L8a3

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a3 at Knotilus!
	[edit L8a3 Quick Notes] L8a3 is $8_{9}^{2}$ in the Rolfsen table of links.

[edit L8a3 Further Notes and Views]

Link Presentations

[edit Notes on L8a3's Link Presentations]

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

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

Polynomial invariants

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

-6

-5

-4

-3

-2

-1

0

1

2

χ

4

1

2

1

-1

0

3

1

2

-2

3

2

-1

-4

2

0

-6

2

3

1

-8

2

0

-10

1

3

2

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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