L9n18

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	[edit L9n18 Quick Notes] L9n18 is $9_{53}^{2}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L9n18's Link Presentations]

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_18,5,9,6 X_7,14,8,15 X_13,16,14,17 X_15,8,16,1 X_6,9,7,10 X_4,17,5,18
Gauss code	{1, -2, 3, -9, 4, -8, -5, 7}, {8, -1, 2, -3, -6, 5, -7, 6, 9, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {\left(t(2)t(1)^{2}+1\right)\left(t(1)t(2)^{2}+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}-{\frac {1}{q^{11/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{11}(-z)+a^{9}z^{5}+6a^{9}z^{3}+8a^{9}z+a^{9}z^{-1}-a^{7}z^{7}-7a^{7}z^{5}-15a^{7}z^{3}-11a^{7}z-a^{7}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{13}+3za^{13}+za^{11}-z^{6}a^{10}+6z^{4}a^{10}-8z^{2}a^{10}-z^{7}a^{9}+7z^{5}a^{9}-14z^{3}a^{9}+9za^{9}-a^{9}z^{-1}-z^{6}a^{8}+6z^{4}a^{8}-8z^{2}a^{8}+a^{8}-z^{7}a^{7}+7z^{5}a^{7}-15z^{3}a^{7}+11za^{7}-a^{7}z^{-1}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-6

1

-8

1

-10

1

-12

1

-14

1

-1

-16

1

-1

-18

1

0

-20

1

-22

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-8$	$i=-6$	$i=-4$
$r=-8$		${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-7$
$r=-6$		${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$		${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$