L11n208

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_5,14,6,15 X_22,18,9,17 X_19,5,20,4 X_21,6,22,7 X_7,17,8,16 X_8,9,1,10 X_18,14,19,13 X_15,21,16,20
Gauss code	{1, -2, 3, 6, -4, 7, -8, -9}, {9, -1, 2, -3, 10, 4, -11, 8, 5, -10, -6, 11, -7, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{3}v^{3}-u^{3}v^{2}+u^{2}v^{2}+uv-v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}+q^{7/2}-{\frac {1}{q^{7/2}}}-q^{5/2}+{\frac {1}{q^{5/2}}}+q^{3/2}-{\frac {2}{q^{3/2}}}-{\frac {1}{q^{11/2}}}-2{\sqrt {q}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{3}z^{5}+5a^{3}z^{3}+6a^{3}z+2a^{3}z^{-1}-az^{7}-7az^{5}+z^{5}a^{-1}-16az^{3}+5z^{3}a^{-1}-14az+6za^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-a^{2}z^{8}-z^{8}a^{-2}-2z^{8}+7az^{7}+7z^{7}a^{-1}+6a^{2}z^{6}+7z^{6}a^{-2}+13z^{6}-2a^{3}z^{5}-18az^{5}-16z^{5}a^{-1}-a^{4}z^{4}-12a^{2}z^{4}-15z^{4}a^{-2}-26z^{4}-a^{5}z^{3}+8a^{3}z^{3}+24az^{3}+15z^{3}a^{-1}-a^{6}z^{2}+2a^{4}z^{2}+11a^{2}z^{2}+10z^{2}a^{-2}+18z^{2}-a^{7}z+a^{5}z-7a^{3}z-16az-7za^{-1}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

8

1

-1

6

0

4

1

0

2

1

0

1

-2

2

1

-4

1

-6

1

0

-8

0

-10

1

0

-12

1

Integral Khovanov Homology

(db, data source)

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