L9a29

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	[edit L9a29 Quick Notes] L9a29 is $9_{19}^{2}$ in the Rolfsen table of links.

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{4}-u^{2}v^{3}+u^{2}v^{2}-u^{2}v-uv^{4}+uv^{3}-uv^{2}+uv-u-v^{3}+v^{2}-v+1}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {3}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {2}{q^{17/2}}}+{\frac {1}{q^{19/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{5}\right)-4a^{7}z^{3}-4a^{7}z-a^{7}z^{-1}+a^{5}z^{7}+6a^{5}z^{5}+12a^{5}z^{3}+10a^{5}z+3a^{5}z^{-1}-a^{3}z^{5}-5a^{3}z^{3}-7a^{3}z-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{2}+2a^{11}z^{3}+3a^{10}z^{4}-2a^{10}z^{2}+4a^{9}z^{5}-7a^{9}z^{3}+2a^{9}z+4a^{8}z^{6}-10a^{8}z^{4}+5a^{8}z^{2}-a^{8}+3a^{7}z^{7}-9a^{7}z^{5}+6a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+a^{6}z^{8}-11a^{6}z^{4}+13a^{6}z^{2}-3a^{6}+4a^{5}z^{7}-19a^{5}z^{5}+27a^{5}z^{3}-14a^{5}z+3a^{5}z^{-1}+a^{4}z^{8}-4a^{4}z^{6}+2a^{4}z^{4}+5a^{4}z^{2}-3a^{4}+a^{3}z^{7}-6a^{3}z^{5}+12a^{3}z^{3}-9a^{3}z+2a^{3}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

0

-4

3

1

2

-6

1

0

-8

3

2

1

-10

2

0

-12

2

0

-14

1

2

1

-16

1

2

-1

-18

1

-20

1

-1

Integral Khovanov Homology

(db, data source)

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