L8n2

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	[edit L8n2 Quick Notes] L8n2 is $8_{15}^{2}$ in the Rolfsen table of links.

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

A Braid Representative

A Morse Link Presentation

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

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

-4

-3

-2

-1

0

1

2

χ

6

1

4

0

2

1

0

1

2

1

-2

1

0

-4

1

0

-6

0

-8

1

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