L9n9

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n9 at Knotilus!
	[edit L9n9 Quick Notes] L9n9 is $9_{60}^{2}$ in the Rolfsen table of links.

[edit L9n9 Further Notes and Views]

Link Presentations

[edit Notes on L9n9's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,14,8,15 X_15,18,16,5 X_11,16,12,17 X_17,12,18,13 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, -3, 7, 9, -2, -5, 6, -7, 3, -4, 5, -6, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{3}+t(2)^{3}-t(2)^{2}-t(1)t(2)+t(1)+1}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{11/2}}}-{\frac {1}{q^{15/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+2za^{9}+2a^{9}z^{-1}+za^{7}-z^{5}a^{5}-5z^{3}a^{5}-5za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+5z^{4}a^{12}-6z^{2}a^{12}+2a^{12}-z^{7}a^{11}+5z^{5}a^{11}-6z^{3}a^{11}+3za^{11}-a^{11}z^{-1}-2z^{6}a^{10}+11z^{4}a^{10}-15z^{2}a^{10}+5a^{10}-z^{7}a^{9}+6z^{5}a^{9}-10z^{3}a^{9}+7za^{9}-2a^{9}z^{-1}-z^{6}a^{8}+6z^{4}a^{8}-8z^{2}a^{8}+3a^{8}+z^{3}a^{7}-za^{7}+z^{2}a^{6}-a^{6}-z^{5}a^{5}+5z^{3}a^{5}-5za^{5}+a^{5}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

1

-8

1

0

-10

1

-12

1

3

1

-14

1

-16

1

0

-18

1

0

-20

0

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$	$i=-2$
$r=-9$		${\mathbb {Z} }$
$r=-8$		${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$		${\mathbb {Z} }$
$r=-6$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$