^2 a^{10}+3 a^{10}-z^7 a^9+7 z^5 a^9-15 z^3 a^9+12 z a^9-3 a^9 z^{-1} -z^6 a^8+6 z^4 a^8-9 z^2 a^8+3 a^8-z^7 a^7+7 z^5 a^7-15 z^3 a^7+11 z a^7-2 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   \ -6 -5 -4 -3 -2 -1 0 χ -6             1 1 -8             1 1 -10         1     1 -12     1         1 -14     1 1       0 -16 1 1           0 -18 1 1           0

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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/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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