L10a80

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_18,10,19,9 X_20,13,7,14 X_14,19,15,20 X_10,16,11,15 X_16,6,17,5 X₂₇₃₈ X_4,11,5,12 X_6,18,1,17
Gauss code	{1, -8, 2, -9, 7, -10}, {8, -1, 3, -6, 9, -2, 4, -5, 6, -7, 10, -3, 5, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{3}-4u^{2}v^{2}+3u^{2}v-u^{2}+uv^{4}-4uv^{3}+5uv^{2}-4uv+u-v^{4}+3v^{3}-4v^{2}+2v}{uv^{2}}}$ (db)
Jones polynomial	$-q^{9/2}+3q^{7/2}-6q^{5/2}+9q^{3/2}-11{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {11}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-3a^{3}z^{3}-z^{3}a^{-3}-6a^{3}z-2a^{3}z^{-1}-za^{-3}+2az^{5}+z^{5}a^{-1}+6az^{3}+z^{3}a^{-1}+6az+2az^{-1}-za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{3}z^{9}-2az^{9}-3a^{4}z^{8}-10a^{2}z^{8}-7z^{8}-a^{5}z^{7}+a^{3}z^{7}-8az^{7}-10z^{7}a^{-1}+12a^{4}z^{6}+34a^{2}z^{6}-9z^{6}a^{-2}+13z^{6}+4a^{5}z^{5}+19a^{3}z^{5}+42az^{5}+21z^{5}a^{-1}-6z^{5}a^{-3}-14a^{4}z^{4}-30a^{2}z^{4}+13z^{4}a^{-2}-3z^{4}a^{-4}-6a^{5}z^{3}-32a^{3}z^{3}-44az^{3}-13z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}+4a^{4}z^{2}+7a^{2}z^{2}-5z^{2}a^{-2}-2z^{2}+4a^{5}z+15a^{3}z+17az+5za^{-1}-za^{-3}-a^{2}-a^{5}z^{-1}-2a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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

3

4

χ

10

1

8

2

-2

6

4

1

3

4

5

2

-3

2

6

4

2

0

6

0

-2

5

0

-4

4

7

3

-6

2

4

-2

-8

1

4

3

-10

2

-2

-12

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$