L9a5

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a5 at Knotilus!
	[edit L9a5 Quick Notes] L9a5 is $9_{30}^{2}$ in the Rolfsen table of links.

[edit L9a5 Further Notes and Views]

Link Presentations

[edit Notes on L9a5's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{3}-3t(2)^{3}-4t(1)t(2)^{2}+5t(2)^{2}+5t(1)t(2)-4t(2)-3t(1)+1}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+6q^{7/2}-{\frac {1}{q^{7/2}}}-8q^{5/2}+{\frac {2}{q^{5/2}}}+9q^{3/2}-{\frac {6}{q^{3/2}}}+q^{11/2}-9{\sqrt {q}}+{\frac {7}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}-2az^{3}+z^{3}a^{-1}-2z^{3}a^{-3}+a^{3}z-2az-za^{-1}-za^{-3}+za^{-5}+a^{3}z^{-1}-2a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-2}-z^{8}-3az^{7}-7z^{7}a^{-1}-4z^{7}a^{-3}-2a^{2}z^{6}-10z^{6}a^{-2}-5z^{6}a^{-4}-7z^{6}-a^{3}z^{5}+5az^{5}+11z^{5}a^{-1}+2z^{5}a^{-3}-3z^{5}a^{-5}+3a^{2}z^{4}+24z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}+19z^{4}+3a^{3}z^{3}-3az^{3}-7z^{3}a^{-1}+2z^{3}a^{-3}+3z^{3}a^{-5}-19z^{2}a^{-2}-5z^{2}a^{-4}+z^{2}a^{-6}-13z^{2}-3a^{3}z+2az+6za^{-1}-za^{-5}-a^{2}+5a^{-2}+2a^{-4}+3+a^{3}z^{-1}-2a^{-1}z^{-1}-a^{-3}z^{-1}$ (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

4

1

-3

6

4

2

4

5

4

-1

2

4

0

4

6

2

-2

2

3

-1

-4

4

-6

1

2

-1

-8

1

Integral Khovanov Homology

(db, data source)

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