L10n40

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_20,10,7,9 X₂₇₃₈ X_15,5,16,4 X_5,13,6,12 X_11,16,12,17 X_6,18,1,17 X_14,19,15,20 X_18,13,19,14
Gauss code	{1, -4, 2, 5, -6, -8}, {4, -1, 3, -2, -7, 6, 10, -9, -5, 7, 8, -10, 9, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{4}-t(2)^{4}-2t(1)t(2)^{3}+t(2)^{3}+t(1)t(2)^{2}+t(1)^{2}t(2)-2t(1)t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+{\frac {1}{q^{7/2}}}+2q^{5/2}-{\frac {2}{q^{5/2}}}-4q^{3/2}+{\frac {3}{q^{3/2}}}+3{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{-3}-a^{3}z+2za^{-3}-a^{3}z^{-1}-z^{5}a^{-1}+2az^{3}-4z^{3}a^{-1}+5az-5za^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-4}-4z^{4}a^{-4}+a^{4}z^{2}+3z^{2}a^{-4}-a^{4}+2z^{7}a^{-3}-9z^{5}a^{-3}+2a^{3}z^{3}+11z^{3}a^{-3}-2a^{3}z-5za^{-3}+a^{3}z^{-1}+z^{8}a^{-2}+a^{2}z^{6}-2z^{6}a^{-2}-3a^{2}z^{4}-4z^{4}a^{-2}+6a^{2}z^{2}+5z^{2}a^{-2}-3a^{2}+2az^{7}+4z^{7}a^{-1}-9az^{5}-18z^{5}a^{-1}+16az^{3}+25z^{3}a^{-1}-11az-14za^{-1}+3az^{-1}+2a^{-1}z^{-1}+z^{8}-2z^{6}-3z^{4}+7z^{2}-3$ (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		\

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

1

6

1

0

4

3

1

2

1

2

1

0

3

2

1

-2

1

2

1

-4

1

2

-1

-6

1

-8

1

-1

Integral Khovanov Homology

(db, data source)

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