L9n24

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	[edit L9n24 Quick Notes] L9n24 is $9_{14}^{3}$ 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_2,14,3,13 X_14,7,15,8 X_11,16,12,17 X_9,11,10,18 X_17,5,18,10 X_15,1,16,4
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 $u$ , $v$ , $w$ , ...)	${\frac {uvw^{2}-2uvw-uw^{2}+uw+v^{2}(-w)+v^{2}+2vw-v}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$2q^{2}-2q+4-3q^{-1}+4q^{-2}-2q^{-3}+2q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{4}-a^{4}+z^{4}a^{2}+3z^{2}a^{2}+a^{2}z^{-2}+4a^{2}-3z^{2}-2z^{-2}-5+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$a^{5}z^{5}-3a^{5}z^{3}+a^{5}z+2a^{4}z^{6}-7a^{4}z^{4}+6a^{4}z^{2}-2a^{4}+a^{3}z^{7}-a^{3}z^{5}-4a^{3}z^{3}+3a^{3}z+4a^{2}z^{6}-14a^{2}z^{4}+17a^{2}z^{2}+3z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-8a^{2}-4a^{-2}+az^{7}-az^{5}+z^{5}a^{-1}-2az^{3}-z^{3}a^{-1}+5az+3za^{-1}-2az^{-1}-2a^{-1}z^{-1}+2z^{6}-7z^{4}+14z^{2}+2z^{-2}-9$ (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

5

2

3

1

0

1

3

1

2

-1

2

3

1

-3

2

1

-5

1

3

2

-7

1

0

-9

1

-11

1

-1

Integral Khovanov Homology

(db, data source)

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