L9n10

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	[edit L9n10 Quick Notes] L9n10 is $9_{58}^{2}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,14,8,15 X_18,16,5,15 X_16,12,17,11 X_12,18,13,17 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, -3, 7, 9, -2, 5, -6, -7, 3, 4, -5, 6, -4}

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

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

2

1

-1

0

3

2

1

-2

3

0

-4

2

0

-6

1

3

2

-8

1

2

-1

-10

2

Integral Khovanov Homology

(db, data source)

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