L9a30

L9a29

L9a31

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a30 at Knotilus!
	L9a30 is $9^2_{3}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a30's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X16,13,17,14 X14,6,15,5 X4,16,5,15 X18,11,7,12 X12,17,13,18
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 8, -9, 5, -6, 7, -5, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{2 u^2 v^2-2 u^2 v-2 u v^2+3 u v-2 u-2 v+2}{u v}$ (db)
Jones polynomial	$-\sqrt{q}+\frac{1}{\sqrt{q}}-\frac{3}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{5}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^7 \left(-z^3\right)-2 a^7 z+a^5 z^5+3 a^5 z^3+2 a^5 z+a^3 z^5+3 a^3 z^3+2 a^3 z+a^3 z^{-1} -a z^3-3 a z-a z^{-1}$ (db)
Kauffman polynomial	$a^{10} z^4-2 a^{10} z^2+2 a^9 z^5-4 a^9 z^3+a^9 z+2 a^8 z^6-3 a^8 z^4+a^8 z^2+2 a^7 z^7-5 a^7 z^5+7 a^7 z^3-2 a^7 z+a^6 z^8-2 a^6 z^6+4 a^6 z^4-a^6 z^2+3 a^5 z^7-9 a^5 z^5+12 a^5 z^3-4 a^5 z+a^4 z^8-3 a^4 z^6+6 a^4 z^4-5 a^4 z^2+a^3 z^7-a^3 z^5-3 a^3 z^3+3 a^3 z-a^3 z^{-1} +a^2 z^6-2 a^2 z^4-a^2 z^2+a^2+a z^5-4 a z^3+4 a z-a z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                   1 1 0                     0 -2               3 1   2 -4             2 1     -1 -6           3 2       1 -8         3 3         0 -10       2 2           0 -12     1 3             2 -14   1 2               -1 -16   1                 1 -18 1                   -1

Integral Khovanov Homology (db, data source)

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

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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