L9a24

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	[edit L9a24 Quick Notes] L9a24 is $9_{21}^{2}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_14,6,15,5 X_16,11,17,12 X_18,13,7,14 X_12,17,13,18 X_4,16,5,15 X₂₇₃₈ X_6,9,1,10
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

0

1

-1

-2

3

1

2

-4

3

2

-1

-6

4

2

-8

3

4

1

-10

3

0

-12

1

3

2

-14

1

3

-2

-16

1

-18

1

-1

Integral Khovanov Homology

(db, data source)

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