L9a53

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	[edit L9a53 Quick Notes] L9a53 is $9_{12}^{3}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_18,10,15,9 X₈₄₉₃ X_16,5,17,6 X_14,17,5,18 X_10,16,11,15 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 $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)^{3}}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{4}-4q^{3}+8q^{2}-9q+12-10q^{-1}+10q^{-2}-6q^{-3}+3q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{4}z^{2}-a^{4}+2a^{2}z^{4}+z^{4}a^{-2}+4a^{2}z^{2}+z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+3a^{2}+a^{-2}-z^{6}-3z^{4}-4z^{2}-2z^{-2}-3$ (db)
Kauffman polynomial	$2a^{2}z^{8}+2z^{8}+4a^{3}z^{7}+11az^{7}+7z^{7}a^{-1}+3a^{4}z^{6}+7a^{2}z^{6}+8z^{6}a^{-2}+12z^{6}+a^{5}z^{5}-5a^{3}z^{5}-17az^{5}-7z^{5}a^{-1}+4z^{5}a^{-3}-6a^{4}z^{4}-22a^{2}z^{4}-11z^{4}a^{-2}+z^{4}a^{-4}-28z^{4}-2a^{5}z^{3}-a^{3}z^{3}+3az^{3}-2z^{3}a^{-3}+5a^{4}z^{2}+17a^{2}z^{2}+5z^{2}a^{-2}+17z^{2}+a^{5}z+3a^{3}z+3az+za^{-1}-2a^{4}-6a^{2}-2a^{-2}-5-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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

9

1

7

3

-3

5

1

4

3

4

3

-1

1

8

5

3

-1

6

8

2

-3

4

0

-5

2

6

4

-7

1

4

-3

-9

2

-11

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$