L11n211

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Planar diagram presentation	X_10,1,11,2 X_15,21,16,20 X_5,14,6,15 X_3,12,4,13 X_13,4,14,5 X_2,19,3,20 X_16,7,17,8 X_8,9,1,10 X_18,12,19,11 X_22,18,9,17 X_21,6,22,7
Gauss code	{1, -6, -4, 5, -3, 11, 7, -8}, {8, -1, 9, 4, -5, 3, -2, -7, 10, -9, 6, 2, -11, -10}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{3}v^{2}+u^{2}v^{5}-2u^{2}v^{4}+3u^{2}v^{3}-3u^{2}v^{2}+2u^{2}v+2uv^{4}-3uv^{3}+3uv^{2}-2uv+u+v^{3}}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	${\frac {8}{q^{9/2}}}-{\frac {8}{q^{7/2}}}+{\frac {5}{q^{5/2}}}-{\frac {3}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {2}{q^{17/2}}}-{\frac {5}{q^{15/2}}}+{\frac {7}{q^{13/2}}}-{\frac {8}{q^{11/2}}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+4z^{3}a^{7}+6za^{7}+2a^{7}z^{-1}-z^{7}a^{5}-6z^{5}a^{5}-14z^{3}a^{5}-13za^{5}-3a^{5}z^{-1}+z^{5}a^{3}+3z^{3}a^{3}+3za^{3}+a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-2z^{6}a^{10}+4z^{4}a^{10}-z^{2}a^{10}-3z^{7}a^{9}+7z^{5}a^{9}-6z^{3}a^{9}+3za^{9}-2z^{8}a^{8}+2z^{6}a^{8}+z^{2}a^{8}-z^{9}a^{7}+z^{7}a^{7}-6z^{5}a^{7}+11z^{3}a^{7}-8za^{7}+2a^{7}z^{-1}-3z^{8}a^{6}+7z^{6}a^{6}-14z^{4}a^{6}+12z^{2}a^{6}-3a^{6}-z^{9}a^{5}+4z^{7}a^{5}-17z^{5}a^{5}+25z^{3}a^{5}-16za^{5}+3a^{5}z^{-1}-z^{8}a^{4}+3z^{6}a^{4}-11z^{4}a^{4}+12z^{2}a^{4}-3a^{4}-3z^{5}a^{3}+5z^{3}a^{3}-3za^{3}+a^{3}z^{-1}-z^{4}a^{2}+2z^{2}a^{2}-a^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

χ

0

1

-1

-2

2

-4

4

2

-2

-6

4

1

3

-8

4

0

-10

4

0

-12

3

4

1

-14

2

4

-2

-16

3

-18

1

2

-1

-20

1

Integral Khovanov Homology

(db, data source)

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