L10n20

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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_11,14,12,15 X₈₄₉₃ X_5,13,6,12 X_13,5,14,20 X_9,16,10,17 X_15,10,16,11 X_2,18,3,17
Gauss code	{1, -10, 5, -3}, {-6, -1, 2, -5, -8, 9, -4, 6, -7, 4, -9, 8, 10, -2, 3, 7}

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

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

1

-1

4

1

3

2

1

-1

0

5

4

1

-2

4

0

-4

2

3

-1

-6

2

4

2

-8

2

-2

-10

2

Integral Khovanov Homology

(db, data source)

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