L11n418

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Planar diagram presentation	X₈₁₉₂ X_7,16,8,17 X_3,10,4,11 X_2,18,3,17 X_18,9,19,10 X_11,20,12,21 X_5,14,6,15 X_15,13,16,22 X_13,6,14,1 X_19,5,20,4 X_21,12,22,7
Gauss code	{1, -4, -3, 10, -7, 9}, {-2, -1, 5, 3, -6, 11}, {-9, 7, -8, 2, 4, -5, -10, 6, -11, 8}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}vw^{3}-2u^{2}vw^{2}+u^{2}vw+u^{2}w^{2}-u^{2}w+uv^{2}w-uvw^{3}+2uvw^{2}-2uvw+uv-uw^{2}+v^{2}w^{2}-v^{2}w-vw^{2}+2vw-v}{uvw^{3/2}}}$ (db)
Jones polynomial	$-1+3q^{-1}-4q^{-2}+7q^{-3}-6q^{-4}+7q^{-5}-5q^{-6}+4q^{-7}-2q^{-8}+q^{-9}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+a^{8}z^{-2}+2a^{8}-2a^{6}z^{4}-7a^{6}z^{2}-2a^{6}z^{-2}-8a^{6}+a^{4}z^{6}+5a^{4}z^{4}+10a^{4}z^{2}+a^{4}z^{-2}+7a^{4}-a^{2}z^{4}-2a^{2}z^{2}-a^{2}$ (db)
Kauffman polynomial	$a^{10}z^{6}-4a^{10}z^{4}+4a^{10}z^{2}-a^{10}+2a^{9}z^{7}-7a^{9}z^{5}+5a^{9}z^{3}+2a^{8}z^{8}-6a^{8}z^{6}+4a^{8}z^{4}-4a^{8}z^{2}-a^{8}z^{-2}+4a^{8}+a^{7}z^{9}-a^{7}z^{7}-4a^{7}z^{5}+6a^{7}z^{3}-6a^{7}z+2a^{7}z^{-1}+4a^{6}z^{8}-16a^{6}z^{6}+28a^{6}z^{4}-27a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+a^{5}z^{9}-2a^{5}z^{7}+a^{5}z^{5}+6a^{5}z^{3}-8a^{5}z+2a^{5}z^{-1}+2a^{4}z^{8}-9a^{4}z^{6}+23a^{4}z^{4}-23a^{4}z^{2}-a^{4}z^{-2}+10a^{4}+a^{3}z^{7}-2a^{3}z^{5}+6a^{3}z^{3}-3a^{3}z+3a^{2}z^{4}-4a^{2}z^{2}+2a^{2}+az^{3}-az$ (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

1

-1

2

-3

3

2

-1

-5

4

1

3

-7

3

4

1

-9

4

3

1

-11

2

4

2

-13

2

3

-1

-15

1

3

2

-17

1

-1

-19

1

Integral Khovanov Homology

(db, data source)

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