L11a264

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{3}t(1)^{3}-3t(2)^{2}t(1)^{3}+2t(2)t(1)^{3}-3t(2)^{3}t(1)^{2}+7t(2)^{2}t(1)^{2}-5t(2)t(1)^{2}+2t(1)^{2}+2t(2)^{3}t(1)-5t(2)^{2}t(1)+7t(2)t(1)-3t(1)+2t(2)^{2}-3t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+6q^{13/2}-10q^{11/2}+12q^{9/2}-15q^{7/2}+14q^{5/2}-12q^{3/2}+9{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+4z^{5}a^{-3}-2z^{5}a^{-5}+az^{3}-6z^{3}a^{-1}+6z^{3}a^{-3}-6z^{3}a^{-5}+z^{3}a^{-7}+2az-3za^{-1}+5za^{-3}-4za^{-5}+2za^{-7}+a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-3z^{9}a^{-1}-6z^{9}a^{-3}-3z^{9}a^{-5}-5z^{8}a^{-2}-7z^{8}a^{-4}-5z^{8}a^{-6}-3z^{8}-az^{7}+8z^{7}a^{-1}+12z^{7}a^{-3}-3z^{7}a^{-5}-6z^{7}a^{-7}+25z^{6}a^{-2}+21z^{6}a^{-4}+3z^{6}a^{-6}-5z^{6}a^{-8}+12z^{6}+4az^{5}+4z^{5}a^{-3}+18z^{5}a^{-5}+7z^{5}a^{-7}-3z^{5}a^{-9}-24z^{4}a^{-2}-12z^{4}a^{-4}+4z^{4}a^{-6}+5z^{4}a^{-8}-z^{4}a^{-10}-14z^{4}-5az^{3}-8z^{3}a^{-1}-13z^{3}a^{-3}-18z^{3}a^{-5}-5z^{3}a^{-7}+3z^{3}a^{-9}+5z^{2}a^{-2}-3z^{2}a^{-6}-2z^{2}a^{-8}+z^{2}a^{-10}+5z^{2}+2az+2za^{-1}+6za^{-3}+9za^{-5}+2za^{-7}-za^{-9}+a^{-4}-a^{-3}z^{-1}-a^{-5}z^{-1}$ (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

7

χ

18

1

-1

16

2

14

4

1

-3

12

6

2

4

10

7

5

-2

8

5

3

6

7

1

4

6

8

-2

2

4

7

3

0

2

5

-3

-2

1

4

3

-4

2

-2

-6

1

Integral Khovanov Homology

(db, data source)

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