L9n1

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	[edit L9n1 Quick Notes] L9n1 is $9_{45}^{2}$ in the Rolfsen table of links.

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)t(2)^{3}-t(1)t(2)^{2}-t(2)+2}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {2}{q^{9/2}}}+{\frac {1}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {2}{q^{15/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{19/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+z^{3}a^{9}+4za^{9}+2a^{9}z^{-1}-z^{5}a^{7}-4z^{3}a^{7}-3za^{7}-z^{5}a^{5}-4z^{3}a^{5}-3za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{2}-2a^{12}+a^{11}z^{3}-a^{11}z+a^{11}z^{-1}+a^{10}z^{6}-5a^{10}z^{4}+10a^{10}z^{2}-5a^{10}+a^{9}z^{7}-5a^{9}z^{5}+9a^{9}z^{3}-5a^{9}z+2a^{9}z^{-1}+2a^{8}z^{6}-8a^{8}z^{4}+9a^{8}z^{2}-3a^{8}+a^{7}z^{7}-4a^{7}z^{5}+4a^{7}z^{3}-a^{7}z+a^{6}z^{6}-3a^{6}z^{4}+a^{6}+a^{5}z^{5}-4a^{5}z^{3}+3a^{5}z-a^{5}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

1

0

-8

1

-10

1

-12

3

1

2

-14

1

-16

1

2

-1

-18

0

-20

1

-1

Integral Khovanov Homology

(db, data source)

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