L9a18

L9a17

L9a19

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a18 at Knotilus!
	L9a18 is is $9^2_{10}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a18's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X18,8,5,7 X16,10,17,9 X14,12,15,11 X10,16,11,15 X8,18,9,17 X2536 X4,14,1,13
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -7, 4, -6, 5, -2, 9, -5, 6, -4, 7, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{3 (t(1)-1) (t(2)-1)}{\sqrt{t(1)} \sqrt{t(2)}}$ (db)
Jones polynomial	$-3 q^{9/2}+4 q^{7/2}-4 q^{5/2}+3 q^{3/2}-\frac{1}{q^{3/2}}+q^{15/2}-2 q^{13/2}+2 q^{11/2}-3 \sqrt{q}+\frac{1}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a^{-7} -z^3 a^{-5} -z a^{-5} -z^3 a^{-3} -z^3 a^{-1} +a z-z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-z^8 a^{-4} -z^8 a^{-6} -z^7 a^{-3} -3 z^7 a^{-5} -2 z^7 a^{-7} -z^6 a^{-2} +3 z^6 a^{-4} +3 z^6 a^{-6} -z^6 a^{-8} -z^5 a^{-1} +z^5 a^{-3} +11 z^5 a^{-5} +9 z^5 a^{-7} -4 z^4 a^{-4} -z^4 a^{-6} +4 z^4 a^{-8} -z^4-a z^3-z^3 a^{-1} -10 z^3 a^{-5} -10 z^3 a^{-7} +3 z^2 a^{-4} -3 z^2 a^{-8} +2 a z+2 z a^{-1} +2 z a^{-5} +2 z a^{-7} +1-a z^{-1} - a^{-1} z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 16                   1 -1 14                 1   1 12               1 1   0 10             2 1     1 8           2 1       -1 6         2 2         0 4       1 2           1 2     2 2             0 0   1 3               2 -2                     0 -4 1                   1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

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.

L9a17 L9a19