L9a41

L9a40

L9a42

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a41 at Knotilus!
	L9a41 is $9^2_{23}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a41's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,5,9,6 X14,8,15,7 X16,14,17,13 X8,16,1,15 X6,9,7,10 X4,17,5,18
Gauss code	{1, -2, 3, -9, 4, -8, 5, -7}, {8, -1, 2, -3, 6, -5, 7, -6, 9, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

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