L11n381

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,10,6,11 X₈₄₉₃ X_22,18,19,17 X_11,20,12,21 X_19,12,20,13 X_18,22,5,21 X_9,16,10,17 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {-8, 7, 9, -6}, {-4, -1, 2, -5, -10, 4, -7, 8, 11, -2, 3, 10, 6, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	0 (db)
Jones polynomial	$-q^{-7}+q^{-6}-q^{-5}+q^{-3}+q^{-2}+q+2q^{-1}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-a^{6}z^{-2}-2a^{6}+a^{4}z^{4}+5a^{4}z^{2}+4a^{4}z^{-2}+7a^{4}-a^{2}z^{4}-5a^{2}z^{2}-5a^{2}z^{-2}-8a^{2}+z^{2}+2z^{-2}+3$ (db)
Kauffman polynomial	$a^{6}z^{8}+a^{4}z^{8}+a^{7}z^{7}+2a^{5}z^{7}+a^{3}z^{7}-6a^{6}z^{6}-7a^{4}z^{6}-a^{2}z^{6}-6a^{7}z^{5}-14a^{5}z^{5}-8a^{3}z^{5}+9a^{6}z^{4}+15a^{4}z^{4}+7a^{2}z^{4}+z^{4}+10a^{7}z^{3}+28a^{5}z^{3}+20a^{3}z^{3}+2az^{3}-4a^{6}z^{2}-15a^{4}z^{2}-15a^{2}z^{2}-4z^{2}-6a^{7}z-19a^{5}z-21a^{3}z-8az+3a^{6}+10a^{4}+11a^{2}+5+a^{7}z^{-1}+5a^{5}z^{-1}+9a^{3}z^{-1}+5az^{-1}-a^{6}z^{-2}-4a^{4}z^{-2}-5a^{2}z^{-2}-2z^{-2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

1

-1

1

4

1

2

-3

1

3

-5

1

3

2

-7

1

-9

1

2

-1

-11

1

0

-13

0

-15

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$	$i=1$
$r=-7$		${\mathbb {Z} }$
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-6}		${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$		${\mathbb {Z} }$
$r=-4$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }^{3}$	${\mathbb {Z} }^{4}$	${\mathbb {Z} }$
$r=1$	${\mathbb {Z} }$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$