L8a19

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8a19 at Knotilus!
	[edit L8a19 Quick Notes] L8a19 is $8_{6}^{3}$ in the Rolfsen table of links.

[edit L8a19 Further Notes and Views]

Link Presentations

[edit Notes on L8a19's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,5,15,6 X_8,12,9,11 X_16,8,11,7 X_10,13,5,14 X_2,9,3,10 X_4,16,1,15
Gauss code	{1, -7, 2, -8}, {3, -1, 5, -4, 7, -6}, {4, -2, 6, -3, 8, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(3)t(2)-t(2)+1)(t(2)t(3)-t(3)+1)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{4}+q^{-4}-3q^{3}-3q^{-3}+5q^{2}+5q^{-2}-5q-5q^{-1}+8$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}+a^{2}z^{4}+z^{4}a^{-2}-4z^{4}+2a^{2}z^{2}+2z^{2}a^{-2}-5z^{2}+a^{2}+a^{-2}-2+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$2az^{7}+2z^{7}a^{-1}+4a^{2}z^{6}+4z^{6}a^{-2}+8z^{6}+3a^{3}z^{5}+az^{5}+z^{5}a^{-1}+3z^{5}a^{-3}+a^{4}z^{4}-7a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}-16z^{4}-4a^{3}z^{3}-4az^{3}-4z^{3}a^{-1}-4z^{3}a^{-3}-a^{4}z^{2}+5a^{2}z^{2}+5z^{2}a^{-2}-z^{2}a^{-4}+12z^{2}+2az+2za^{-1}-2a^{2}-2a^{-2}-3-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

χ

9

1

7

2

-2

5

3

1

2

3

0

1

5

2

3

-1

2

5

3

-3

3

0

-5

1

3

2

-7

2

-2

-9

1

Integral Khovanov Homology

(db, data source)

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