L7n2

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	[edit L7n2 Quick Notes] L7n2 is $7_{8}^{2}$ in the Rolfsen table of links.

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_13,1,14,4 X_5,10,6,11 X₃₈₄₉ X_9,14,10,5 X_2,12,3,11
Gauss code	{1, -7, -5, 3}, {-4, -1, 2, 5, -6, 4, 7, -2, -3, 6}

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	$-{\frac {2}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {2}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {1}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}(-z)-a^{5}z^{-1}+a^{3}z^{3}+3a^{3}z+3a^{3}z^{-1}-2az-2az^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}-3a^{6}z^{2}+a^{6}+a^{5}z^{5}-3a^{5}z^{3}+2a^{5}z-a^{5}z^{-1}+2a^{4}z^{4}-5a^{4}z^{2}+3a^{4}+a^{3}z^{5}-3a^{3}z^{3}+5a^{3}z-3a^{3}z^{-1}+a^{2}z^{4}-2a^{2}z^{2}+3a^{2}+3az-2az^{-1}$ (db)

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

-5

-4

-3

-2

-1

0

χ

0

2

-2

1

2

1

-4

1

-6

1

-8

1

0

-10

0

-12

1

-1

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