L11n223

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_18,8,19,7 X_16,9,17,10 X_22,15,9,16 X_21,5,22,4 X_5,14,6,15 X_13,20,14,21 X_8,18,1,17 X_6,20,7,19
Gauss code	{1, -2, 3, 7, -8, -11, 4, -10}, {5, -1, 2, -3, -9, 8, 6, -5, 10, -4, 11, 9, -7, -6}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(u^{2}v+2uv^{2}+2u+v\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-2{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-2za^{7}+z^{5}a^{5}+2z^{3}a^{5}-a^{5}z^{-1}+2z^{5}a^{3}+7z^{3}a^{3}+7za^{3}+3a^{3}z^{-1}-2z^{3}a-5za-2az^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}-2a^{10}z^{2}+2a^{9}z^{5}-3a^{9}z^{3}+3a^{8}z^{6}-5a^{8}z^{4}+3a^{8}z^{2}+4a^{7}z^{7}-11a^{7}z^{5}+15a^{7}z^{3}-4a^{7}z+3a^{6}z^{8}-8a^{6}z^{6}+12a^{6}z^{4}-5a^{6}z^{2}+a^{6}+a^{5}z^{9}+a^{5}z^{7}-6a^{5}z^{5}+7a^{5}z^{3}-a^{5}z^{-1}+4a^{4}z^{8}-13a^{4}z^{6}+20a^{4}z^{4}-16a^{4}z^{2}+3a^{4}+a^{3}z^{9}-3a^{3}z^{7}+10a^{3}z^{5}-21a^{3}z^{3}+13a^{3}z-3a^{3}z^{-1}+a^{2}z^{8}-2a^{2}z^{6}+2a^{2}z^{4}-6a^{2}z^{2}+3a^{2}+3az^{5}-10az^{3}+9az-2az^{-1}$ (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

1

2

χ

2

0

1

-1

-2

5

2

3

-4

4

3

-1

-6

4

3

1

-8

4

0

-10

3

4

-1

-12

1

4

3

-14

1

3

-2

-16

1

-18

1

-1

Integral Khovanov Homology

(db, data source)

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