L9n12

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	[edit L9n12 Quick Notes] L9n12 is $9_{59}^{2}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L9n12's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,17,12,16 X_7,15,8,14 X_15,9,16,8 X_13,5,14,18 X_17,13,18,12 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, -4, 5, 9, -2, -3, 7, -6, 4, -5, 3, -7, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(2)^{5}-t(1)t(2)^{3}+t(2)^{3}+t(1)t(2)^{2}-t(2)^{2}-t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-q^{9/2}+q^{7/2}-q^{5/2}+q^{3/2}-{\sqrt {q}}-{\frac {1}{\sqrt {q}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-5}+2za^{-3}+2a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+4az-9za^{-1}+3az^{-1}-5a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-3z^{2}a^{-6}+a^{-6}+z^{5}a^{-5}-3z^{3}a^{-5}+za^{-5}+z^{4}a^{-4}-2z^{2}a^{-4}+z^{3}a^{-3}-3za^{-3}+2a^{-3}z^{-1}+z^{6}a^{-2}-7z^{4}a^{-2}+13z^{2}a^{-2}-5a^{-2}+az^{7}+z^{7}a^{-1}-7az^{5}-8z^{5}a^{-1}+15az^{3}+19z^{3}a^{-1}-12az-16za^{-1}+3az^{-1}+5a^{-1}z^{-1}+z^{6}-7z^{4}+12z^{2}-5$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

0

8

1

0

6

1

0

4

1

0

2

1

2

1

0

2

-2

1

-4

1

-6

1

Integral Khovanov Homology

(db, data source)

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