L9n14

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	[edit L9n14 Quick Notes] L9n14 is $9_{50}^{2}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L9n14's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_11,17,12,16 X_3,10,4,11 X_15,3,16,2 X_5,13,6,12 X₆₇₁₈ X_14,10,15,9 X_18,14,7,13 X_17,4,18,5
Gauss code	{1, 4, -3, 9, -5, -6}, {6, -1, 7, 3, -2, 5, 8, -7, -4, 2, -9, -8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

0

6

1

0

4

1

2

1

0

2

1

-2

1

-4

1

0

-6

1

Integral Khovanov Homology

(db, data source)

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