L11n159

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_7,17,8,16 X_10,4,11,3 X_2,15,3,16 X_14,10,15,9 X_11,19,12,18 X_5,13,6,12 X_21,1,22,6 X_20,14,21,13 X_17,7,18,22 X_4,20,5,19
Gauss code	{1, -4, 3, -11, -7, 8}, {-2, -1, 5, -3, -6, 7, 9, -5, 4, 2, -10, 6, 11, -9, -8, 10}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{4}-2u^{2}v^{3}+u^{2}v^{2}-uv^{4}+uv^{2}-u+v^{2}-2v+1}{uv^{2}}}$ (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 2 q^{15/2}-2 q^{13/2}+2 q^{11/2}-4 q^{9/2}+2 q^{7/2}-3 q^{5/2}+2 q^{3/2}-\sqrt{q}} (db)
Signature	5 (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^7 a^{-5} +z^5 a^{-3} -5 z^5 a^{-5} +z^5 a^{-7} +4 z^3 a^{-3} -6 z^3 a^{-5} +4 z^3 a^{-7} +3 z a^{-3} -2 z a^{-5} +z a^{-7} -z a^{-9} + a^{-3} z^{-1} -2 a^{-7} z^{-1} + a^{-9} 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 -z^9 a^{-5} -z^9 a^{-7} -2 z^8 a^{-4} -3 z^8 a^{-6} -z^8 a^{-8} -z^7 a^{-3} +3 z^7 a^{-5} +4 z^7 a^{-7} +10 z^6 a^{-4} +15 z^6 a^{-6} +5 z^6 a^{-8} +5 z^5 a^{-3} +4 z^5 a^{-5} -z^5 a^{-7} -12 z^4 a^{-4} -19 z^4 a^{-6} -7 z^4 a^{-8} -7 z^3 a^{-3} -12 z^3 a^{-5} -5 z^3 a^{-7} +2 z^2 a^{-4} +8 z^2 a^{-6} +7 z^2 a^{-8} +z^2 a^{-10} +4 z a^{-3} +6 z a^{-5} +3 z a^{-7} +z a^{-9} + a^{-4} -3 a^{-6} -5 a^{-8} -2 a^{-10} - a^{-3} z^{-1} +2 a^{-7} z^{-1} + a^{-9} z^{-1} } (db)

Khovanov Homology

The coefficients of the monomials 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 t^rq^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		\

-2

-1

0

1

2

3

4

5

χ

16

2

-2

14

1

0

12

2

0

10

2

1

2

8

1

3

2

6

2

1

4

1

2

1

2

1

-1

0

1

Integral Khovanov Homology

(db, data source)

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