L10a53

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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_10,4,11,3 X_4,15,5,16 X_16,12,17,11 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 $u$ , $v$ , $w$ , ...)	${\frac {\left(uv^{2}-2uv+u+2v-1\right)\left(uv^{2}-2uv-v^{2}+2v-1\right)}{uv^{2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {4}{q^{9/2}}}-4q^{7/2}+{\frac {8}{q^{7/2}}}+8q^{5/2}-{\frac {13}{q^{5/2}}}-12q^{3/2}+{\frac {15}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+15{\sqrt {q}}-{\frac {17}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-2a^{3}z^{3}+z^{3}a^{-3}-a^{3}z+za^{-3}+a^{3}z^{-1}+az^{7}+4az^{5}-2z^{5}a^{-1}+6az^{3}-5z^{3}a^{-1}+2az-3za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}+4a^{5}z^{5}-2a^{5}z^{3}+8a^{4}z^{6}+z^{6}a^{-4}-7a^{4}z^{4}-2z^{4}a^{-4}+a^{4}z^{2}+z^{2}a^{-4}+11a^{3}z^{7}+4z^{7}a^{-3}-16a^{3}z^{5}-10z^{5}a^{-3}+9a^{3}z^{3}+7z^{3}a^{-3}-a^{3}z-2za^{-3}-a^{3}z^{-1}+9a^{2}z^{8}+6z^{8}a^{-2}-11a^{2}z^{6}-14z^{6}a^{-2}+2a^{2}z^{4}+7z^{4}a^{-2}+a^{2}+3az^{9}+3z^{9}a^{-1}+12az^{7}+5z^{7}a^{-1}-39az^{5}-29z^{5}a^{-1}+29az^{3}+25z^{3}a^{-1}-5az-6za^{-1}-az^{-1}+15z^{8}-34z^{6}+19z^{4}-2z^{2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

5

1

-4

4

7

3

4

2

8

5

-3

0

9

7

2

-2

7

9

2

-4

6

8

-2

-6

3

8

5

-8

1

5

-4

-10

3

-12

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$