L10n91

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,20,14,17 X_7,18,8,19 X_17,10,18,11 X_9,15,10,14 X_15,9,16,8 X_19,16,20,5 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -9, 2, -10}, {-5, 4, -8, 3}, {9, -1, -4, 7, -6, 5, 10, -2, -3, 6, -7, 8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(3)-1)(t(3)+1)(-2t(1)t(2)+t(1)t(3)t(2)-2t(3)+1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$1-q^{-1}+q^{-2}-q^{-3}+q^{-5}+q^{-6}+2q^{-7}-q^{-8}+q^{-9}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}-2a^{8}z^{-2}-4a^{8}+z^{4}a^{6}+6z^{2}a^{6}+a^{6}z^{-2}+6a^{6}-z^{6}a^{4}-5z^{4}a^{4}-6z^{2}a^{4}-4a^{4}+z^{4}a^{2}+4z^{2}a^{2}+2a^{2}$ (db)
Kauffman polynomial	$a^{10}z^{6}-5a^{10}z^{4}+6a^{10}z^{2}+a^{10}z^{-2}-4a^{10}+a^{9}z^{7}-4a^{9}z^{5}+4a^{9}z-2a^{9}z^{-1}+a^{8}z^{8}-5a^{8}z^{6}+3a^{8}z^{4}+6a^{8}z^{2}+2a^{8}z^{-2}-5a^{8}+2a^{7}z^{7}-12a^{7}z^{5}+16a^{7}z^{3}-4a^{7}z-2a^{7}z^{-1}+a^{6}z^{8}-5a^{6}z^{6}+2a^{6}z^{4}+6a^{6}z^{2}+a^{6}z^{-2}-2a^{6}+2a^{5}z^{7}-13a^{5}z^{5}+22a^{5}z^{3}-12a^{5}z+2a^{4}z^{6}-11a^{4}z^{4}+12a^{4}z^{2}-2a^{4}+a^{3}z^{7}-5a^{3}z^{5}+6a^{3}z^{3}-4a^{3}z+a^{2}z^{6}-5a^{2}z^{4}+6a^{2}z^{2}-2a^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

0

-3

1

2

1

0

-5

1

0

-7

2

1

-1

-9

1

-11

1

4

1

2

-13

1

3

-15

1

-17

1

0

-19

1

Integral Khovanov Homology

(db, data source)

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