L10a13

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_16,13,17,14 X_14,7,15,8 X_8,15,9,16 X_20,18,5,17 X_18,12,19,11 X_12,20,13,19 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 4, -5, 10, -2, 7, -8, 3, -4, 5, -3, 6, -7, 8, -6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{5}+2t(1)t(2)^{4}-4t(2)^{4}-5t(1)t(2)^{3}+7t(2)^{3}+7t(1)t(2)^{2}-5t(2)^{2}-4t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{5/2}-3q^{3/2}+6{\sqrt {q}}-{\frac {10}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {13}{q^{5/2}}}+{\frac {11}{q^{7/2}}}-{\frac {10}{q^{9/2}}}+{\frac {6}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+3a^{5}z^{3}+6a^{5}z+4a^{5}z^{-1}-2a^{3}z^{5}-6a^{3}z^{3}-8a^{3}z-4a^{3}z^{-1}-az^{5}-az^{3}+z^{3}a^{-1}+az^{-1}+za^{-1}$ (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-3a^{6}z^{8}-7a^{4}z^{8}-4a^{2}z^{8}-3a^{7}z^{7}-9a^{5}z^{7}-12a^{3}z^{7}-6az^{7}-a^{8}z^{6}+3a^{6}z^{6}+7a^{4}z^{6}-2a^{2}z^{6}-5z^{6}+9a^{7}z^{5}+32a^{5}z^{5}+33a^{3}z^{5}+7az^{5}-3z^{5}a^{-1}+3a^{8}z^{4}+10a^{6}z^{4}+16a^{4}z^{4}+15a^{2}z^{4}-z^{4}a^{-2}+5z^{4}-9a^{7}z^{3}-32a^{5}z^{3}-31a^{3}z^{3}-5az^{3}+3z^{3}a^{-1}-3a^{8}z^{2}-13a^{6}z^{2}-21a^{4}z^{2}-15a^{2}z^{2}+z^{2}a^{-2}-3z^{2}+4a^{7}z+17a^{5}z+16a^{3}z+2az-za^{-1}+a^{8}+4a^{6}+7a^{4}+4a^{2}+1-a^{7}z^{-1}-4a^{5}z^{-1}-4a^{3}z^{-1}-az^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

2

4

1

-3

0

6

2

4

-2

7

5

-2

-4

6

5

1

-6

5

7

2

-8

5

6

-1

-10

2

6

4

-12

1

4

-3

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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