L10n55

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_11,17,12,16 X_8,9,1,10 X_17,9,18,20 X_3,12,4,13 X_7,14,8,15 X_13,6,14,7 X_5,18,6,19 X_19,4,20,5 X_15,2,16,3
Gauss code	{1, 10, -5, 9, -8, 7, -6, -3}, {3, -1, -2, 5, -7, 6, -10, 2, -4, 8, -9, 4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}(-v)+u^{2}v^{3}-3u^{2}v^{2}+3u^{2}v-2u^{2}-2uv^{3}+3uv^{2}-3uv+u-v^{2}}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	${\frac {7}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {4}{q^{5/2}}}-{\frac {2}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {2}{q^{17/2}}}-{\frac {5}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {7}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z+a^{9}z^{-1}-2a^{7}z^{3}-3a^{7}z-a^{7}z^{-1}+a^{5}z^{5}+2a^{5}z^{3}+a^{5}z-2a^{3}z^{3}-3a^{3}z$ (db)
Kauffman polynomial	$a^{11}z^{5}-3a^{11}z^{3}+2a^{11}z+2a^{10}z^{6}-4a^{10}z^{4}+a^{10}z^{2}+3a^{9}z^{7}-8a^{9}z^{5}+9a^{9}z^{3}-7a^{9}z+a^{9}z^{-1}+a^{8}z^{8}+3a^{8}z^{6}-10a^{8}z^{4}+6a^{8}z^{2}-a^{8}+5a^{7}z^{7}-11a^{7}z^{5}+12a^{7}z^{3}-7a^{7}z+a^{7}z^{-1}+a^{6}z^{8}+2a^{6}z^{6}-4a^{6}z^{4}+3a^{6}z^{2}+2a^{5}z^{7}-2a^{5}z^{5}+3a^{5}z^{3}-a^{5}z+a^{4}z^{6}+2a^{4}z^{4}-2a^{4}z^{2}+3a^{3}z^{3}-3a^{3}z$ (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

-2

2

-4

3

1

-2

-6

3

1

2

-8

4

3

-1

-10

3

0

-12

3

4

1

-14

2

3

-1

-16

3

-18

1

2

-1

-20

1

Integral Khovanov Homology

(db, data source)

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