L11n24

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_9,21,10,20 X₈₄₉₃ X_21,18,22,19 X_11,14,12,15 X_5,13,6,12 X_13,5,14,22 X_19,11,20,10 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {-8, -1, 2, -5, -4, 10, -7, 8, -9, 7, 11, -2, 3, 6, -10, 4, -6, 9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}+{\frac {1}{q^{9/2}}}-{\frac {3}{q^{7/2}}}-q^{5/2}+{\frac {4}{q^{5/2}}}+3q^{3/2}-{\frac {5}{q^{3/2}}}-q^{11/2}-5{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$az^{5}-a^{3}z^{3}+3az^{3}-2z^{3}a^{-1}+z^{3}a^{-3}-a^{3}z+3az-5za^{-1}+4za^{-3}-za^{-5}+2az^{-1}-4a^{-1}z^{-1}+3a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{7}a^{-5}-6z^{5}a^{-5}+9z^{3}a^{-5}-4za^{-5}+a^{-5}z^{-1}+z^{8}a^{-4}+a^{4}z^{6}-7z^{6}a^{-4}-3a^{4}z^{4}+13z^{4}a^{-4}+a^{4}z^{2}-9z^{2}a^{-4}+a^{-4}+3a^{3}z^{7}+z^{7}a^{-3}-11a^{3}z^{5}-10z^{5}a^{-3}+8a^{3}z^{3}+21z^{3}a^{-3}-3a^{3}z-14za^{-3}+3a^{-3}z^{-1}+3a^{2}z^{8}+2z^{8}a^{-2}-11a^{2}z^{6}-15z^{6}a^{-2}+7a^{2}z^{4}+31z^{4}a^{-2}-a^{2}z^{2}-20z^{2}a^{-2}+a^{2}+3a^{-2}+az^{9}+z^{9}a^{-1}-3z^{7}a^{-1}-15az^{5}-8z^{5}a^{-1}+23az^{3}+27z^{3}a^{-1}-11az-18za^{-1}+2az^{-1}+4a^{-1}z^{-1}+4z^{8}-20z^{6}+28z^{4}-13z^{2}+2$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

12

1

10

0

8

1

-1

6

2

1

4

2

1

-2

2

4

3

1

2

0

3

4

1

-2

2

3

2

1

-4

2

3

1

-6

1

2

-1

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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