L9a27

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	[edit L9a27 Quick Notes] L9a27 is $9_{12}^{2}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_18,10,7,9 X₂₇₃₈ X_16,13,17,14 X_6,12,1,11 X_4,16,5,15 X_14,6,15,5 X_12,17,13,18
Gauss code	{1, -4, 2, -7, 8, -6}, {4, -1, 3, -2, 6, -9, 5, -8, 7, -5, 9, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

2

8

3

1

-2

6

5

2

3

4

0

2

5

4

1

0

3

5

2

-2

2

4

-2

-4

1

3

2

-6

2

-2

-8

1

Integral Khovanov Homology

(db, data source)

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