L9a33

L9a32

L9a34

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a33 at Knotilus!
	L9a33 is $9^2_{24}$ in the Rolfsen table of links.

Symmetric form		Alternate symmetric version, with three lines touching at center		Alternate symmetric version, with three lines touching at circumference
Form made from 45-degree lines and circular arcs.		Depiction obtained by knotilus.		With an hypotrochoid [1].
Mexican book.

Link Presentations

[edit Notes on L9a33's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{u^2 v^2-3 u^2 v+3 u^2-3 u v^2+7 u v-3 u+3 v^2-3 v+1}{u v}$ (db)
Jones polynomial	$-\frac{6}{q^{9/2}}+\frac{7}{q^{7/2}}+q^{5/2}-\frac{9}{q^{5/2}}-4 q^{3/2}+\frac{10}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+6 \sqrt{q}-\frac{8}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^7 z^{-1} -3 a^5 z-a^5 z^{-1} +3 a^3 z^3+3 a^3 z-a z^5-2 a z^3+z^3 a^{-1} -3 a z$ (db)
Kauffman polynomial	$-a^4 z^8-a^2 z^8-3 a^5 z^7-7 a^3 z^7-4 a z^7-2 a^6 z^6-7 a^4 z^6-11 a^2 z^6-6 z^6-a^7 z^5+5 a^5 z^5+9 a^3 z^5-a z^5-4 z^5 a^{-1} +3 a^6 z^4+18 a^4 z^4+24 a^2 z^4-z^4 a^{-2} +8 z^4+3 a^7 z^3-3 a^5 z^3-2 a^3 z^3+8 a z^3+4 z^3 a^{-1} -11 a^4 z^2-14 a^2 z^2-3 z^2-3 a^7 z+a^5 z+2 a^3 z-2 a z-a^6+a^7 z^{-1} +a^5 z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                   1 -1 4                 3   3 2               3 1   -2 0             5 3     2 -2           6 4       -2 -4         3 4         -1 -6       4 6           2 -8     2 3             -1 -10     4               4 -12 1 2                 -1 -14 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L9a32 L9a34