L9n13

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	[edit L9n13 Quick Notes] L9n13 is $9_{54}^{2}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L9n13's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_11,17,12,16 X_3,10,4,11 X_15,3,16,2 X_5,13,6,12 X₆₇₁₈ X_9,14,10,15 X_13,18,14,7 X_17,4,18,5
Gauss code	{1, 4, -3, 9, -5, -6}, {6, -1, -7, 3, -2, 5, -8, 7, -4, 2, -9, 8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-6

-5

-4

-3

-2

-1

0

1

χ

2

1

-1

0

2

-2

1

2

1

-4

3

1

2

-6

1

2

1

-8

1

2

-1

-10

1

0

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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