L10a122

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Link Presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {-t(1)t(3)^{3}+t(1)t(2)t(3)^{3}-2t(2)t(3)^{3}+t(3)^{3}+4t(1)t(3)^{2}-3t(1)t(2)t(3)^{2}+6t(2)t(3)^{2}-3t(3)^{2}-6t(1)t(3)+3t(1)t(2)t(3)-4t(2)t(3)+3t(3)+2t(1)-t(1)t(2)+t(2)-1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-2q^{-5}+q^{4}+6q^{-4}-4q^{3}-9q^{-3}+9q^{2}+13q^{-2}-11q-14q^{-1}+14$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-3z^{2}a^{4}-3a^{4}z^{-2}-5a^{4}+3z^{4}a^{2}+7z^{2}a^{2}+4a^{2}z^{-2}+8a^{2}-z^{6}-3z^{4}-6z^{2}-3z^{-2}-6+z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-4a^{6}z^{4}+6a^{6}z^{2}+a^{6}z^{-2}-4a^{6}+2a^{5}z^{7}-5a^{5}z^{5}+3a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+2a^{4}z^{8}+a^{4}z^{6}-16a^{4}z^{4}+z^{4}a^{-4}+24a^{4}z^{2}+3a^{4}z^{-2}-14a^{4}+a^{3}z^{9}+7a^{3}z^{7}-20a^{3}z^{5}+4z^{5}a^{-3}+12a^{3}z^{3}-z^{3}a^{-3}+a^{3}z-a^{3}z^{-1}+7a^{2}z^{8}-2a^{2}z^{6}+9z^{6}a^{-2}-30a^{2}z^{4}-11z^{4}a^{-2}+40a^{2}z^{2}+6z^{2}a^{-2}+4a^{2}z^{-2}+a^{-2}z^{-2}-21a^{2}-4a^{-2}+az^{9}+15az^{7}+10z^{7}a^{-1}-32az^{5}-13z^{5}a^{-1}+14az^{3}+4z^{3}a^{-1}+az+za^{-1}-az^{-1}-a^{-1}z^{-1}+5z^{8}+7z^{6}-30z^{4}+28z^{2}+3z^{-2}-14$ (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

9

1

7

3

-3

5

6

1

5

3

5

3

-2

1

9

6

3

-1

8

0

-3

5

6

-1

-5

4

8

4

-7

2

5

-3

-9

1

5

4

-11

1

-1

-13

1

Integral Khovanov Homology

(db, data source)

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