L11n139

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_11,19,12,18 X_3,10,4,11 X_17,3,18,2 X_5,13,6,12 X₆₇₁₈ X_16,10,17,9 X_20,16,21,15 X_22,14,7,13 X_14,22,15,21 X_19,4,20,5
Gauss code	{1, 4, -3, 11, -5, -6}, {6, -1, 7, 3, -2, 5, 9, -10, 8, -7, -4, 2, -11, -8, 10, -9}

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} , $v$ , $w$ , ...)	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{u^2 v^2-u^2 v+u v^2-u v+u-v+1}{u v}} (db)
Jones polynomial	$q^{13/2}-q^{11/2}+q^{9/2}-q^{7/2}-{\sqrt {q}}-{\frac {1}{q^{3/2}}}$ (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 z^3 a^{-5} +3 z a^{-5} + a^{-5} z^{-1} -z^5 a^{-3} -5 z^3 a^{-3} -6 z a^{-3} -2 a^{-3} z^{-1} +a z+z a^{-1} +a z^{-1} } (db)
Kauffman polynomial	$z^{8}a^{-6}-7z^{6}a^{-6}+15z^{4}a^{-6}-11z^{2}a^{-6}+2a^{-6}+z^{9}a^{-5}-7z^{7}a^{-5}+15z^{5}a^{-5}-12z^{3}a^{-5}+4za^{-5}-a^{-5}z^{-1}+2z^{8}a^{-4}-14z^{6}a^{-4}+29z^{4}a^{-4}-21z^{2}a^{-4}+5a^{-4}+z^{9}a^{-3}-7z^{7}a^{-3}+15z^{5}a^{-3}-14z^{3}a^{-3}+8za^{-3}-2a^{-3}z^{-1}+z^{8}a^{-2}-7z^{6}a^{-2}+14z^{4}a^{-2}-10z^{2}a^{-2}+3a^{-2}+az^{3}-z^{3}a^{-1}-3az+za^{-1}+az^{-1}-1$ (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

j

, alternation over

r

).

\		r
	\
j		\

-2

-1

0

1

2

3

4

5

6

7

χ

14

1

-1

12

0

10

1

0

8

1

0

6

1

4

1

2

1

0

2

1

0

2

1

-2

1

-4

1

Integral Khovanov Homology

(db, data source)

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