L9n14

L9n13

L9n15

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n14 at Knotilus!
	L9n14 is [math]\displaystyle{ 9^2_{50} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n14's Link Presentations]

Planar diagram presentation	X8192 X11,17,12,16 X3,10,4,11 X15,3,16,2 X5,13,6,12 X6718 X14,10,15,9 X18,14,7,13 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{u^2 v^2-u^2 v+u v-v+1}{u v} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-q^{7/2}+q^{5/2}-\frac{1}{q^{5/2}}-2 q^{3/2}+\frac{1}{q^{3/2}}+\sqrt{q}-\frac{2}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{-3} +3 z a^{-3} + a^{-3} z^{-1} -z^5 a^{-1} +a z^3-5 z^3 a^{-1} +3 a z-7 z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^7 a^{-1} -z^7 a^{-3} -2 z^6 a^{-2} -z^6 a^{-4} -z^6+5 z^5 a^{-1} +5 z^5 a^{-3} +9 z^4 a^{-2} +5 z^4 a^{-4} +4 z^4-2 a z^3-9 z^3 a^{-1} -7 z^3 a^{-3} -a^2 z^2-10 z^2 a^{-2} -6 z^2 a^{-4} -5 z^2-a^3 z+4 a z+9 z a^{-1} +4 z a^{-3} +3 a^{-2} + a^{-4} +3-2 a z^{-1} -3 a^{-1} z^{-1} - 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   \ -2 -1 0 1 2 3 4 5 χ 10               1 -1 8                 0 6           1 1   0 4         1       1 2         1       1 0     2 1         1 -2     1           1 -4 1 1             0 -6 1               1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/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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