L11n162

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_17,13,18,12 X_14,5,15,6 X_4,13,5,14 X_11,19,12,18 X_19,7,20,22 X_15,21,16,20 X_21,17,22,16 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, -6, 3, 5, -4, -8, 9, -3, 6, -7, 8, -9, 7}

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{t(1) t(2)^4+t(2)^4+t(1)^2 t(2)^3-2 t(1) t(2)^3-t(1)^2 t(2)^2+3 t(1) t(2)^2-t(2)^2-2 t(1) t(2)+t(2)+t(1)^2+t(1)}{t(1) t(2)^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 q^{9/2}-2 q^{7/2}+3 q^{5/2}-\frac{2}{q^{5/2}}-4 q^{3/2}+\frac{3}{q^{3/2}}-\frac{1}{q^{11/2}}+4 \sqrt{q}-\frac{4}{\sqrt{q}}} (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 z a^5+a^5 z^{-1} -z^5 a-4 z^3 a-5 z a-2 a z^{-1} -z^5 a^{-1} -3 z^3 a^{-1} -z a^{-1} + a^{-1} z^{-1} +z^3 a^{-3} +2 z a^{-3} } (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 a^5 z^7-7 a^5 z^5+14 a^5 z^3-9 a^5 z+a^5 z^{-1} +z^6 a^{-4} -2 a^4 z^4-4 z^4 a^{-4} +4 a^4 z^2+3 z^2 a^{-4} -a^4+2 z^7 a^{-3} -2 a^3 z^5-8 z^5 a^{-3} +6 a^3 z^3+7 z^3 a^{-3} -3 a^3 z-2 z a^{-3} +a^2 z^8+2 z^8 a^{-2} -7 a^2 z^6-8 z^6 a^{-2} +15 a^2 z^4+8 z^4 a^{-2} -10 a^2 z^2-5 z^2 a^{-2} +3 a^2+2 a^{-2} +a z^9+z^9 a^{-1} -6 a z^7-3 z^7 a^{-1} +13 a z^5-13 a z^3+2 z^3 a^{-1} +8 a z-2 a z^{-1} - a^{-1} z^{-1} +3 z^8-16 z^6+29 z^4-22 z^2+5} (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 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} ).

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

1

6

2

1

-1

4

2

1

2

0

1

3

2

0

-2

2

3

1

-4

1

2

-1

-6

2

-8

1

0

-10

1

-12

1

Integral Khovanov Homology

(db, data source)

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