L6a2

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	[edit L6a2 Quick Notes] L6a2 is $6_{2}^{2}$ in the Rolfsen table of links.

Mongolian ornament (4 crossings are unnecessary)

Planar diagram presentation	X₈₁₉₂ X_12,5,7,6 X_10,3,11,4 X_4,11,5,12 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -5, 3, -4, 2, -6}, {5, -1, 6, -3, 4, -2}

A Braid Representative

A Morse Link Presentation

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

-2

1

-4

1

0

-6

1

-8

1

0

-10

1

0

-12

1

-14

1

0

-16

1

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