L11n83

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_5,14,6,15 X_10,4,11,3 X_11,16,12,17 X_15,12,16,13 X_13,22,14,5 X_18,9,19,10 X_2,18,3,17 X_8,19,9,20
Gauss code	{1, -10, 5, -3}, {-4, -1, 2, -11, 9, -5, -6, 7, -8, 4, -7, 6, 10, -9, 11, -2, 3, 8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in 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 u} , 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 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{2 (t(1)-1) (t(2)-1) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}}} (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 -\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{8}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{2}{q^{17/2}}} (db)
Signature	-3 (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^9 z^{-1} -a^7 z^3+2 a^7 z^{-1} +a^5 z^5+2 a^5 z^3+a^5 z-a^5 z^{-1} +a^3 z^5+2 a^3 z^3+a^3 z+a^3 z^{-1} -a z^3-2 a z-a z^{-1} } (db)
Kauffman 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 3 a^{10} z^4-8 a^{10} z^2+3 a^{10}+a^9 z^7-a^9 z^5-a^9 z^3+a^9 z-a^9 z^{-1} +2 a^8 z^8-8 a^8 z^6+19 a^8 z^4-20 a^8 z^2+7 a^8+a^7 z^9-a^7 z^7-a^7 z^5+5 a^7 z^3+2 a^7 z-2 a^7 z^{-1} +4 a^6 z^8-13 a^6 z^6+22 a^6 z^4-14 a^6 z^2+4 a^6+a^5 z^9-2 a^5 z^5+3 a^5 z^3+a^5 z-a^5 z^{-1} +2 a^4 z^8-3 a^4 z^6+2 a^4 z^4-2 a^4 z^2+2 a^3 z^7-a^3 z^5-6 a^3 z^3+3 a^3 z-a^3 z^{-1} +2 a^2 z^6-4 a^2 z^4+a^2+a z^5-3 a z^3+3 a z-a z^{-1} } (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums 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 \chi} (fixed 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 j} , alternation over

r

).

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

1

-1

-2

4

1

3

-4

4

3

-1

-6

4

2

-8

4

0

-10

4

0

-12

1

4

3

-14

2

4

-2

-16

1

-18

2

-2

Integral Khovanov Homology

(db, data source)

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