L10a70

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X_14,6,15,5 X_16,12,17,11 X_10,4,11,3 X_12,7,13,8 X_20,15,7,16 X_6,14,1,13 X_4,19,5,20 X_2,18,3,17
Gauss code	{1, -10, 5, -9, 3, -8}, {6, -1, 2, -5, 4, -6, 8, -3, 7, -4, 10, -2, 9, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+t(2)^{4}-3t(1)^{2}t(2)^{3}+7t(1)t(2)^{3}-4t(2)^{3}+5t(1)^{2}t(2)^{2}-9t(1)t(2)^{2}+5t(2)^{2}-4t(1)^{2}t(2)+7t(1)t(2)-3t(2)+t(1)^{2}-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {4}{q^{9/2}}}-5q^{7/2}+{\frac {9}{q^{7/2}}}+9q^{5/2}-{\frac {14}{q^{5/2}}}-14q^{3/2}+{\frac {17}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+17{\sqrt {q}}-{\frac {19}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-2a^{3}z^{3}+z^{3}a^{-3}-2a^{3}z-a^{-3}z^{-1}+az^{7}+4az^{5}-2z^{5}a^{-1}+7az^{3}-4z^{3}a^{-1}+4az-za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}+4a^{5}z^{5}-a^{5}z^{3}+9a^{4}z^{6}+z^{6}a^{-4}-8a^{4}z^{4}-z^{4}a^{-4}+3a^{4}z^{2}+13a^{3}z^{7}+5z^{7}a^{-3}-19a^{3}z^{5}-11z^{5}a^{-3}+12a^{3}z^{3}+5z^{3}a^{-3}-3a^{3}z+2za^{-3}-a^{-3}z^{-1}+11a^{2}z^{8}+8z^{8}a^{-2}-13a^{2}z^{6}-19z^{6}a^{-2}+a^{2}z^{4}+11z^{4}a^{-2}+a^{2}z^{2}-z^{2}a^{-2}+a^{-2}+4az^{9}+4z^{9}a^{-1}+13az^{7}+5z^{7}a^{-1}-42az^{5}-30z^{5}a^{-1}+27az^{3}+19z^{3}a^{-1}-5az-a^{-1}z^{-1}+19z^{8}-42z^{6}+22z^{4}-3z^{2}$ (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

4

6

5

1

-4

4

9

4

5

2

9

6

-3

0

10

8

2

-2

8

10

2

-4

6

9

-3

-6

3

8

5

-8

1

6

-5

-10

3

-12

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$