L10n79

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₈₄₉₃ X_2,16,3,15 X_16,7,17,8 X_9,18,10,19 X_11,20,12,13 X_13,12,14,5 X_4,17,1,18 X_19,10,20,11
Gauss code	{1, -4, 3, -9}, {-2, -1, 5, -3, -6, 10, -7, 8}, {-8, 2, 4, -5, 9, 6, -10, 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} , $w$ , ...)	$-{\frac {(u-1)(v-1)(w-1)(vw+1)}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$2q^{-2}-2q^{-3}+5q^{-4}-5q^{-5}+6q^{-6}-5q^{-7}+4q^{-8}-2q^{-9}+q^{-10}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$z^{4}a^{8}+3z^{2}a^{8}+a^{8}z^{-2}+3a^{8}-z^{6}a^{6}-5z^{4}a^{6}-10z^{2}a^{6}-2a^{6}z^{-2}-9a^{6}+2z^{4}a^{4}+7z^{2}a^{4}+a^{4}z^{-2}+6a^{4}$ (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^{12} z^4-2 a^{12} z^2+a^{12}+2 a^{11} z^5-3 a^{11} z^3+2 a^{10} z^6-a^{10} z^4-2 a^{10} z^2+2 a^9 z^7-3 a^9 z^5+3 a^9 z^3+a^8 z^8-a^8 z^6+3 a^8 z^4-3 a^8 z^2-a^8 z^{-2} +3 a^8+3 a^7 z^7-8 a^7 z^5+13 a^7 z^3-9 a^7 z+2 a^7 z^{-1} +a^6 z^8-3 a^6 z^6+8 a^6 z^4-12 a^6 z^2-2 a^6 z^{-2} +9 a^6+a^5 z^7-3 a^5 z^5+7 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +3 a^4 z^4-9 a^4 z^2-a^4 z^{-2} +6 a^4} (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

2

-5

2

0

-7

3

-9

2

0

-11

4

3

1

-13

1

2

1

-15

3

4

-1

-17

1

3

2

-19

1

-1

-21

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=-5}	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=-3}
$r=-8$	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}}
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=-7}	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	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}}
$r=-6$	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}^{3}\oplus{\mathbb Z}_2}	${\mathbb {Z} }^{3}$
$r=-5$	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}^{4}\oplus{\mathbb Z}_2}	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	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}^{4}}
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	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}}	${\mathbb {Z} }^{2}$