L10a9

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)^{3}\left(t(2)^{2}+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$9q^{9/2}-10q^{7/2}+10q^{5/2}-{\frac {1}{q^{5/2}}}-10q^{3/2}+{\frac {2}{q^{3/2}}}-q^{15/2}+3q^{13/2}-6q^{11/2}+7{\sqrt {q}}-{\frac {5}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-5}-3z^{3}a^{-5}-3za^{-5}-a^{-5}z^{-1}+z^{7}a^{-3}+5z^{5}a^{-3}+10z^{3}a^{-3}+10za^{-3}+3a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-8z^{3}a^{-1}+3az-10za^{-1}+2az^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-9}+3z^{4}a^{-8}+6z^{5}a^{-7}-4z^{3}a^{-7}+za^{-7}+9z^{6}a^{-6}-14z^{4}a^{-6}+7z^{2}a^{-6}-a^{-6}+9z^{7}a^{-5}-18z^{5}a^{-5}+10z^{3}a^{-5}-4za^{-5}+a^{-5}z^{-1}+5z^{8}a^{-4}-4z^{6}a^{-4}-15z^{4}a^{-4}+14z^{2}a^{-4}-3a^{-4}+z^{9}a^{-3}+11z^{7}a^{-3}-44z^{5}a^{-3}+45z^{3}a^{-3}-19za^{-3}+3a^{-3}z^{-1}+7z^{8}a^{-2}-21z^{6}a^{-2}+11z^{4}a^{-2}+6z^{2}a^{-2}-3a^{-2}+z^{9}a^{-1}+az^{7}+3z^{7}a^{-1}-5az^{5}-25z^{5}a^{-1}+9az^{3}+39z^{3}a^{-1}-7az-21za^{-1}+2az^{-1}+4a^{-1}z^{-1}+2z^{8}-8z^{6}+9z^{4}-z^{2}-2$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

16

1

14

2

-2

12

4

1

3

10

5

2

-3

8

5

4

1

6

5

0

4

5

0

2

4

7

3

0

1

3

-2

1

4

3

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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