L11n202

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Planar diagram presentation	X_10,1,11,2 X_16,8,17,7 X_11,18,12,19 X_19,3,20,2 X_3,12,4,13 X_13,21,14,20 X_5,15,6,14 X_6,9,7,10 X_15,22,16,9 X_8,18,1,17 X_21,4,22,5
Gauss code	{1, 4, -5, 11, -7, -8, 2, -10}, {8, -1, -3, 5, -6, 7, -9, -2, 10, 3, -4, 6, -11, 9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v^{2}+uv+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{7/2}-2q^{5/2}+2q^{3/2}-4{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {2}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{3}z^{5}+4a^{3}z^{3}+3a^{3}z-az^{7}-6az^{5}+z^{5}a^{-1}-11az^{3}+4z^{3}a^{-1}-6az+3za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{5}z^{7}-5a^{5}z^{5}+6a^{5}z^{3}-a^{5}z+2a^{4}z^{8}-11a^{4}z^{6}+17a^{4}z^{4}-7a^{4}z^{2}+z^{2}a^{-4}+a^{3}z^{9}-4a^{3}z^{7}+3a^{3}z^{5}-a^{3}z^{3}+2z^{3}a^{-3}+2a^{3}z-za^{-3}+3a^{2}z^{8}-15a^{2}z^{6}+22a^{2}z^{4}+2z^{4}a^{-2}-11a^{2}z^{2}-z^{2}a^{-2}+az^{9}-5az^{7}+11az^{5}+3z^{5}a^{-1}-15az^{3}-6z^{3}a^{-1}+6az+2za^{-1}+az^{-1}+a^{-1}z^{-1}+z^{8}-4z^{6}+7z^{4}-6z^{2}-1$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

8

1

-1

6

1

4

1

0

2

3

1

2

0

2

3

1

-2

2

1

-4

1

2

1

-6

1

2

-1

-8

1

0

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

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