L9a3

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a3 at Knotilus!
	[edit L9a3 Quick Notes] L9a3 is $9_{33}^{2}$ in the Rolfsen table of links.

[edit L9a3 Further Notes and Views]

Link Presentations

[edit Notes on L9a3's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_12,6,13,5 X₈₄₉₃ X_14,10,15,9 X_10,14,11,13 X_18,12,5,11 X_2,16,3,15
Gauss code	{1, -9, 5, -3}, {4, -1, 2, -5, 6, -7, 8, -4, 7, -6, 9, -2, 3, -8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-6q^{9/2}+8q^{7/2}-10q^{5/2}+{\frac {1}{q^{5/2}}}+9q^{3/2}-{\frac {3}{q^{3/2}}}-q^{13/2}+4q^{11/2}-9{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}+z^{5}a^{-3}-az^{3}+2z^{3}a^{-1}+z^{3}a^{-3}-z^{3}a^{-5}-az+3za^{-1}-2za^{-3}+2a^{-1}z^{-1}-3a^{-3}z^{-1}+a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-2z^{8}a^{-2}-2z^{8}a^{-4}-4z^{7}a^{-1}-9z^{7}a^{-3}-5z^{7}a^{-5}-4z^{6}a^{-2}-4z^{6}a^{-4}-4z^{6}a^{-6}-4z^{6}-3az^{5}+2z^{5}a^{-1}+16z^{5}a^{-3}+10z^{5}a^{-5}-z^{5}a^{-7}-a^{2}z^{4}+9z^{4}a^{-2}+13z^{4}a^{-4}+8z^{4}a^{-6}+3z^{4}+4az^{3}+3z^{3}a^{-1}-6z^{3}a^{-3}-4z^{3}a^{-5}+z^{3}a^{-7}+a^{2}z^{2}-2z^{2}a^{-2}-3z^{2}a^{-4}-2z^{2}a^{-6}-2az-5za^{-1}-3za^{-3}-3a^{-2}-3a^{-4}-a^{-6}+2a^{-1}z^{-1}+3a^{-3}z^{-1}+a^{-5}z^{-1}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

3

1

2

8

5

3

-2

6

5

3

2

4

5

1

2

5

0

2

6

4

-2

1

3

-2

-4

2

-6

1

-1

Integral Khovanov Homology

(db, data source)

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