L10a43

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Planar diagram presentation	X₆₁₇₂ X_2,9,3,10 X_12,3,13,4 X_20,12,5,11 X_18,6,19,5 X_10,20,11,19 X_4,17,1,18 X_16,14,17,13 X_14,8,15,7 X_8,16,9,15
Gauss code	{1, -2, 3, -7}, {5, -1, 9, -10, 2, -6, 4, -3, 8, -9, 10, -8, 7, -5, 6, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{5}-4uv^{4}+8uv^{3}-8uv^{2}+4uv+4v^{4}-8v^{3}+8v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-9q^{9/2}+13q^{7/2}-{\frac {1}{q^{7/2}}}-16q^{5/2}+{\frac {3}{q^{5/2}}}+17q^{3/2}-{\frac {8}{q^{3/2}}}-q^{13/2}+5q^{11/2}-16{\sqrt {q}}+{\frac {11}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-5z^{5}a^{-1}+2z^{5}a^{-3}+3az^{3}-12z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+5az-12za^{-1}+5za^{-3}+3az^{-1}-5a^{-1}z^{-1}+2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}+5z^{6}a^{-6}-6z^{4}a^{-6}+a^{-6}+9z^{7}a^{-5}-15z^{5}a^{-5}+5z^{3}a^{-5}+7z^{8}a^{-4}-4z^{6}a^{-4}-9z^{4}a^{-4}+4z^{2}a^{-4}+2z^{9}a^{-3}+15z^{7}a^{-3}+a^{3}z^{5}-34z^{5}a^{-3}-2a^{3}z^{3}+18z^{3}a^{-3}+a^{3}z-5za^{-3}+2a^{-3}z^{-1}+12z^{8}a^{-2}+3a^{2}z^{6}-13z^{6}a^{-2}-4a^{2}z^{4}-10z^{4}a^{-2}+a^{2}z^{2}+17z^{2}a^{-2}-5a^{-2}+2z^{9}a^{-1}+6az^{7}+12z^{7}a^{-1}-12az^{5}-31z^{5}a^{-1}+14az^{3}+29z^{3}a^{-1}-10az-16za^{-1}+3az^{-1}+5a^{-1}z^{-1}+5z^{8}-z^{6}-11z^{4}+14z^{2}-5$ (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

6

χ

14

1

12

4

-4

10

5

1

4

8

4

-4

6

8

5

3

4

9

8

-1

2

7

8

-1

0

5

10

5

-2

3

6

-3

-4

5

-6

1

3

-2

-8

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-4$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$