L8n3

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	[edit L8n3 Quick Notes] L8n3 is $8_{7}^{3}$ in the Rolfsen table of links.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{2}-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{-3}+q^{-5}+q^{-7}+q^{-9}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}+a^{10}-z^{4}a^{8}-5z^{2}a^{8}-2a^{8}z^{-2}-6a^{8}+z^{6}a^{6}+6z^{4}a^{6}+10z^{2}a^{6}+a^{6}z^{-2}+5a^{6}$ (db)
Kauffman polynomial	$a^{12}+a^{10}z^{2}+a^{10}z^{-2}-3a^{10}+a^{9}z^{5}-5a^{9}z^{3}+6a^{9}z-2a^{9}z^{-1}+a^{8}z^{6}-6a^{8}z^{4}+11a^{8}z^{2}+2a^{8}z^{-2}-8a^{8}+a^{7}z^{5}-5a^{7}z^{3}+6a^{7}z-2a^{7}z^{-1}+a^{6}z^{6}-6a^{6}z^{4}+10a^{6}z^{2}+a^{6}z^{-2}-5a^{6}$ (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

-5

1

-7

1

-9

1

-11

1

-13

2

1

-15

1

-17

2

1

-19

1

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-7$	$i=-5$	$i=-3$
$r=-6$	${\mathbb {Z} }$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$