L10n69

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,14,3,13 X_14,7,15,8 X_9,18,10,19 X_11,16,12,17 X_17,20,18,11 X_15,1,16,4 X_19,10,20,5
Gauss code	{1, -4, -3, 9}, {-2, -1, 5, 3, -6, 10}, {-7, 2, 4, -5, -9, 7, -8, 6, -10, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2uvw^{2}-2uvw-uw^{2}+2uw-2v^{2}w+v^{2}+2vw-2v}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{-9}-2q^{-8}+3q^{-7}-4q^{-6}+5q^{-5}-4q^{-4}+5q^{-3}-2q^{-2}+2q^{-1}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{2}a^{8}+a^{8}-z^{4}a^{6}-2z^{2}a^{6}+a^{6}z^{-2}-a^{6}-z^{4}a^{4}-2z^{2}a^{4}-2a^{4}z^{-2}-3a^{4}+2z^{2}a^{2}+a^{2}z^{-2}+3a^{2}$ (db)
Kauffman polynomial	$a^{10}z^{6}-4a^{10}z^{4}+4a^{10}z^{2}-a^{10}+2a^{9}z^{7}-8a^{9}z^{5}+8a^{9}z^{3}-2a^{9}z+a^{8}z^{8}-a^{8}z^{6}-6a^{8}z^{4}+6a^{8}z^{2}-a^{8}+4a^{7}z^{7}-14a^{7}z^{5}+14a^{7}z^{3}-6a^{7}z+a^{6}z^{8}-7a^{6}z^{4}+8a^{6}z^{2}+a^{6}z^{-2}-3a^{6}+2a^{5}z^{7}-5a^{5}z^{5}+5a^{5}z^{3}-2a^{5}z^{-1}+2a^{4}z^{6}-5a^{4}z^{4}+9a^{4}z^{2}+2a^{4}z^{-2}-6a^{4}+a^{3}z^{5}-a^{3}z^{3}+4a^{3}z-2a^{3}z^{-1}+3a^{2}z^{2}+a^{2}z^{-2}-4a^{2}$ (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

χ

-1

2

-3

2

0

-5

3

-7

2

3

1

-9

3

2

1

-11

1

2

1

-13

2

3

-1

-15

1

2

1

-17

1

-1

-19

1

Integral Khovanov Homology

(db, data source)

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