L11n167

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,5,15,6 X_11,19,12,18 X_22,19,7,20 X_20,15,21,16 X_16,21,17,22 X_17,13,18,12 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, -5, 9, 11, -4, 7, -8, -9, 5, 6, -7, 8, -6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	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{t(1)^2 t(2)^4-t(1) t(2)^4-3 t(1)^2 t(2)^3+3 t(1) t(2)^3-t(2)^3+2 t(1)^2 t(2)^2-5 t(1) t(2)^2+2 t(2)^2-t(1)^2 t(2)+3 t(1) t(2)-3 t(2)-t(1)+1}{t(1) t(2)^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 -\frac{2}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{9}{q^{13/2}}+\frac{9}{q^{15/2}}-\frac{8}{q^{17/2}}+\frac{5}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{1}{q^{23/2}}} (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 a^9 \left(-z^5\right)-3 a^9 z^3-2 a^9 z+a^7 z^7+5 a^7 z^5+9 a^7 z^3+7 a^7 z+a^7 z^{-1} -2 a^5 z^5-8 a^5 z^3-8 a^5 z-a^5 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^4 a^{14}+z^2 a^{14}-3 z^5 a^{13}+4 z^3 a^{13}-z a^{13}-4 z^6 a^{12}+4 z^4 a^{12}-4 z^7 a^{11}+4 z^5 a^{11}-2 z a^{11}-3 z^8 a^{10}+4 z^6 a^{10}-4 z^4 a^{10}+z^2 a^{10}-z^9 a^9-2 z^7 a^9+5 z^5 a^9-3 z^3 a^9-4 z^8 a^8+10 z^6 a^8-11 z^4 a^8+6 z^2 a^8-z^9 a^7+2 z^7 a^7-5 z^5 a^7+11 z^3 a^7-7 z a^7+a^7 z^{-1} -z^8 a^6+2 z^6 a^6-2 z^4 a^6+4 z^2 a^6-a^6-3 z^5 a^5+10 z^3 a^5-8 z a^5+a^5 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

j

, alternation over

r

).

\		r
	\
j		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

2

-6

2

1

-1

-8

4

1

3

-10

4

2

-2

-12

5

4

1

-14

4

0

-16

4

5

-1

-18

2

5

3

-20

1

3

-2

-22

2

-24

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$