L10n22

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_9,16,10,17 X_20,17,5,18 X_18,13,19,14 X_14,19,15,20 X_15,8,16,9 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 3, 8, -4, -2, 10, -3, 6, -7, -8, 4, 5, -6, 7, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)^{3}}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	${\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {4}{q^{3/2}}}-{\frac {6}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {2}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+2a^{5}z^{3}+4a^{5}z+2a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-3a^{3}z+az^{3}-az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-4a^{8}z^{4}+5a^{8}z^{2}-2a^{8}+2a^{7}z^{7}-7a^{7}z^{5}+6a^{7}z^{3}-2a^{7}z+a^{7}z^{-1}+a^{6}z^{8}+a^{6}z^{6}-13a^{6}z^{4}+14a^{6}z^{2}-5a^{6}+5a^{5}z^{7}-16a^{5}z^{5}+14a^{5}z^{3}-7a^{5}z+2a^{5}z^{-1}+a^{4}z^{8}+2a^{4}z^{6}-12a^{4}z^{4}+11a^{4}z^{2}-3a^{4}+3a^{3}z^{7}-9a^{3}z^{5}+12a^{3}z^{3}-6a^{3}z+2a^{2}z^{6}-3a^{2}z^{4}+3a^{2}z^{2}+a^{2}+4az^{3}-az-az^{-1}+z^{2}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

0

1

χ

2

1

-1

0

3

-2

3

0

-4

3

1

2

-6

2

3

1

-8

3

0

-10

1

2

1

-12

1

3

-2

-14

1

-16

1

-1

Integral Khovanov Homology

(db, data source)

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