L11n387

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,12,6,13 X₈₄₉₃ X_11,19,12,22 X_21,18,22,5 X_9,20,10,21 X_17,11,18,10 X_19,17,20,16 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {-10, 8, -7, 6}, {-4, -1, 2, -5, -8, 9, -6, 4, 11, -2, 3, 10, -9, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)^{3}}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{3}+4q^{2}-6q+11-10q^{-1}+11q^{-2}-9q^{-3}+7q^{-4}-4q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{2}z^{6}+a^{4}z^{4}-3a^{2}z^{4}+2z^{4}+a^{4}z^{2}-3a^{2}z^{2}-z^{2}a^{-2}+3z^{2}+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$2a^{3}z^{9}+2az^{9}+5a^{4}z^{8}+9a^{2}z^{8}+4z^{8}+4a^{5}z^{7}+3a^{3}z^{7}+az^{7}+2z^{7}a^{-1}+a^{6}z^{6}-13a^{4}z^{6}-27a^{2}z^{6}-13z^{6}-11a^{5}z^{5}-23a^{3}z^{5}-16az^{5}-4z^{5}a^{-1}-2a^{6}z^{4}+5a^{4}z^{4}+25a^{2}z^{4}+4z^{4}a^{-2}+22z^{4}+6a^{5}z^{3}+20a^{3}z^{3}+23az^{3}+10z^{3}a^{-1}+z^{3}a^{-3}+a^{6}z^{2}-a^{4}z^{2}-9a^{2}z^{2}-4z^{2}a^{-2}-11z^{2}-a^{5}z-4a^{3}z-6az-4za^{-1}-za^{-3}+1-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

7

1

-1

5

3

1

-2

1

8

3

5

-1

6

7

1

-3

5

1

-5

4

6

2

-7

3

5

-2

-9

1

4

3

-11

3

-3

-13

1

Integral Khovanov Homology

(db, data source)

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