L11a377

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Planar diagram presentation	X_12,1,13,2 X_14,4,15,3 X_22,7,11,8 X_8,11,9,12 X_20,14,21,13 X_16,6,17,5 X_18,10,19,9 X_4,16,5,15 X_6,18,7,17 X_10,20,1,19 X_2,21,3,22
Gauss code	{1, -11, 2, -8, 6, -9, 3, -4, 7, -10}, {4, -1, 5, -2, 8, -6, 9, -7, 10, -5, 11, -3}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{4}t(1)^{4}-t(2)^{3}t(1)^{4}-2t(2)^{4}t(1)^{3}+5t(2)^{3}t(1)^{3}-4t(2)^{2}t(1)^{3}+t(2)t(1)^{3}+t(2)^{4}t(1)^{2}-5t(2)^{3}t(1)^{2}+7t(2)^{2}t(1)^{2}-5t(2)t(1)^{2}+t(1)^{2}+t(2)^{3}t(1)-4t(2)^{2}t(1)+5t(2)t(1)-2t(1)-t(2)+1}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+6q^{13/2}-10q^{11/2}+13q^{9/2}-15q^{7/2}+14q^{5/2}-13q^{3/2}+9{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{9}a^{-3}+z^{7}a^{-1}-7z^{7}a^{-3}+z^{7}a^{-5}+5z^{5}a^{-1}-18z^{5}a^{-3}+5z^{5}a^{-5}+8z^{3}a^{-1}-20z^{3}a^{-3}+8z^{3}a^{-5}+5za^{-1}-8za^{-3}+4za^{-5}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}-z^{2}a^{-10}+3z^{5}a^{-9}-3z^{3}a^{-9}+za^{-9}+5z^{6}a^{-8}-5z^{4}a^{-8}+2z^{2}a^{-8}+6z^{7}a^{-7}-6z^{5}a^{-7}+z^{3}a^{-7}+za^{-7}+6z^{8}a^{-6}-8z^{6}a^{-6}+3z^{4}a^{-6}+5z^{9}a^{-5}-10z^{7}a^{-5}+11z^{5}a^{-5}-11z^{3}a^{-5}+4za^{-5}+2z^{10}a^{-4}+2z^{8}a^{-4}-16z^{6}a^{-4}+16z^{4}a^{-4}-6z^{2}a^{-4}+9z^{9}a^{-3}-32z^{7}a^{-3}+40z^{5}a^{-3}-27z^{3}a^{-3}+9za^{-3}-a^{-3}z^{-1}+2z^{10}a^{-2}-z^{8}a^{-2}-15z^{6}a^{-2}+20z^{4}a^{-2}-7z^{2}a^{-2}+a^{-2}+4z^{9}a^{-1}+az^{7}-15z^{7}a^{-1}-4az^{5}+16z^{5}a^{-1}+4az^{3}-8z^{3}a^{-1}-az+4za^{-1}-a^{-1}z^{-1}+3z^{8}-12z^{6}+13z^{4}-4z^{2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

18

1

-1

16

2

14

4

1

-3

12

6

2

4

10

7

4

-3

8

6

2

6

7

8

1

4

6

7

-1

2

4

8

4

0

2

5

-3

-2

1

4

3

-4

2

-2

-6

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=2$	$i=4$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$