L9a25

L9a24

L9a26

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a25 at Knotilus!
	L9a25 is [math]\displaystyle{ 9^2_{8} }[/math] in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a25's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X14,6,15,5 X18,11,7,12 X16,13,17,14 X12,17,13,18 X4,16,5,15 X2738 X6,9,1,10
Gauss code	{1, -8, 2, -7, 3, -9}, {8, -1, 9, -2, 4, -6, 5, -3, 7, -5, 6, -4}

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{2 t(2) t(1)^2-2 t(1)^2+2 t(2)^2 t(1)-5 t(2) t(1)+2 t(1)-2 t(2)^2+2 t(2)}{t(1) t(2)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{5}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{6}{q^{5/2}}-q^{3/2}+\frac{5}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{3}{q^{11/2}}+2 \sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7+z^3 a^5+2 z^3 a^3+2 z a^3+a^3 z^{-1} +z^3 a-z a-a z^{-1} -z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^8+4 z^4 a^8-4 z^2 a^8-2 z^7 a^7+8 z^5 a^7-9 z^3 a^7+3 z a^7-z^8 a^6+z^6 a^6+4 z^4 a^6-3 z^2 a^6-4 z^7 a^5+11 z^5 a^5-7 z^3 a^5+2 z a^5-z^8 a^4-z^6 a^4+5 z^4 a^4-z^2 a^4-2 z^7 a^3+6 z^3 a^3-5 z a^3+a^3 z^{-1} -3 z^6 a^2+3 z^4 a^2-z^2 a^2-a^2-3 z^5 a+3 z^3 a-3 z a+a z^{-1} -2 z^4+z^2-z^3 a^{-1} +z a^{-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 4                   1 1 2                 1   -1 0               3 1   2 -2             3 2     -1 -4           3 2       1 -6         3 4         1 -8       2 2           0 -10     1 3             2 -12   1 2               -1 -14   1                 1 -16 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=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

L9a24 L9a26