L9a9

L9a8

L9a10

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a9 at Knotilus!
	L9a9 is $9^2_{37}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a9's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X16,9,17,10 X8,15,9,16 X4,17,1,18 X12,6,13,5 X10,4,11,3 X18,12,5,11 X2,14,3,13
Gauss code	{1, -9, 7, -5}, {6, -1, 2, -4, 3, -7, 8, -6, 9, -2, 4, -3, 5, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(2)-1) \left(t(2)^2+1\right) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-\frac{3}{q^{9/2}}-q^{7/2}+\frac{5}{q^{7/2}}+3 q^{5/2}-\frac{7}{q^{5/2}}-5 q^{3/2}+\frac{8}{q^{3/2}}+\frac{1}{q^{11/2}}+6 \sqrt{q}-\frac{9}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^3 z^5-3 a^3 z^3-2 a^3 z+a z^7+5 a z^5-z^5 a^{-1} +8 a z^3-3 z^3 a^{-1} +4 a z-2 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-2 a^2 z^8-2 z^8-4 a^3 z^7-8 a z^7-4 z^7 a^{-1} -4 a^4 z^6-3 z^6 a^{-2} +z^6-3 a^5 z^5+7 a^3 z^5+22 a z^5+11 z^5 a^{-1} -z^5 a^{-3} -a^6 z^4+5 a^4 z^4+3 a^2 z^4+7 z^4 a^{-2} +4 z^4+4 a^5 z^3-8 a^3 z^3-24 a z^3-10 z^3 a^{-1} +2 z^3 a^{-3} +a^6 z^2-a^4 z^2-3 a^2 z^2-2 z^2 a^{-2} -3 z^2+4 a^3 z+8 a z+4 z a^{-1} +1-a z^{-1} - a^{-1} z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                   1 1 6                 2   -2 4               3 1   2 2             3 2     -1 0           6 3       3 -2         4 5         1 -4       3 4           -1 -6     2 4             2 -8   1 3               -2 -10   2                 2 -12 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

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