L9a21

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	[edit L9a21 Quick Notes] L9a21 is $9_{34}^{2}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,8,15,7 X_18,14,7,13 X_6,17,1,18 X_16,11,17,12 X_12,6,13,5 X_4,16,5,15
Gauss code	{1, -2, 3, -9, 8, -6}, {4, -1, 2, -3, 7, -8, 5, -4, 9, -7, 6, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {\left(t(1)t(2)^{2}-t(1)t(2)+t(2)+t(1)-1\right)\left(t(1)t(2)^{2}-t(2)^{2}-t(1)t(2)+t(2)-1\right)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+3q^{5/2}-5q^{3/2}+7{\sqrt {q}}-{\frac {9}{\sqrt {q}}}+{\frac {8}{q^{3/2}}}-{\frac {8}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-3a^{3}z^{3}-2a^{3}z+a^{3}z^{-1}+az^{7}+5az^{5}-z^{5}a^{-1}+8az^{3}-3z^{3}a^{-1}+3az-2za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{8}-2z^{8}-4a^{3}z^{7}-8az^{7}-4z^{7}a^{-1}-4a^{4}z^{6}-a^{2}z^{6}-3z^{6}a^{-2}-3a^{5}z^{5}+5a^{3}z^{5}+19az^{5}+10z^{5}a^{-1}-z^{5}a^{-3}-a^{6}z^{4}+4a^{4}z^{4}+4a^{2}z^{4}+7z^{4}a^{-2}+6z^{4}+4a^{5}z^{3}-3a^{3}z^{3}-17az^{3}-8z^{3}a^{-1}+2z^{3}a^{-3}+a^{6}z^{2}-2a^{2}z^{2}-3z^{2}a^{-2}-4z^{2}-a^{5}z+4az+3za^{-1}-a^{2}+a^{3}z^{-1}+az^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

2

-2

4

3

1

2

4

2

-2

0

5

3

2

-2

4

5

1

-4

4

0

-6

2

5

3

-8

1

3

-2

-10

2

-12

1

-1

Integral Khovanov Homology

(db, data source)

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