L10n48

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X₆₇₁₈ X_20,14,7,13 X_12,5,13,6 X_3,10,4,11 X_15,5,16,4 X_11,16,12,17 X_14,20,15,19 X_2,18,3,17
Gauss code	{1, -10, -6, 7, 5, -3}, {3, -1, 2, 6, -8, -5, 4, -9, -7, 8, 10, -2, 9, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-6

-5

-4

-3

-2

-1

0

1

2

χ

4

1

2

1

-1

0

3

1

2

-2

2

0

-4

3

2

1

-6

2

3

1

-8

1

2

-1

-10

1

2

1

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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