L8n4

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n4 at Knotilus!
	[edit L8n4 Quick Notes] L8n4 is $8_{8}^{3}$ in the Rolfsen table of links.

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,14,3,13 X_14,7,15,8 X_9,16,10,11 X_11,10,12,5 X_15,1,16,4
Gauss code	{1, -4, -3, 8}, {-2, -1, 5, 3, -6, 7}, {-7, 2, 4, -5, -8, 6}

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

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

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

-6

-5

-4

-3

-2

-1

0

χ

-1

2

-3

1

2

1

-5

2

-7

1

2

1

-9

1

0

-11

1

-13

1

0

-15

1

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