L11n249

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Planar diagram presentation	X_12,1,13,2 X_9,19,10,18 X_5,14,6,15 X_11,6,12,7 X_15,11,16,22 X_7,20,8,21 X₃₉₄₈ X_21,17,22,16 X_17,4,18,5 X_10,13,1,14 X_19,3,20,2
Gauss code	{1, 11, -7, 9, -3, 4, -6, 7, -2, -10}, {-4, -1, 10, 3, -5, 8, -9, 2, -11, 6, -8, 5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}v^{3}-3u^{3}v^{2}+2u^{3}v-2u^{2}v^{3}+7u^{2}v^{2}-7u^{2}v+u^{2}+uv^{3}-7uv^{2}+7uv-2u+2v^{2}-3v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {3}{q^{9/2}}}-4q^{7/2}+{\frac {7}{q^{7/2}}}+8q^{5/2}-{\frac {12}{q^{5/2}}}-12q^{3/2}+{\frac {14}{q^{3/2}}}+15{\sqrt {q}}-{\frac {16}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}+z^{3}a^{-3}-4a^{3}z+za^{-3}-a^{3}z^{-1}+az^{7}+4az^{5}-2z^{5}a^{-1}+6az^{3}-5z^{3}a^{-1}+3az-3za^{-1}$ (db)
Kauffman polynomial	$-3az^{9}-3z^{9}a^{-1}-8a^{2}z^{8}-6z^{8}a^{-2}-14z^{8}-8a^{3}z^{7}-8az^{7}-4z^{7}a^{-1}-4z^{7}a^{-3}-3a^{4}z^{6}+13a^{2}z^{6}+14z^{6}a^{-2}-z^{6}a^{-4}+31z^{6}+13a^{3}z^{5}+27az^{5}+24z^{5}a^{-1}+10z^{5}a^{-3}-3a^{4}z^{4}-10a^{2}z^{4}-7z^{4}a^{-2}+2z^{4}a^{-4}-16z^{4}-6a^{5}z^{3}-15a^{3}z^{3}-18az^{3}-16z^{3}a^{-1}-7z^{3}a^{-3}+a^{4}z^{2}+3a^{2}z^{2}-z^{2}a^{-4}+3z^{2}+5a^{5}z+7a^{3}z+4az+3za^{-1}+za^{-3}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

5

1

-4

4

7

3

4

2

8

5

-3

0

8

7

1

-2

7

9

2

-4

5

7

-2

-6

2

7

5

-8

1

5

-4

-10

3

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-4$	${\mathbb {Z} }^{3}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$