L9n10

L9n9

L9n11

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n10 at Knotilus!
	L9n10 is $9^2_{58}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9n10's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,14,8,15 X18,16,5,15 X16,12,17,11 X12,18,13,17 X13,8,14,9 X2536 X4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, -3, 7, 9, -2, 5, -6, -7, 3, 4, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{t(1) t(2)^3-t(2)^3-2 t(1) t(2)^2+3 t(2)^2+3 t(1) t(2)-2 t(2)-t(1)+1}{\sqrt{t(1)} t(2)^{3/2}}$ (db)
Jones polynomial	$-\frac{2}{q^{9/2}}+\frac{3}{q^{7/2}}+q^{5/2}-\frac{5}{q^{5/2}}-3 q^{3/2}+\frac{5}{q^{3/2}}+4 \sqrt{q}-\frac{5}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^5 z^{-1} +z^3 a^3-a^3 z^{-1} -z^5 a-3 z^3 a-3 z a+z^3 a^{-1} +z a^{-1}$ (db)
Kauffman polynomial	$3 a^5 z^3-4 a^5 z+a^5 z^{-1} +a^4 z^6+a^4 z^2-a^4+a^3 z^7+2 a^3 z^3-3 a^3 z+a^3 z^{-1} +4 a^2 z^6-5 a^2 z^4+z^4 a^{-2} +2 a^2 z^2-z^2 a^{-2} +a z^7+3 a z^5+3 z^5 a^{-1} -6 a z^3-5 z^3 a^{-1} +2 a z+z a^{-1} +3 z^6-4 z^4$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 χ 6               1 -1 4             2   2 2           2 1   -1 0         3 2     1 -2       3 3       0 -4     2 2         0 -6   1 3           2 -8 1 2             -1 -10 2               2

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

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.

L9n9 L9n11