L11n46

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_5,14,6,15 X₈₄₉₃ X_9,18,10,19 X_11,20,12,21 X_13,22,14,5 X_19,10,20,11 X_21,12,22,13 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 9, -7, 10, -8, 4, 11, -2, 3, 6, -9, 7, -10, 8}

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)-1) (t(2)-1) \left(t(2)^4-t(2)^3+t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}} (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{2}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{7}{q^{15/2}}-\frac{5}{q^{17/2}}+\frac{4}{q^{19/2}}-\frac{2}{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)-4 a^9 z^3-5 a^9 z-3 a^9 z^{-1} +a^7 z^7+6 a^7 z^5+14 a^7 z^3+16 a^7 z+7 a^7 z^{-1} -2 a^5 z^5-9 a^5 z^3-11 a^5 z-4 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 a^{14} z^4-2 a^{14} z^2+a^{14}+2 a^{13} z^5-3 a^{13} z^3+2 a^{12} z^6-a^{12} z^4-2 a^{12} z^2+2 a^{11} z^7-2 a^{11} z^5+a^{11} z^3+2 a^{10} z^8-5 a^{10} z^6+7 a^{10} z^4-a^{10} z^2+a^9 z^9-2 a^9 z^7+3 a^9 z^5-3 a^9 z^3+6 a^9 z-3 a^9 z^{-1} +3 a^8 z^8-11 a^8 z^6+18 a^8 z^4-16 a^8 z^2+7 a^8+a^7 z^9-4 a^7 z^7+10 a^7 z^5-19 a^7 z^3+18 a^7 z-7 a^7 z^{-1} +a^6 z^8-4 a^6 z^6+9 a^6 z^4-15 a^6 z^2+7 a^6+3 a^5 z^5-12 a^5 z^3+12 a^5 z-4 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 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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

2

-6

2

0

-8

3

-10

2

0

-12

5

3

2

-14

2

0

-16

3

5

-2

-18

1

2

1

-20

1

3

-2

-22

1

-24

1

-1

Integral Khovanov Homology

(db, data source)

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