L11n323

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Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₃₈₄₉ X_15,2,16,3 X_16,7,17,8 X_9,18,10,19 X_11,21,12,20 X_19,22,20,13 X_13,12,14,5 X_4,17,1,18 X_21,11,22,10
Gauss code	{1, 4, -3, -10}, {-2, -1, 5, 3, -6, 11, -7, 9}, {-9, 2, -4, -5, 10, 6, -8, 7, -11, 8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(3)^{3}t(2)^{3}-t(1)t(3)^{2}t(2)^{3}-t(1)t(3)^{3}t(2)^{2}+t(2)+t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-10}+q^{-9}+2q^{-7}-q^{-6}+2q^{-5}-q^{-4}+2q^{-3}-q^{-2}+q^{-1}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{10}\left(-z^{2}\right)-2a^{10}+a^{8}z^{6}+6a^{8}z^{4}+10a^{8}z^{2}+a^{8}z^{-2}+6a^{8}-a^{6}z^{8}-7a^{6}z^{6}-16a^{6}z^{4}-16a^{6}z^{2}-2a^{6}z^{-2}-9a^{6}+a^{4}z^{6}+6a^{4}z^{4}+10a^{4}z^{2}+a^{4}z^{-2}+5a^{4}$ (db)
Kauffman polynomial	$za^{13}+z^{2}a^{12}-a^{12}+za^{11}+z^{4}a^{10}-3z^{2}a^{10}+z^{7}a^{9}-5z^{5}a^{9}+6z^{3}a^{9}-3za^{9}+2z^{8}a^{8}-13z^{6}a^{8}+27z^{4}a^{8}-24z^{2}a^{8}-a^{8}z^{-2}+9a^{8}+z^{9}a^{7}-5z^{7}a^{7}+4z^{5}a^{7}+6z^{3}a^{7}-8za^{7}+2a^{7}z^{-1}+3z^{8}a^{6}-20z^{6}a^{6}+42z^{4}a^{6}-35z^{2}a^{6}-2a^{6}z^{-2}+13a^{6}+z^{9}a^{5}-6z^{7}a^{5}+9z^{5}a^{5}-5za^{5}+2a^{5}z^{-1}+z^{8}a^{4}-7z^{6}a^{4}+16z^{4}a^{4}-15z^{2}a^{4}-a^{4}z^{-2}+6a^{4}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

-1

1

-3

0

-5

2

1

-7

1

-9

2

1

-11

2

1

-13

1

4

2

1

-15

1

2

-17

2

1

-19

1

0

-21

1

-1

Integral Khovanov Homology

(db, data source)

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