L9a30

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a30 at Knotilus!
	[edit L9a30 Quick Notes] L9a30 is $9_{3}^{2}$ in the Rolfsen table of links.

[edit L9a30 Further Notes and Views]

Link Presentations

[edit Notes on L9a30's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X₆₇₁₈ X_16,13,17,14 X_14,6,15,5 X_4,16,5,15 X_18,11,7,12 X_12,17,13,18
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 8, -9, 5, -6, 7, -5, 9, -8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{2}-2u^{2}v-2uv^{2}+3uv-2u-2v+2}{uv}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {1}{\sqrt {q}}}-{\frac {3}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {5}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {3}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-2a^{7}z+a^{5}z^{5}+3a^{5}z^{3}+2a^{5}z+a^{3}z^{5}+3a^{3}z^{3}+2a^{3}z+a^{3}z^{-1}-az^{3}-3az-az^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}-2a^{10}z^{2}+2a^{9}z^{5}-4a^{9}z^{3}+a^{9}z+2a^{8}z^{6}-3a^{8}z^{4}+a^{8}z^{2}+2a^{7}z^{7}-5a^{7}z^{5}+7a^{7}z^{3}-2a^{7}z+a^{6}z^{8}-2a^{6}z^{6}+4a^{6}z^{4}-a^{6}z^{2}+3a^{5}z^{7}-9a^{5}z^{5}+12a^{5}z^{3}-4a^{5}z+a^{4}z^{8}-3a^{4}z^{6}+6a^{4}z^{4}-5a^{4}z^{2}+a^{3}z^{7}-a^{3}z^{5}-3a^{3}z^{3}+3a^{3}z-a^{3}z^{-1}+a^{2}z^{6}-2a^{2}z^{4}-a^{2}z^{2}+a^{2}+az^{5}-4az^{3}+4az-az^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

-2

3

1

2

-4

2

1

-1

-6

3

2

1

-8

3

0

-10

2

0

-12

1

3

2

-14

1

2

-1

-16

1

-18

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{3}$
$r=1$	${\mathbb {Z} }$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$