L10n13

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_9,14,10,15 X₃₈₄₉ X_5,13,6,12 X_13,5,14,20 X_11,16,12,17 X_15,10,16,11 X_17,2,18,3
Gauss code	{1, 10, -5, -3}, {-6, -1, 2, 5, -4, 9, -8, 6, -7, 4, -9, 8, -10, -2, 3, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{5}-2t(1)t(2)^{4}+t(1)t(2)^{3}-t(2)^{3}-t(1)t(2)^{2}+t(2)^{2}-2t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {3}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {3}{q^{5/2}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}z+a^{9}z^{-1}-a^{7}z^{5}-5a^{7}z^{3}-7a^{7}z-3a^{7}z^{-1}+a^{5}z^{7}+6a^{5}z^{5}+12a^{5}z^{3}+11a^{5}z+4a^{5}z^{-1}-a^{3}z^{5}-5a^{3}z^{3}-7a^{3}z-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{11}z+2a^{10}z^{2}-a^{10}+a^{9}z^{5}-a^{9}z^{3}-a^{9}z+a^{9}z^{-1}+3a^{8}z^{6}-11a^{8}z^{4}+10a^{8}z^{2}-3a^{8}+3a^{7}z^{7}-13a^{7}z^{5}+17a^{7}z^{3}-11a^{7}z+3a^{7}z^{-1}+a^{6}z^{8}-a^{6}z^{6}-9a^{6}z^{4}+12a^{6}z^{2}-3a^{6}+4a^{5}z^{7}-20a^{5}z^{5}+30a^{5}z^{3}-18a^{5}z+4a^{5}z^{-1}+a^{4}z^{8}-4a^{4}z^{6}+2a^{4}z^{4}+4a^{4}z^{2}-2a^{4}+a^{3}z^{7}-6a^{3}z^{5}+12a^{3}z^{3}-9a^{3}z+2a^{3}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

0

-4

3

1

2

-6

1

0

-8

2

0

-10

1

2

1

0

-12

2

0

-14

1

2

1

-16

1

-1

-18

1

Integral Khovanov Homology

(db, data source)

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