L10a62

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{4}-2u^{2}v^{3}+2u^{2}v^{2}-2u^{2}v+u^{2}-2uv^{4}+3uv^{3}-3uv^{2}+3uv-2u+v^{4}-2v^{3}+2v^{2}-2v+1}{uv^{2}}}$ (db)
Jones polynomial	$8q^{9/2}-9q^{7/2}+9q^{5/2}-{\frac {1}{q^{5/2}}}-9q^{3/2}+{\frac {2}{q^{3/2}}}-q^{15/2}+3q^{13/2}-5q^{11/2}+6{\sqrt {q}}-{\frac {5}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-5}-3z^{3}a^{-5}-2za^{-5}+z^{7}a^{-3}+5z^{5}a^{-3}+9z^{3}a^{-3}+7za^{-3}+a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-8z^{3}a^{-1}+3az-9za^{-1}+2az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-9}+3z^{4}a^{-8}-z^{2}a^{-8}+5z^{5}a^{-7}-3z^{3}a^{-7}+7z^{6}a^{-6}-10z^{4}a^{-6}+4z^{2}a^{-6}+7z^{7}a^{-5}-14z^{5}a^{-5}+9z^{3}a^{-5}-4za^{-5}+4z^{8}a^{-4}-3z^{6}a^{-4}-13z^{4}a^{-4}+11z^{2}a^{-4}-a^{-4}+z^{9}a^{-3}+8z^{7}a^{-3}-35z^{5}a^{-3}+36z^{3}a^{-3}-13za^{-3}+a^{-3}z^{-1}+6z^{8}a^{-2}-18z^{6}a^{-2}+8z^{4}a^{-2}+7z^{2}a^{-2}-3a^{-2}+z^{9}a^{-1}+az^{7}+2z^{7}a^{-1}-5az^{5}-21z^{5}a^{-1}+9az^{3}+32z^{3}a^{-1}-7az-16za^{-1}+2az^{-1}+3a^{-1}z^{-1}+2z^{8}-8z^{6}+8z^{4}+z^{2}-3$ (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

5

6

χ

16

1

14

2

-2

12

3

1

2

10

5

2

-3

8

4

3

1

6

5

0

4

0

2

3

6

3

0

2

3

-1

-2

1

4

3

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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