L10n17

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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_5,12,6,13 X₃₈₄₉ X_13,17,14,16 X_9,15,10,14 X_15,11,16,10 X_11,20,12,5 X_17,2,18,3
Gauss code	{1, 10, -5, -3}, {-4, -1, 2, 5, -7, 8, -9, 4, -6, 7, -8, 6, -10, -2, 3, 9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-uv^{5}+2uv^{4}-3uv^{3}+uv^{2}+v^{3}-3v^{2}+2v-1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-2{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+a^{5}z^{5}+5a^{5}z^{3}+8a^{5}z+4a^{5}z^{-1}-a^{3}z^{7}-6a^{3}z^{5}-13a^{3}z^{3}-13a^{3}z-4a^{3}z^{-1}+az^{5}+4az^{3}+4az+az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{2}-a^{8}+3a^{7}z^{3}-2a^{7}z+a^{7}z^{-1}+2a^{6}z^{6}-6a^{6}z^{4}+10a^{6}z^{2}-4a^{6}+3a^{5}z^{7}-12a^{5}z^{5}+20a^{5}z^{3}-14a^{5}z+4a^{5}z^{-1}+a^{4}z^{8}+a^{4}z^{6}-13a^{4}z^{4}+18a^{4}z^{2}-7a^{4}+5a^{3}z^{7}-20a^{3}z^{5}+26a^{3}z^{3}-17a^{3}z+4a^{3}z^{-1}+a^{2}z^{8}-11a^{2}z^{4}+13a^{2}z^{2}-4a^{2}+2az^{7}-8az^{5}+9az^{3}-5az+az^{-1}+z^{6}-4z^{4}+4z^{2}-1$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

1

0

2

1

-1

-2

3

1

2

-4

2

3

1

-6

3

2

1

-8

1

2

1

-10

2

3

-1

-12

2

-14

1

-1

Integral Khovanov Homology

(db, data source)

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