L11n426

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,8,17,7 X_14,6,15,5 X_3,10,4,11 X_4,14,5,13 X_17,2,18,3 X_9,19,10,18 X_12,21,7,22 X_22,11,13,12 X_20,16,21,15 X_6,19,1,20
Gauss code	{1, 6, -4, -5, 3, -11}, {2, -1, -7, 4, 9, -8}, {5, -3, 10, -2, -6, 7, 11, -10, 8, -9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , 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 w} , ...)	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 \frac{(t(1)-t(2))^2 (t(3)-1)}{t(1) t(2) \sqrt{t(3)}}} (db)
Jones polynomial	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 q^{-5} - q^{-4} -q^3+ q^{-3} +q^2+ q^{-1} +2} (db)
Signature	1 (db)
HOMFLY-PT polynomial	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 a^4 z^2+a^4 z^{-2} +2 a^4-a^2 z^4-5 a^2 z^2-2 a^2 z^{-2} -z^2 a^{-2} -7 a^2-2 a^{-2} +z^4+5 z^2+ z^{-2} +7} (db)
Kauffman polynomial	$a^{4}z^{8}-7a^{4}z^{6}+15a^{4}z^{4}-12a^{4}z^{2}-a^{4}z^{-2}+5a^{4}+a^{3}z^{9}-7a^{3}z^{7}+14a^{3}z^{5}-7a^{3}z^{3}+z^{3}a^{-3}-3a^{3}z-2za^{-3}+2a^{3}z^{-1}+2a^{2}z^{8}-15a^{2}z^{6}+35a^{2}z^{4}+z^{4}a^{-2}-32a^{2}z^{2}-4z^{2}a^{-2}-2a^{2}z^{-2}+13a^{2}+4a^{-2}+az^{9}-7az^{7}+13az^{5}-z^{5}a^{-1}-3az^{3}+5z^{3}a^{-1}-7az-6za^{-1}+2az^{-1}+z^{8}-8z^{6}+21z^{4}-24z^{2}-z^{-2}+13$ (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

0

3

1

2

-1

1

3

-3

1

2

1

-5

1

2

1

-7

1

0

-9

0

-11

1

Integral Khovanov Homology

(db, data source)

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