L11n344

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Planar diagram presentation	X₆₁₇₂ X_2,16,3,15 X_3,10,4,11 X_5,14,6,15 X_11,22,12,13 X_13,12,14,5 X_21,1,22,4 X_20,17,21,18 X_16,7,17,8 X_8,20,9,19 X_18,10,19,9
Gauss code	{1, -2, -3, 7}, {-4, -1, 9, -10, 11, 3, -5, 6}, {-6, 4, 2, -9, 8, -11, 10, -8, -7, 5}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{3}-3uv^{2}w^{2}+2uv^{2}w-uvw^{3}+3uvw^{2}-3uvw-uw^{2}+uw+v^{3}\left(-w^{2}\right)+v^{3}w+3v^{2}w^{2}-3v^{2}w+v^{2}-2vw^{2}+3vw-v}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$2q-3+7q^{-1}-9q^{-2}+11q^{-3}-9q^{-4}+9q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{4}\right)-2a^{6}z^{2}-2a^{6}+a^{4}z^{6}+4a^{4}z^{4}+8a^{4}z^{2}+a^{4}z^{-2}+7a^{4}-3a^{2}z^{4}-9a^{2}z^{2}-2a^{2}z^{-2}-9a^{2}+2z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-6z^{4}a^{8}+3z^{2}a^{8}-a^{8}+4z^{7}a^{7}-6z^{5}a^{7}-z^{3}a^{7}+za^{7}+3z^{8}a^{6}-2z^{6}a^{6}-4z^{4}a^{6}+2z^{2}a^{6}+z^{9}a^{5}+4z^{7}a^{5}-10z^{5}a^{5}+9z^{3}a^{5}-3za^{5}+5z^{8}a^{4}-12z^{6}a^{4}+21z^{4}a^{4}-21z^{2}a^{4}-a^{4}z^{-2}+9a^{4}+z^{9}a^{3}+z^{7}a^{3}-4z^{5}a^{3}+11z^{3}a^{3}-8za^{3}+2a^{3}z^{-1}+2z^{8}a^{2}-7z^{6}a^{2}+22z^{4}a^{2}-28z^{2}a^{2}-2a^{2}z^{-2}+13a^{2}+z^{7}a-z^{5}a+3z^{3}a-5za+2az^{-1}+3z^{4}-8z^{2}-z^{-2}+6$ (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

3

2

1

-1

6

2

4

-3

6

4

-2

-5

5

3

2

-7

4

6

2

-9

5

0

-11

2

5

3

-13

1

4

-3

-15

2

-17

1

-1

Integral Khovanov Homology

(db, data source)

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