L9a48

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	[edit L9a48 Quick Notes] L9a48 is $9_{5}^{3}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_18,14,11,13 X_8,16,9,15 X_14,8,15,7 X_16,10,17,9 X_10,18,5,17 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -8, 2, -9}, {8, -1, 5, -4, 6, -7}, {9, -2, 3, -5, 4, -6, 7, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{2}-uv^{2}w-uvw^{2}+2uvw-uv-uw+2u-2v^{2}w^{2}+v^{2}w+vw^{2}-2vw+v+w-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{6}+3q^{5}-4q^{4}+6q^{3}-6q^{2}+6q-4+4q^{-1}-q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-4}-2z^{2}a^{-4}+z^{6}a^{-2}+4z^{4}a^{-2}+a^{2}z^{2}+5z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+3a^{2}+3a^{-2}-2z^{4}-7z^{2}-2z^{-2}-6$ (db)
Kauffman polynomial	$z^{8}a^{-2}+z^{8}+az^{7}+4z^{7}a^{-1}+3z^{7}a^{-3}+a^{2}z^{6}+2z^{6}a^{-2}+4z^{6}a^{-4}-z^{6}-2az^{5}-10z^{5}a^{-1}-4z^{5}a^{-3}+4z^{5}a^{-5}-5a^{2}z^{4}-8z^{4}a^{-2}-4z^{4}a^{-4}+3z^{4}a^{-6}-6z^{4}-3az^{3}+2z^{3}a^{-1}+z^{3}a^{-3}-3z^{3}a^{-5}+z^{3}a^{-7}+8a^{2}z^{2}+5z^{2}a^{-2}-2z^{2}a^{-6}+11z^{2}+6az+6za^{-1}-5a^{2}-3a^{-2}+a^{-4}-8-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

13

1

-1

11

2

9

2

1

-1

7

4

2

5

4

0

3

2

0

1

3

5

2

-1

1

0

-3

3

-5

1

0

-7

1

Integral Khovanov Homology

(db, data source)

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