L11a202

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X₆₇₁₈ X_18,11,19,12 X_16,6,17,5 X_4,18,5,17 X_20,13,21,14 X_22,15,7,16 X_12,19,13,20 X_14,21,15,22
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -10, 8, -11, 9, -6, 7, -5, 10, -8, 11, -9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{6}-t(1)t(2)^{6}-t(1)^{2}t(2)^{5}+t(1)t(2)^{5}-t(2)^{5}+t(1)^{2}t(2)^{4}-t(1)t(2)^{4}+t(2)^{4}-t(1)^{2}t(2)^{3}+t(1)t(2)^{3}-t(2)^{3}+t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(2)^{2}-t(1)^{2}t(2)+t(1)t(2)-t(2)-t(1)+1}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {5}{q^{15/2}}}+{\frac {5}{q^{17/2}}}-{\frac {4}{q^{19/2}}}+{\frac {3}{q^{21/2}}}-{\frac {2}{q^{23/2}}}+{\frac {1}{q^{25/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-z^{7}a^{9}-6z^{5}a^{9}-11z^{3}a^{9}-7za^{9}-2a^{9}z^{-1}+z^{9}a^{7}+8z^{7}a^{7}+23z^{5}a^{7}+30z^{3}a^{7}+19za^{7}+5a^{7}z^{-1}-z^{7}a^{5}-7z^{5}a^{5}-16z^{3}a^{5}-13za^{5}-3a^{5}z^{-1}$ (db)
Kauffman polynomial	$a^{16}z^{2}+2a^{15}z^{3}+3a^{14}z^{4}-2a^{14}z^{2}+4a^{13}z^{5}-6a^{13}z^{3}+5a^{12}z^{6}-13a^{12}z^{4}+6a^{12}z^{2}-a^{12}+5a^{11}z^{7}-17a^{11}z^{5}+13a^{11}z^{3}-3a^{11}z+4a^{10}z^{8}-15a^{10}z^{6}+11a^{10}z^{4}+a^{10}z^{2}+3a^{9}z^{9}-14a^{9}z^{7}+18a^{9}z^{5}-9a^{9}z^{3}+6a^{9}z-2a^{9}z^{-1}+a^{8}z^{10}-2a^{8}z^{8}-11a^{8}z^{6}+31a^{8}z^{4}-22a^{8}z^{2}+5a^{8}+4a^{7}z^{9}-27a^{7}z^{7}+62a^{7}z^{5}-59a^{7}z^{3}+25a^{7}z-5a^{7}z^{-1}+a^{6}z^{10}-6a^{6}z^{8}+9a^{6}z^{6}+4a^{6}z^{4}-14a^{6}z^{2}+5a^{6}+a^{5}z^{9}-8a^{5}z^{7}+23a^{5}z^{5}-29a^{5}z^{3}+16a^{5}z-3a^{5}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{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

1

2

χ

-2

1

-4

0

-6

3

1

2

-8

1

0

-10

4

2

-12

2

0

-14

3

0

-16

2

0

-18

2

3

-1

-20

1

2

1

-22

1

2

-1

-24

1

-26

1

-1

Integral Khovanov Homology

(db, data source)

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