L10a99

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_20,12,9,11 X_2,9,3,10 X_18,14,19,13 X_14,7,15,8 X_16,5,17,6 X_6,15,7,16 X_4,17,5,18 X_8,20,1,19
Gauss code	{1, -4, 2, -9, 7, -8, 6, -10}, {4, -1, 3, -2, 5, -6, 8, -7, 9, -5, 10, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{2}t(1)^{3}-2t(2)t(1)^{3}+t(1)^{3}+t(2)^{3}t(1)^{2}-3t(2)^{2}t(1)^{2}+3t(2)t(1)^{2}-2t(1)^{2}-2t(2)^{3}t(1)+3t(2)^{2}t(1)-3t(2)t(1)+t(1)+t(2)^{3}-2t(2)^{2}+t(2)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{9/2}+3q^{7/2}-5q^{5/2}+7q^{3/2}-8{\sqrt {q}}+{\frac {8}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-2a^{3}z^{3}-z^{3}a^{-3}-4a^{3}z-a^{3}z^{-1}-za^{-3}+az^{5}+z^{5}a^{-1}+2az^{3}+2z^{3}a^{-1}+az+za^{-1}$ (db)
Kauffman polynomial	$a^{5}z^{7}-5a^{5}z^{5}+8a^{5}z^{3}+z^{3}a^{-5}-5a^{5}z+a^{5}z^{-1}+2a^{4}z^{8}-9a^{4}z^{6}+11a^{4}z^{4}+3z^{4}a^{-4}-2a^{4}z^{2}-z^{2}a^{-4}-a^{4}+a^{3}z^{9}-13a^{3}z^{5}+5z^{5}a^{-3}+21a^{3}z^{3}-4z^{3}a^{-3}-9a^{3}z+za^{-3}+a^{3}z^{-1}+5a^{2}z^{8}-17a^{2}z^{6}+6z^{6}a^{-2}+14a^{2}z^{4}-8z^{4}a^{-2}-2a^{2}z^{2}+3z^{2}a^{-2}+az^{9}+4az^{7}+5z^{7}a^{-1}-20az^{5}-7z^{5}a^{-1}+19az^{3}+z^{3}a^{-1}-6az-za^{-1}+3z^{8}-2z^{6}-8z^{4}+4z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

10

1

8

2

-2

6

3

1

2

4

2

-2

2

4

3

1

0

5

0

-2

3

0

-4

2

5

3

-6

2

3

-1

-8

1

3

2

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

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