L11a244

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Planar diagram presentation	X₈₁₉₂ X_10,8,11,7 X_12,3,13,4 X_18,5,19,6 X_16,12,17,11 X_14,20,15,19 X_20,14,21,13 X_2,15,3,16 X_22,18,7,17 X_4,21,5,22 X_6,9,1,10
Gauss code	{1, -8, 3, -10, 4, -11}, {2, -1, 11, -2, 5, -3, 7, -6, 8, -5, 9, -4, 6, -7, 10, -9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}-5t(1)^{2}t(2)^{3}+11t(1)t(2)^{3}-6t(2)^{3}+9t(1)^{2}t(2)^{2}-19t(1)t(2)^{2}+9t(2)^{2}-6t(1)^{2}t(2)+11t(1)t(2)-5t(2)-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {11}{q^{9/2}}}-5q^{7/2}+{\frac {17}{q^{7/2}}}+11q^{5/2}-{\frac {25}{q^{5/2}}}-18q^{3/2}+{\frac {28}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+25{\sqrt {q}}-{\frac {28}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z+2a^{5}z^{-1}-2a^{3}z^{5}-4a^{3}z^{3}+z^{3}a^{-3}-6a^{3}z-3a^{3}z^{-1}+az^{7}+3az^{5}-2z^{5}a^{-1}+6az^{3}-3z^{3}a^{-1}+4az-2za^{-1}+az^{-1}$ (db)
Kauffman polynomial	$-3a^{2}z^{10}-3z^{10}-10a^{3}z^{9}-19az^{9}-9z^{9}a^{-1}-13a^{4}z^{8}-24a^{2}z^{8}-10z^{8}a^{-2}-21z^{8}-10a^{5}z^{7}+3a^{3}z^{7}+26az^{7}+8z^{7}a^{-1}-5z^{7}a^{-3}-4a^{6}z^{6}+18a^{4}z^{6}+62a^{2}z^{6}+21z^{6}a^{-2}-z^{6}a^{-4}+62z^{6}-a^{7}z^{5}+16a^{5}z^{5}+17a^{3}z^{5}+5az^{5}+14z^{5}a^{-1}+9z^{5}a^{-3}+3a^{6}z^{4}-8a^{4}z^{4}-46a^{2}z^{4}-13z^{4}a^{-2}+z^{4}a^{-4}-49z^{4}+a^{7}z^{3}-15a^{5}z^{3}-16a^{3}z^{3}-9az^{3}-13z^{3}a^{-1}-4z^{3}a^{-3}-3a^{4}z^{2}+9a^{2}z^{2}+2z^{2}a^{-2}+14z^{2}+8a^{5}z+7a^{3}z+za^{-1}+3a^{4}+3a^{2}+1-2a^{5}z^{-1}-3a^{3}z^{-1}-az^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

4

6

7

1

-6

4

11

4

7

2

14

7

-7

0

14

11

3

-2

15

0

-4

10

13

-3

-6

7

15

8

-8

4

10

-6

-10

7

-12

1

4

-3

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{15}$
$r=0$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$