L9n26

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	[edit L9n26 Quick Notes] L9n26 is $9_{19}^{3}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,17,12,16 X_7,14,8,15 X_13,8,14,9 X_15,13,16,18 X_17,5,18,12 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -8, 2, -9}, {-5, 4, -6, 3, -7, 6}, {8, -1, -4, 5, 9, -2, -3, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(2)-1)(t(3)-1)\left(t(3)^{2}-t(1)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{-5}+3q^{-4}-2q^{-3}+3q^{-2}-q-2q^{-1}+3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-2a^{4}z^{2}-2a^{4}z^{-2}-4a^{4}+a^{2}z^{4}+3a^{2}z^{2}+a^{2}z^{-2}+3a^{2}-z^{2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-5a^{6}z^{4}+8a^{6}z^{2}+a^{6}z^{-2}-5a^{6}+a^{5}z^{7}-3a^{5}z^{5}-a^{5}z^{3}+5a^{5}z-2a^{5}z^{-1}+4a^{4}z^{6}-17a^{4}z^{4}+20a^{4}z^{2}+2a^{4}z^{-2}-10a^{4}+a^{3}z^{7}-a^{3}z^{5}-6a^{3}z^{3}+7a^{3}z-2a^{3}z^{-1}+3a^{2}z^{6}-12a^{2}z^{4}+15a^{2}z^{2}+a^{2}z^{-2}-7a^{2}+2az^{5}-5az^{3}+3az+za^{-1}+3z^{2}-1$ (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

χ

3

1

-1

1

2

-1

2

3

1

-3

1

-5

1

2

1

-7

2

1

-9

1

3

2

-11

0

-13

1

Integral Khovanov Homology

(db, data source)

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