L8a10

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	[edit L8a10 Quick Notes] L8a10 is $8_{5}^{2}$ in the Rolfsen table of links.

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Symmetric version

Mongolian ornament

Link Presentations

[edit Notes on L8a10's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_12,15,13,16 X_14,5,15,6 X_4,13,5,14 X_16,11,7,12 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -7, 2, -5, 4, -8}, {7, -1, 8, -2, 6, -3, 5, -4, 3, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(2)t(1)^{2}-t(1)^{2}+2t(2)^{2}t(1)-3t(2)t(1)+2t(1)-t(2)^{2}+2t(2)}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z-a^{7}z^{3}+a^{7}z^{-1}-2a^{5}z^{3}-3a^{5}z-a^{5}z^{-1}-a^{3}z^{3}-a^{3}z$ (db)
Kauffman polynomial	$-z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-2z^{6}a^{10}+6z^{4}a^{10}-4z^{2}a^{10}-z^{7}a^{9}+4z^{3}a^{9}-za^{9}-4z^{6}a^{8}+8z^{4}a^{8}-3z^{2}a^{8}-z^{7}a^{7}-2z^{5}a^{7}+6z^{3}a^{7}-4za^{7}+a^{7}z^{-1}-2z^{6}a^{6}+2z^{2}a^{6}-a^{6}-3z^{5}a^{5}+4z^{3}a^{5}-4za^{5}+a^{5}z^{-1}-2z^{4}a^{4}+z^{2}a^{4}-z^{3}a^{3}+za^{3}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

1

-4

2

1

-1

-6

2

-8

2

0

-10

3

2

1

-12

1

2

1

-14

2

3

-1

-16

1

2

1

-18

1

-1

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$