L9n27

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	[edit L9n27 Quick Notes] L9n27 is $9_{21}^{3}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L9n27's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_9,18,10,15 X₈₄₉₃ X_5,17,6,16 X_17,5,18,14 X_15,10,16,11 X_2,12,3,11
Gauss code	{1, -9, 5, -3}, {-8, 6, -7, 4}, {-6, -1, 2, -5, -4, 8, 9, -2, 3, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	0 (db)
Jones polynomial	$q^{3}-q^{2}+q+1+q^{-1}+q^{-2}+q^{-4}-q^{-5}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-z^{2}a^{4}-2a^{4}+z^{4}a^{2}+5z^{2}a^{2}+a^{2}z^{-2}+6a^{2}-z^{4}-5z^{2}-2z^{-2}-6+z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$az^{7}+z^{7}a^{-1}+a^{4}z^{6}+2a^{2}z^{6}+z^{6}a^{-2}+2z^{6}+a^{5}z^{5}+a^{3}z^{5}-5az^{5}-5z^{5}a^{-1}-5a^{4}z^{4}-13a^{2}z^{4}-5z^{4}a^{-2}-13z^{4}-4a^{5}z^{3}-6a^{3}z^{3}+2az^{3}+4z^{3}a^{-1}+6a^{4}z^{2}+22a^{2}z^{2}+6z^{2}a^{-2}+22z^{2}+2a^{5}z+6a^{3}z+6az+2za^{-1}-4a^{4}-12a^{2}-4a^{-2}-11-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

0

3

1

0

1

3

1

2

-1

1

4

1

2

-3

1

2

-5

1

-7

1

-9

0

-11

1

-1

Integral Khovanov Homology

(db, data source)

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