L8n7

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L8n7 at Knotilus!
	[edit L8n7 Quick Notes] L8n7 is $8_{2}^{4}$ in the Rolfsen table of links.

Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_11,13,12,16 X_3,11,4,10 X_9,1,10,4 X_7,15,8,14 X_13,5,14,8 X_15,9,16,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 8}, {-7, 6, -8, 3}

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

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)-t(1)t(3)t(2)+t(3)t(2)-t(1)t(4)t(2)-t(3)+t(1)t(4)+t(3)t(4)-t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}{\sqrt {t(4)}}}}$ (db)
Jones polynomial	$-q^{13/2}+q^{11/2}-4q^{9/2}+q^{7/2}-4q^{5/2}+2q^{3/2}-3{\sqrt {q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-a^{-7}z^{-3}-a^{-7}z^{-1}+3a^{-5}z^{-3}+3za^{-5}+5a^{-5}z^{-1}-2z^{3}a^{-3}-3a^{-3}z^{-3}-6za^{-3}-7a^{-3}z^{-1}+a^{-1}z^{-3}+3za^{-1}+3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-4z^{3}a^{-7}+a^{-7}z^{-3}+6za^{-7}-4a^{-7}z^{-1}+z^{6}a^{-6}-z^{4}a^{-6}-6z^{2}a^{-6}-3a^{-6}z^{-2}+8a^{-6}+5z^{5}a^{-5}-16z^{3}a^{-5}+3a^{-5}z^{-3}+14za^{-5}-9a^{-5}z^{-1}+z^{6}a^{-4}+2z^{4}a^{-4}-12z^{2}a^{-4}-6a^{-4}z^{-2}+15a^{-4}+4z^{5}a^{-3}-12z^{3}a^{-3}+3a^{-3}z^{-3}+14za^{-3}-9a^{-3}z^{-1}+3z^{4}a^{-2}-6z^{2}a^{-2}-3a^{-2}z^{-2}+8a^{-2}+a^{-1}z^{-3}+6za^{-1}-4a^{-1}z^{-1}$ (db)

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

0

1

2

3

4

5

6

χ

14

1

12

0

10

4

1

3

8

1

4

3

6

3

4

1

2

4

3

1

0

3

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