L10a10

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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_18,16,19,15 X_16,9,17,10 X_8,17,9,18 X_20,14,5,13 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, 7, -8, 4, -5, 6, -4, 8, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2(t(1)-1)(t(2)-1)^{3}}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{13/2}+3q^{11/2}-6q^{9/2}+8q^{7/2}-10q^{5/2}+11q^{3/2}-10{\sqrt {q}}+{\frac {7}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-za^{-5}-a^{-5}z^{-1}+z^{5}a^{-3}+2z^{3}a^{-3}+a^{3}z+3za^{-3}+a^{3}z^{-1}+2a^{-3}z^{-1}+z^{5}a^{-1}-2az^{3}+z^{3}a^{-1}-3az-az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-6z^{8}a^{-2}-4z^{8}a^{-4}-2z^{8}-2az^{7}-3z^{7}a^{-1}-6z^{7}a^{-3}-5z^{7}a^{-5}-2a^{2}z^{6}+15z^{6}a^{-2}+9z^{6}a^{-4}-3z^{6}a^{-6}+z^{6}-a^{3}z^{5}+8z^{5}a^{-1}+21z^{5}a^{-3}+13z^{5}a^{-5}-z^{5}a^{-7}+4a^{2}z^{4}-20z^{4}a^{-2}-9z^{4}a^{-4}+6z^{4}a^{-6}-z^{4}+3a^{3}z^{3}+7az^{3}-10z^{3}a^{-1}-27z^{3}a^{-3}-11z^{3}a^{-5}+2z^{3}a^{-7}-a^{2}z^{2}+11z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}a^{-6}+5z^{2}-3a^{3}z-5az+6za^{-1}+13za^{-3}+5za^{-5}-a^{2}-3a^{-2}-a^{-4}-2+a^{3}z^{-1}+az^{-1}-a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

2

-2

10

4

1

3

8

4

2

-2

6

4

2

4

5

4

-1

2

5

6

-1

0

4

7

3

-2

1

3

-2

-4

1

4

3

-6

1

-1

-8

1

Integral Khovanov Homology

(db, data source)

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