L9n17

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	[edit L9n17 Quick Notes] L9n17 is $9_{52}^{2}$ 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 X_7,14,8,15 X_13,18,14,7 X_17,1,18,6 X_16,11,17,12 X_5,12,6,13 X_4,16,5,15
Gauss code	{1, -2, 3, -9, -8, 6}, {-4, -1, 2, -3, 7, 8, -5, 4, 9, -7, -6, 5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

χ

-2

2

-4

1

0

-6

3

1

2

-8

2

0

-10

2

0

-12

1

2

1

-14

1

2

-1

-16

1

-18

1

-1

Integral Khovanov Homology

(db, data source)

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