L9a31

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	[edit L9a31 Quick Notes] L9a31 is $9_{39}^{2}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,9,17,10 X₆₇₁₈ X_18,13,7,14 X_10,4,11,3 X_14,6,15,5 X_4,12,5,11 X_12,17,13,18 X_2,16,3,15
Gauss code	{1, -9, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -8, 4, -6, 9, -2, 8, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {\left(v^{2}-v+1\right)(uv-u+1)(uv-v+1)}{uv^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+3q^{5/2}-6q^{3/2}+7{\sqrt {q}}-{\frac {10}{\sqrt {q}}}+{\frac {9}{q^{3/2}}}-{\frac {8}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-3a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}+az^{7}+5az^{5}-z^{5}a^{-1}+9az^{3}-3z^{3}a^{-1}+7az-3za^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}-a^{6}z^{2}+3a^{5}z^{5}-3a^{5}z^{3}+5a^{4}z^{6}-7a^{4}z^{4}+4a^{4}z^{2}-a^{4}+5a^{3}z^{7}-8a^{3}z^{5}+z^{5}a^{-3}+8a^{3}z^{3}-2z^{3}a^{-3}-4a^{3}z+za^{-3}+a^{3}z^{-1}+2a^{2}z^{8}+4a^{2}z^{6}+3z^{6}a^{-2}-12a^{2}z^{4}-6z^{4}a^{-2}+10a^{2}z^{2}+2z^{2}a^{-2}-3a^{2}+9az^{7}+4z^{7}a^{-1}-20az^{5}-8z^{5}a^{-1}+18az^{3}+5z^{3}a^{-1}-9az-4za^{-1}+3az^{-1}+2a^{-1}z^{-1}+2z^{8}+2z^{6}-10z^{4}+7z^{2}-3$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

2

-2

4

1

3

2

4

3

-1

0

6

3

-2

4

5

1

-4

4

5

-1

-6

2

4

2

-8

1

4

-3

-10

2

-12

1

-1

Integral Khovanov Homology

(db, data source)

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