L10n112

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_11,19,12,18 X_3,11,4,10 X_9,1,10,4 X_7,15,8,14 X_13,5,14,8 X_15,17,16,20 X_19,13,20,16 X_17,9,18,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 10}, {-7, 6, -8, 9}, {-10, 3, -9, 8}

A Braid Representative	{{{braid_table}}}
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} , $v$ , $w$ , ...)	${\frac {-t(1)t(2)+t(1)t(3)t(2)-t(3)t(2)+t(1)t(4)t(2)-t(1)t(3)t(4)t(2)+t(3)t(4)t(2)+2t(1)t(5)t(2)-t(1)t(3)t(5)t(2)+t(3)t(5)t(2)-t(1)t(4)t(5)t(2)-t(5)t(2)+t(3)-t(1)t(4)+t(1)t(3)t(4)-2t(3)t(4)+t(4)-t(1)t(5)-t(3)t(5)+t(1)t(4)t(5)+t(3)t(4)t(5)-t(4)t(5)+t(5)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}{\sqrt {t(4)}}{\sqrt {t(5)}}}}$ (db)
Jones polynomial	$q^{9}-q^{8}+6q^{7}-5q^{6}+11q^{5}-5q^{4}+10q^{3}-5q^{2}+4q$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$a^{-10}z^{-4}+a^{-10}z^{-2}-4a^{-8}z^{-4}-7a^{-8}z^{-2}-4a^{-8}+6a^{-6}z^{-4}+6z^{2}a^{-6}+15a^{-6}z^{-2}+14a^{-6}-3z^{4}a^{-4}-4a^{-4}z^{-4}-10z^{2}a^{-4}-13a^{-4}z^{-2}-16a^{-4}+a^{-2}z^{-4}+4z^{2}a^{-2}+4a^{-2}z^{-2}+6a^{-2}$ (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 z^8 a^{-6} +z^8 a^{-8} +5 z^7 a^{-5} +6 z^7 a^{-7} +z^7 a^{-9} +10 z^6 a^{-4} +12 z^6 a^{-6} +3 z^6 a^{-8} +z^6 a^{-10} +6 z^5 a^{-3} -6 z^5 a^{-7} -25 z^4 a^{-4} -39 z^4 a^{-6} -19 z^4 a^{-8} -5 z^4 a^{-10} -10 z^3 a^{-3} -30 z^3 a^{-5} -30 z^3 a^{-7} -10 z^3 a^{-9} +10 z^2 a^{-2} +30 z^2 a^{-4} +40 z^2 a^{-6} +30 z^2 a^{-8} +10 z^2 a^{-10} +20 z a^{-3} +55 z a^{-5} +55 z a^{-7} +20 z a^{-9} -10 a^{-2} -25 a^{-4} -31 a^{-6} -25 a^{-8} -10 a^{-10} -15 a^{-3} z^{-1} -41 a^{-5} z^{-1} -41 a^{-7} z^{-1} -15 a^{-9} z^{-1} +5 a^{-2} z^{-2} +14 a^{-4} z^{-2} +18 a^{-6} z^{-2} +14 a^{-8} z^{-2} +5 a^{-10} z^{-2} +4 a^{-3} z^{-3} +12 a^{-5} z^{-3} +12 a^{-7} z^{-3} +4 a^{-9} z^{-3} - a^{-2} z^{-4} -4 a^{-4} z^{-4} -6 a^{-6} z^{-4} -4 a^{-8} z^{-4} - a^{-10} z^{-4} } (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\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		\

0

1

2

3

4

5

6

7

8

χ

19

1

17

1

0

15

5

13

1

11

5

6

9

4

10

6

7

6

1

5

1

4

5

3

5

6

-1

1

4

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$	$i=5$
$r=0$	${\mathbb {Z} }^{4}$	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}^{5}}	${\mathbb {Z} }$
$r=1$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{11}$
$r=5$	${\mathbb {Z} }^{5}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=8$	${\mathbb {Z} }$	${\mathbb {Z} }$