L9n23

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	[edit L9n23 Quick Notes] L9n23 is $9_{13}^{3}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_13,2,14,3 X_14,7,15,8 X_11,16,12,17 X_9,11,10,18 X_17,5,18,10 X_4,15,1,16
Gauss code	{1, 4, -3, -9}, {-2, -1, 5, 3, -7, 8}, {-6, 2, -4, -5, 9, 6, -8, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(3)^{2}t(2)^{2}+t(1)t(3)t(2)^{2}+t(1)t(3)^{2}t(2)-t(2)-t(3)+1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$1-q^{-1}+2q^{-2}-q^{-3}+3q^{-4}-q^{-5}+2q^{-6}-q^{-7}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{8}+z^{4}a^{6}+4z^{2}a^{6}+a^{6}z^{-2}+3a^{6}-z^{6}a^{4}-5z^{4}a^{4}-7z^{2}a^{4}-2a^{4}z^{-2}-5a^{4}+z^{4}a^{2}+4z^{2}a^{2}+a^{2}z^{-2}+3a^{2}$ (db)
Kauffman polynomial	$za^{9}+2z^{2}a^{8}-2a^{8}+z^{5}a^{7}-3z^{3}a^{7}+3za^{7}+2z^{6}a^{6}-9z^{4}a^{6}+13z^{2}a^{6}+a^{6}z^{-2}-8a^{6}+z^{7}a^{5}-3z^{5}a^{5}-z^{3}a^{5}+5za^{5}-2a^{5}z^{-1}+3z^{6}a^{4}-14z^{4}a^{4}+18z^{2}a^{4}+2a^{4}z^{-2}-9a^{4}+z^{7}a^{3}-4z^{5}a^{3}+2z^{3}a^{3}+3za^{3}-2a^{3}z^{-1}+z^{6}a^{2}-5z^{4}a^{2}+7z^{2}a^{2}+a^{2}z^{-2}-4a^{2}$ (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

χ

1

-1

0

-3

2

1

-5

1

-7

3

1

2

-9

1

2

-11

2

1

-13

1

-15

1

-1

Integral Khovanov Homology

(db, data source)

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