L8n1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n1 at Knotilus!
	[edit L8n1 Quick Notes] L8n1 is $8_{16}^{2}$ in the Rolfsen table of links.

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_9,12,10,13 X₃₈₄₉ X_5,11,6,10 X_11,5,12,16 X_13,2,14,3
Gauss code	{1, 8, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 4, -8, -2, 3, 7}

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

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

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

-4

-3

-2

-1

0

1

2

χ

2

1

0

-2

2

1

-4

1

0

-6

1

0

-8

1

0

-10

1

0

-12

2

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