L9a4

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	[edit L9a4 Quick Notes] L9a4 is $9_{18}^{2}$ in the Rolfsen table of links.

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2(u-1)(v-1)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+6q^{13/2}-7q^{11/2}+8q^{9/2}-9q^{7/2}+6q^{5/2}-5q^{3/2}+2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+za^{-7}+a^{-7}z^{-1}-z^{5}a^{-5}-2z^{3}a^{-5}-2za^{-5}-2a^{-5}z^{-1}-z^{5}a^{-3}-2z^{3}a^{-3}-za^{-3}+z^{3}a^{-1}+2za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-4}-z^{8}a^{-6}-2z^{7}a^{-3}-6z^{7}a^{-5}-4z^{7}a^{-7}-2z^{6}a^{-2}-4z^{6}a^{-4}-7z^{6}a^{-6}-5z^{6}a^{-8}-z^{5}a^{-1}+z^{5}a^{-3}+10z^{5}a^{-5}+5z^{5}a^{-7}-3z^{5}a^{-9}+4z^{4}a^{-2}+11z^{4}a^{-4}+16z^{4}a^{-6}+8z^{4}a^{-8}-z^{4}a^{-10}+3z^{3}a^{-1}+4z^{3}a^{-3}-6z^{3}a^{-5}-4z^{3}a^{-7}+3z^{3}a^{-9}-z^{2}a^{-2}-8z^{2}a^{-4}-13z^{2}a^{-6}-5z^{2}a^{-8}+z^{2}a^{-10}-3za^{-1}-2za^{-3}+3za^{-5}+2za^{-7}-a^{-2}+3a^{-4}+5a^{-6}+2a^{-8}+a^{-1}z^{-1}-2a^{-5}z^{-1}-a^{-7}z^{-1}$ (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

18

1

-1

16

2

14

4

1

-3

12

3

2

1

10

5

4

-1

8

4

3

1

6

2

5

3

4

3

4

-1

2

1

4

3

0

1

-1

-2

1

Integral Khovanov Homology

(db, data source)

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