L9a39

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a39 at Knotilus!
	[edit L9a39 Quick Notes] L9a39 is $9_{2}^{2}$ in the Rolfsen table of links.

[edit L9a39 Further Notes and Views]

Link Presentations

[edit Notes on L9a39's Link Presentations]

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_14,5,15,6 X_6,9,7,10 X_16,8,17,7 X_18,16,9,15 X_8,18,1,17 X_4,13,5,14
Gauss code	{1, -2, 3, -9, 4, -5, 6, -8}, {5, -1, 2, -3, 9, -4, 7, -6, 8, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{3}t(1)^{3}-t(2)^{2}t(1)^{3}-t(2)^{3}t(1)^{2}+3t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(1)+3t(2)t(1)-t(1)-t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {4}{q^{9/2}}}-{\frac {5}{q^{7/2}}}+{\frac {4}{q^{5/2}}}+q^{3/2}-{\frac {4}{q^{3/2}}}-{\frac {1}{q^{15/2}}}+{\frac {2}{q^{13/2}}}-{\frac {3}{q^{11/2}}}-2{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{3}z^{7}+a^{5}z^{5}-6a^{3}z^{5}+az^{5}+4a^{5}z^{3}-12a^{3}z^{3}+4az^{3}+4a^{5}z-9a^{3}z+3az+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{9}+za^{9}-2z^{4}a^{8}+2z^{2}a^{8}-2z^{5}a^{7}+z^{3}a^{7}-2z^{6}a^{6}+2z^{4}a^{6}-z^{2}a^{6}-2z^{7}a^{5}+5z^{5}a^{5}-7z^{3}a^{5}+5za^{5}-a^{5}z^{-1}-z^{8}a^{4}+z^{6}a^{4}+3z^{4}a^{4}-4z^{2}a^{4}+a^{4}-4z^{7}a^{3}+16z^{5}a^{3}-20z^{3}a^{3}+10za^{3}-a^{3}z^{-1}-z^{8}a^{2}+2z^{6}a^{2}+3z^{4}a^{2}-4z^{2}a^{2}-2z^{7}a+9z^{5}a-11z^{3}a+4za-z^{6}+4z^{4}-3z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

1

0

1

0

-2

3

1

2

-4

2

0

-6

3

2

1

-8

1

2

1

-10

2

3

-1

-12

1

2

1

-14

1

-1

-16

1

Integral Khovanov Homology

(db, data source)

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