L9a16

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a16 at Knotilus!
	[edit L9a16 Quick Notes] L9a16 is $9_{28}^{2}$ in the Rolfsen table of links.

[edit L9a16 Further Notes and Views]

Link Presentations

[edit Notes on L9a16's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-2

3

1

2

-4

4

3

-1

-6

4

2

-8

3

4

1

-10

4

0

-12

1

4

3

-14

1

3

-2

-16

1

-18

1

-1

Integral Khovanov Homology

(db, data source)

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