L10n4

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_9,14,10,15 X₃₈₄₉ X_5,11,6,10 X_13,5,14,20 X_11,19,12,18 X_19,13,20,12 X_15,2,16,3
Gauss code	{1, 10, -5, -3}, {-6, -1, 2, 5, -4, 6, -8, 9, -7, 4, -10, -2, 3, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{5}-2uv^{4}+2uv^{3}-2uv^{2}-2v^{3}+2v^{2}-2v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+2q^{5/2}-3q^{3/2}+4{\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {5}{q^{3/2}}}-{\frac {4}{q^{5/2}}}+{\frac {2}{q^{7/2}}}-{\frac {2}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-a^{3}z^{5}+6az^{5}-z^{5}a^{-1}-5a^{3}z^{3}+12az^{3}-4z^{3}a^{-1}+a^{5}z-8a^{3}z+9az-4za^{-1}+2a^{5}z^{-1}-4a^{3}z^{-1}+3az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$3a^{5}z^{3}-7a^{5}z+2a^{5}z^{-1}+a^{4}z^{6}-2a^{4}z^{4}+3a^{4}z^{2}-2a^{4}+2a^{3}z^{7}-9a^{3}z^{5}+z^{5}a^{-3}+20a^{3}z^{3}-3z^{3}a^{-3}-16a^{3}z+za^{-3}+4a^{3}z^{-1}+a^{2}z^{8}-2a^{2}z^{6}+2z^{6}a^{-2}-6z^{4}a^{-2}+7a^{2}z^{2}+3z^{2}a^{-2}-3a^{2}-a^{-2}+4az^{7}+2z^{7}a^{-1}-16az^{5}-6z^{5}a^{-1}+25az^{3}+5z^{3}a^{-1}-13az-3za^{-1}+3az^{-1}+a^{-1}z^{-1}+z^{8}-z^{6}-4z^{4}+7z^{2}-3$ (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		\

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

1

-1

4

2

1

2

1

-1

0

3

2

1

-2

3

0

-4

1

2

-1

-6

1

3

2

-8

1

0

-10

2

Integral Khovanov Homology

(db, data source)

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