L11n347

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Planar diagram presentation	X₆₁₇₂ X_3,15,4,14 X_11,20,12,21 X_7,18,8,19 X_17,22,18,13 X_16,9,17,10 X_10,15,11,16 X_19,12,20,5 X_21,8,22,9 X₂₅₃₆ X_13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -4, 9, 6, -7, -3, 8}, {-11, 2, 7, -6, -5, 4, -8, 3, -9, 5}

A Braid Representative

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Polynomial invariants

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

2

χ

1

-1

1

-1

-3

3

1

2

-5

2

3

1

-7

1

3

1

-9

1

2

1

-11

1

4

3

0

-13

2

-15

1

2

-1

-17

1

-19

1

Integral Khovanov Homology

(db, data source)

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