L11n48

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_19,1,20,4 X_5,14,6,15 X₃₈₄₉ X_9,16,10,17 X_15,10,16,11 X_11,20,12,21 X_13,22,14,5 X_21,12,22,13 X_2,18,3,17
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -6, 7, -8, 10, -9, 4, -7, 6, 11, -2, -3, 8, -10, 9}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (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{1}{q^{3/2}}-\frac{2}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{21/2}}} (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)-3a^{9}z-2a^{9}z^{-1}+a^{7}z^{5}+5a^{7}z^{3}+8a^{7}z+4a^{7}z^{-1}-a^{5}z^{3}-2a^{5}z-a^{5}z^{-1}-a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}$ (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^{12} z^6-5 a^{12} z^4+6 a^{12} z^2-2 a^{12}+a^{11} z^7-4 a^{11} z^5+2 a^{11} z^3+a^{11} z+a^{10} z^8-4 a^{10} z^6+2 a^{10} z^4+2 a^{10} z^2-a^{10}+a^9 z^9-6 a^9 z^7+13 a^9 z^5-15 a^9 z^3+8 a^9 z-2 a^9 z^{-1} +2 a^8 z^8-12 a^8 z^6+24 a^8 z^4-20 a^8 z^2+6 a^8+a^7 z^9-7 a^7 z^7+18 a^7 z^5-22 a^7 z^3+15 a^7 z-4 a^7 z^{-1} +a^6 z^8-7 a^6 z^6+17 a^6 z^4-15 a^6 z^2+5 a^6+a^5 z^5-4 a^5 z^3+5 a^5 z-a^5 z^{-1} +a^4 z^2-a^4+a^3 z^3-3 a^3 z+a^3 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

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

1

-4

1

2

1

-6

1

2

-8

2

0

-10

2

1

-12

1

3

1

-14

2

0

-16

1

0

-18

1

2

-1

-20

0

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	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 i=-4}	$i=-2$
$r=-9$		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}}
$r=-8$		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}_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}}
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}		${\mathbb {Z} }^{2}$
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=-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}\oplus{\mathbb Z}_2^{2}}	${\mathbb {Z} }^{2}$
$r=-5$	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}}	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
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=-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}	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	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}}	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$		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}_2}	${\mathbb {Z} }$
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=0}	${\mathbb {Z} }$	${\mathbb {Z} }^{2}$	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}}