L9a23

L9a22

L9a24

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a23 at Knotilus!
	L9a23 is $9_{22}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a23's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ 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                 2   2 2               3 1   -2 0             4 2     2 -2           5 4       -1 -4         3 3         0 -6       3 5           2 -8     2 3             -1 -10     3               3 -12 1 2                 -1 -14 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L9a22 L9a24