L9a15

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	[edit L9a15 Quick Notes] L9a15 is $9_{15}^{2}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L9a15's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

1

14

3

1

-2

12

3

1

2

10

4

3

-1

8

3

0

6

2

4

2

4

3

0

2

1

4

3

0

1

-1

-2

1

Integral Khovanov Homology

(db, data source)

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