L9a24

L9a23

L9a25

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a24 at Knotilus!
	L9a24 is $9_{21}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a24's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{3}-u^{2}v^{2}+u^{2}v-u^{2}+uv^{4}-3uv^{3}+3uv^{2}-3uv+u-v^{4}+v^{3}-v^{2}+v}{uv^{2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {2}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {6}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-2za^{7}-a^{7}z^{-1}+z^{5}a^{5}+3z^{3}a^{5}+4za^{5}+3a^{5}z^{-1}+z^{5}a^{3}+2z^{3}a^{3}-za^{3}-2a^{3}z^{-1}-z^{3}a-2za$ (db)
Kauffman polynomial	$a^{10}z^{4}-2a^{10}z^{2}+2a^{9}z^{5}-3a^{9}z^{3}+3a^{8}z^{6}-6a^{8}z^{4}+5a^{8}z^{2}-a^{8}+3a^{7}z^{7}-8a^{7}z^{5}+13a^{7}z^{3}-7a^{7}z+a^{7}z^{-1}+a^{6}z^{8}+2a^{6}z^{6}-8a^{6}z^{4}+11a^{6}z^{2}-3a^{6}+5a^{5}z^{7}-15a^{5}z^{5}+22a^{5}z^{3}-14a^{5}z+3a^{5}z^{-1}+a^{4}z^{8}+a^{4}z^{6}-6a^{4}z^{4}+6a^{4}z^{2}-3a^{4}+2a^{3}z^{7}-4a^{3}z^{5}+3a^{3}z^{3}-5a^{3}z+2a^{3}z^{-1}+2a^{2}z^{6}-5a^{2}z^{4}+2a^{2}z^{2}+az^{5}-3az^{3}+2az$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                   1 1 0                 1   -1 -2               3 1   2 -4             3 2     -1 -6           4 2       2 -8         3 4         1 -10       3 3           0 -12     1 3             2 -14   1 3               -2 -16   1                 1 -18 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

L9a23 L9a25