L11n425

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,8,17,7 X_5,14,6,15 X_3,10,4,11 X_13,4,14,5 X_17,2,18,3 X_9,19,10,18 X_21,7,22,12 X_11,13,12,22 X_20,16,21,15 X_6,19,1,20
Gauss code	{1, 6, -4, 5, -3, -11}, {2, -1, -7, 4, -9, 8}, {-5, 3, 10, -2, -6, 7, 11, -10, -8, 9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(2)^{2}t(3)^{3}+t(1)t(2)t(3)^{3}-t(2)t(3)^{3}+t(1)t(2)^{2}t(3)^{2}-t(2)^{2}t(3)^{2}-2t(1)t(2)t(3)^{2}+t(2)t(3)^{2}+t(1)^{2}t(3)-t(1)t(3)-t(1)^{2}t(2)t(3)+2t(1)t(2)t(3)+t(1)+t(1)^{2}t(2)-t(1)t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$2q^{2}-3q+5-5q^{-1}+6q^{-2}-4q^{-3}+4q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{2}z^{6}+a^{4}z^{4}-5a^{2}z^{4}+z^{4}+3a^{4}z^{2}-9a^{2}z^{2}+2z^{2}+3a^{4}-6a^{2}+a^{-2}+2+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+3a^{4}z^{8}+4a^{2}z^{8}+z^{8}+2a^{5}z^{7}-a^{3}z^{7}-3az^{7}+a^{6}z^{6}-14a^{4}z^{6}-19a^{2}z^{6}-4z^{6}-7a^{5}z^{5}-8a^{3}z^{5}+z^{5}a^{-1}-4a^{6}z^{4}+22a^{4}z^{4}+33a^{2}z^{4}+7z^{4}+4a^{5}z^{3}+14a^{3}z^{3}+10az^{3}+3a^{6}z^{2}-19a^{4}z^{2}-27a^{2}z^{2}+2z^{2}a^{-2}-3z^{2}-9a^{3}z-9az+7a^{4}+11a^{2}-2a^{-2}+3+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

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

χ

5

2

3

2

1

-1

1

3

1

2

-1

4

0

-3

2

1

-5

2

4

2

-7

2

0

-9

2

-11

1

2

-1

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$	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 {\mathbb Z}}
$r=-5$	${\mathbb {Z} }^{2}$
$r=-4$	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}^{2}\oplus{\mathbb Z}_2^{2}}	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
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 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}\oplus{\mathbb Z}_2^{4}}	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$
$r=1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$