L9n21

L9n20

L9n22

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n21 at Knotilus!
	L9n21 is [math]\displaystyle{ 9^3_{17} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n21's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X16,11,17,12 X18,15,9,16 X12,17,13,18 X2536 X9,1,10,4
Gauss code	{1, -8, -2, 9}, {8, -1, -3, 4}, {-9, 2, 5, -7, -4, 3, 6, -5, 7, -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{-u v-u w^2+u w-v w^2+v w+w^3}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-5} + q^{-4} +q^3- q^{-3} + q^{-2} +q+2 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^2 a^4-a^4 z^{-2} -2 a^4+z^4 a^2+5 z^2 a^2+4 a^2 z^{-2} +8 a^2-z^4-6 z^2-5 z^{-2} -9+z^2 a^{-2} +2 a^{-2} z^{-2} +3 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^4 z^6+a^2 z^6+z^6 a^{-2} +z^6+a^5 z^5+2 a^3 z^5+2 a z^5+z^5 a^{-1} -4 a^4 z^4-6 a^2 z^4-6 z^4 a^{-2} -8 z^4-4 a^5 z^3-10 a^3 z^3-13 a z^3-7 z^3 a^{-1} +3 a^4 z^2+11 a^2 z^2+10 z^2 a^{-2} +18 z^2+3 a^5 z+13 a^3 z+21 a z+11 z a^{-1} -2 a^4-10 a^2-7 a^{-2} -14-a^5 z^{-1} -5 a^3 z^{-1} -9 a z^{-1} -5 a^{-1} z^{-1} +a^4 z^{-2} +4 a^2 z^{-2} +2 a^{-2} z^{-2} +5 z^{-2} }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                   1 1 3               1     1 1           3         3 -1         1 4 1       2 -3       1             1 -5       1 1           0 -7   1 1               0 -9                     0 -11 1                   -1

Integral Khovanov Homology (db, data source)

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