L9a14

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	[edit L9a14 Quick Notes] L9a14 is $9_{13}^{2}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,8,15,7 X_16,10,17,9 X_18,12,5,11 X_8,16,9,15 X_10,18,11,17 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -6, 4, -7, 5, -2, 9, -3, 6, -4, 7, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{2}-v+1\right)\left(v^{2}+v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-4q^{9/2}+2q^{7/2}-3q^{5/2}+q^{3/2}+q^{19/2}-2q^{17/2}+3q^{15/2}-3q^{13/2}+4q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{5}a^{-7}+4z^{3}a^{-7}+4za^{-7}+2a^{-7}z^{-1}-z^{7}a^{-5}-6z^{5}a^{-5}-12z^{3}a^{-5}-11za^{-5}-5a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+7za^{-3}+3a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-4}-z^{8}a^{-6}-z^{7}a^{-3}-4z^{7}a^{-5}-3z^{7}a^{-7}+4z^{6}a^{-4}+z^{6}a^{-6}-3z^{6}a^{-8}+6z^{5}a^{-3}+20z^{5}a^{-5}+11z^{5}a^{-7}-3z^{5}a^{-9}-z^{4}a^{-4}+8z^{4}a^{-6}+6z^{4}a^{-8}-3z^{4}a^{-10}-12z^{3}a^{-3}-30z^{3}a^{-5}-12z^{3}a^{-7}+4z^{3}a^{-9}-2z^{3}a^{-11}-8z^{2}a^{-4}-12z^{2}a^{-6}+3z^{2}a^{-10}-z^{2}a^{-12}+10za^{-3}+17za^{-5}+7za^{-7}+5a^{-4}+5a^{-6}-a^{-10}-3a^{-3}z^{-1}-5a^{-5}z^{-1}-2a^{-7}z^{-1}$ (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

6

7

χ

20

1

-1

18

1

16

2

1

-1

14

1

0

12

3

2

-1

10

1

0

8

1

3

2

6

2

1

4

1

3

2

0

1

Integral Khovanov Homology

(db, data source)

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