L11n321

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Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₃₈₄₉ X_2,16,3,15 X_16,7,17,8 X_9,18,10,19 X_17,1,18,4 X_19,13,20,22 X_13,10,14,11 X_21,5,22,12 X_11,21,12,20
Gauss code	{1, -4, -3, 7}, {-2, -1, 5, 3, -6, 9, -11, 10}, {-9, 2, 4, -5, -7, 6, -8, 11, -10, 8}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-uv^{2}w^{3}+2uv^{2}w^{2}-uv^{2}w+uvw^{3}-4uvw^{2}+2uvw+uw^{2}-uw+v^{3}w^{2}-v^{3}w-2v^{2}w^{2}+4v^{2}w-v^{2}+vw^{2}-2vw+v}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-2q^{3}+5q^{2}-6q+9-8q^{-1}+9q^{-2}-6q^{-3}+4q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{2}z^{6}+a^{4}z^{4}-5a^{2}z^{4}+3z^{4}+3a^{4}z^{2}-11a^{2}z^{2}-2z^{2}a^{-2}+9z^{2}+3a^{4}-9a^{2}-2a^{-2}+8+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-4a^{6}z^{4}+4a^{6}z^{2}-a^{6}+2a^{5}z^{7}-7a^{5}z^{5}+6a^{5}z^{3}-a^{5}z+2a^{4}z^{8}-5a^{4}z^{6}+2a^{4}z^{4}-2a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+a^{3}z^{9}+a^{3}z^{7}-10a^{3}z^{5}+13a^{3}z^{3}+3z^{3}a^{-3}-8a^{3}z-2za^{-3}+2a^{3}z^{-1}+5a^{2}z^{8}-16a^{2}z^{6}+z^{6}a^{-2}+24a^{2}z^{4}+4z^{4}a^{-2}-23a^{2}z^{2}-7z^{2}a^{-2}-2a^{2}z^{-2}+11a^{2}+3a^{-2}+az^{9}+2az^{7}+3z^{7}a^{-1}-9az^{5}-6z^{5}a^{-1}+15az^{3}+11z^{3}a^{-1}-12az-7za^{-1}+2az^{-1}+3z^{8}-9z^{6}+22z^{4}-24z^{2}-z^{-2}+11$ (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

7

2

-2

5

3

2

-1

1

6

3

-1

5

6

1

-3

4

3

1

-5

2

5

3

-7

2

4

-2

-9

1

3

2

-11

1

-1

-13

1

Integral Khovanov Homology

(db, data source)

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