L10n14

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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_9,14,10,15 X₈₄₉₃ X_5,13,6,12 X_13,5,14,20 X_11,16,12,17 X_15,10,16,11 X_2,18,3,17
Gauss code	{1, -10, 5, -3}, {-6, -1, 2, -5, -4, 9, -8, 6, -7, 4, -9, 8, 10, -2, 3, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}}}$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-{\sqrt {q}}-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {2}{q^{5/2}}}+{\frac {2}{q^{7/2}}}-{\frac {2}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+2za^{5}+a^{5}z^{-1}-z^{5}a^{3}-5z^{3}a^{3}-8za^{3}-3a^{3}z^{-1}+z^{5}a+6z^{3}a+9za+4az^{-1}-z^{3}a^{-1}-3za^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+a^{7}z+2a^{6}z^{6}-7a^{6}z^{4}+4a^{6}z^{2}-a^{6}+a^{5}z^{7}-3a^{5}z^{5}+a^{5}z^{3}-2a^{5}z+a^{5}z^{-1}+2a^{4}z^{6}-9a^{4}z^{4}+10a^{4}z^{2}-3a^{4}+2a^{3}z^{7}-13a^{3}z^{5}+24a^{3}z^{3}-15a^{3}z+3a^{3}z^{-1}+a^{2}z^{8}-6a^{2}z^{6}+6a^{2}z^{4}+5a^{2}z^{2}-3a^{2}+2az^{7}+z^{7}a^{-1}-15az^{5}-6z^{5}a^{-1}+30az^{3}+10z^{3}a^{-1}-19az-7za^{-1}+4az^{-1}+2a^{-1}z^{-1}+z^{8}-6z^{6}+8z^{4}-z^{2}-2$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

0

2

1

0

3

1

2

-2

1

2

1

0

-4

2

1

-6

1

0

-8

1

2

1

0

-10

1

0

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$	$i=0$
$r=-6$		${\mathbb {Z} }$
$r=-5$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$		${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$
$r=1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=3$	${\mathbb {Z} }$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$