L11n131

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Planar diagram presentation	X₈₁₉₂ X_11,19,12,18 X_3,10,4,11 X_17,3,18,2 X_5,13,6,12 X₆₇₁₈ X_16,10,17,9 X_20,14,21,13 X_22,16,7,15 X_19,4,20,5 X_14,22,15,21
Gauss code	{1, 4, -3, 10, -5, -6}, {6, -1, 7, 3, -2, 5, 8, -11, 9, -7, -4, 2, -10, -8, 11, -9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}-2t(1)^{2}t(2)^{3}+2t(1)t(2)^{3}-t(2)^{3}+2t(1)^{2}t(2)^{2}-3t(1)t(2)^{2}+2t(2)^{2}-t(1)^{2}t(2)+2t(1)t(2)-2t(2)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{15/2}-2q^{13/2}+4q^{11/2}-5q^{9/2}+6q^{7/2}-7q^{5/2}+5q^{3/2}-5{\sqrt {q}}+{\frac {2}{\sqrt {q}}}-{\frac {1}{q^{3/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-3}+z^{5}a^{-1}-6z^{5}a^{-3}+z^{5}a^{-5}+4z^{3}a^{-1}-13z^{3}a^{-3}+4z^{3}a^{-5}+6za^{-1}-12za^{-3}+5za^{-5}+3a^{-1}z^{-1}-5a^{-3}z^{-1}+2a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-4z^{4}a^{-8}+4z^{2}a^{-8}-a^{-8}+2z^{7}a^{-7}-7z^{5}a^{-7}+5z^{3}a^{-7}+2z^{8}a^{-6}-6z^{6}a^{-6}+3z^{4}a^{-6}+z^{9}a^{-5}-2z^{7}a^{-5}+2z^{5}a^{-5}-7z^{3}a^{-5}+6za^{-5}-2a^{-5}z^{-1}+3z^{8}a^{-4}-10z^{6}a^{-4}+15z^{4}a^{-4}-14z^{2}a^{-4}+5a^{-4}+z^{9}a^{-3}-4z^{7}a^{-3}+13z^{5}a^{-3}-20z^{3}a^{-3}+15za^{-3}-5a^{-3}z^{-1}+z^{8}a^{-2}-3z^{6}a^{-2}+10z^{4}a^{-2}-11z^{2}a^{-2}+5a^{-2}+4z^{5}a^{-1}+az^{3}-7z^{3}a^{-1}-az+8za^{-1}-3a^{-1}z^{-1}+2z^{4}-z^{2}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

1

12

3

1

-2

10

2

1

8

4

3

-1

6

3

2

1

4

2

4

2

3

0

3

-2

1

2

-1

-4

1

Integral Khovanov Homology

(db, data source)

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