L10n33

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,16,8,17 X_20,18,5,17 X_18,11,19,12 X_10,19,11,20 X_9,14,10,15 X_15,8,16,9 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, -3, 8, -7, -6, 5, -2, 10, 7, -8, 3, 4, -5, 6, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

3

2

3

1

-2

0

5

3

2

-2

5

0

-4

4

3

1

-6

2

5

3

-8

2

4

-2

-10

2

-12

2

-2

Integral Khovanov Homology

(db, data source)

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