L9a23

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	[edit L9a23 Quick Notes] L9a23 is $9_{22}^{2}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L9a23's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_18,16,7,15 X_14,5,15,6 X_4,13,5,14 X_12,18,13,17 X_16,12,17,11 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -8, 2, -5, 4, -9}, {8, -1, 9, -2, 7, -6, 5, -4, 3, -7, 6, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

2

3

1

-2

0

4

2

-2

5

4

-1

-4

3

0

-6

3

5

2

-8

2

3

-1

-10

3

-12

1

2

-1

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1}	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=0$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{4}$
$r=1$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$