L11n420

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_7,16,8,17 X_3,10,4,11 X_17,2,18,3 X_18,9,19,10 X_11,20,12,21 X_14,6,15,5 X_22,15,13,16 X_6,14,1,13 X_4,19,5,20 X_21,12,22,7
Gauss code	{1, 4, -3, -10, 7, -9}, {-2, -1, 5, 3, -6, 11}, {9, -7, 8, 2, -4, -5, 10, 6, -11, -8}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{2}w^{3}-u^{2}v^{2}w^{2}-uv^{2}w^{3}+uvw^{2}-uvw+u+w-1}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{-7}+2q^{-5}-q^{-4}+2q^{-3}-q^{-2}+q^{-1}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{10}\left(-z^{2}\right)-a^{10}+a^{8}z^{6}+6a^{8}z^{4}+9a^{8}z^{2}+a^{8}z^{-2}+4a^{8}-a^{6}z^{8}-7a^{6}z^{6}-16a^{6}z^{4}-16a^{6}z^{2}-2a^{6}z^{-2}-8a^{6}+a^{4}z^{6}+6a^{4}z^{4}+10a^{4}z^{2}+a^{4}z^{-2}+5a^{4}$ (db)
Kauffman polynomial	$a^{11}(-z)-2a^{10}z^{2}+a^{10}+a^{9}z^{7}-6a^{9}z^{5}+8a^{9}z^{3}-3a^{9}z+2a^{8}z^{8}-13a^{8}z^{6}+25a^{8}z^{4}-20a^{8}z^{2}-a^{8}z^{-2}+8a^{8}+a^{7}z^{9}-5a^{7}z^{7}+3a^{7}z^{5}+8a^{7}z^{3}-8a^{7}z+2a^{7}z^{-1}+3a^{6}z^{8}-20a^{6}z^{6}+41a^{6}z^{4}-33a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+a^{5}z^{9}-6a^{5}z^{7}+9a^{5}z^{5}-6a^{5}z+2a^{5}z^{-1}+a^{4}z^{8}-7a^{4}z^{6}+16a^{4}z^{4}-15a^{4}z^{2}-a^{4}z^{-2}+6a^{4}$ (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		\

-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

2

-13

2

1

-15

1

-17

1

0

Integral Khovanov Homology

(db, data source)

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