L11a362

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{3}t(1)^{4}-t(2)^{2}t(1)^{4}+t(2)^{4}t(1)^{3}-4t(2)^{3}t(1)^{3}+3t(2)^{2}t(1)^{3}-t(2)t(1)^{3}-t(2)^{4}t(1)^{2}+3t(2)^{3}t(1)^{2}-5t(2)^{2}t(1)^{2}+3t(2)t(1)^{2}-t(1)^{2}-t(2)^{3}t(1)+3t(2)^{2}t(1)-4t(2)t(1)+t(1)-t(2)^{2}+t(2)}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$q^{23/2}-3q^{21/2}+5q^{19/2}-8q^{17/2}+10q^{15/2}-11q^{13/2}+10q^{11/2}-9q^{9/2}+6q^{7/2}-4q^{5/2}+2q^{3/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-5}-z^{7}a^{-7}+z^{5}a^{-3}-4z^{5}a^{-5}-4z^{5}a^{-7}+z^{5}a^{-9}+4z^{3}a^{-3}-3z^{3}a^{-5}-4z^{3}a^{-7}+3z^{3}a^{-9}+3za^{-3}+za^{-5}-2za^{-7}+za^{-9}+a^{-5}z^{-1}-a^{-7}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-14}-z^{2}a^{-14}+3z^{5}a^{-13}-4z^{3}a^{-13}+za^{-13}+4z^{6}a^{-12}-4z^{4}a^{-12}+4z^{7}a^{-11}-3z^{5}a^{-11}-2z^{3}a^{-11}+2za^{-11}+4z^{8}a^{-10}-7z^{6}a^{-10}+6z^{4}a^{-10}-z^{2}a^{-10}+3z^{9}a^{-9}-7z^{7}a^{-9}+9z^{5}a^{-9}-6z^{3}a^{-9}+za^{-9}+z^{10}a^{-8}+2z^{8}a^{-8}-13z^{6}a^{-8}+16z^{4}a^{-8}-6z^{2}a^{-8}+5z^{9}a^{-7}-19z^{7}a^{-7}+25z^{5}a^{-7}-17z^{3}a^{-7}+6za^{-7}-a^{-7}z^{-1}+z^{10}a^{-6}-11z^{6}a^{-6}+16z^{4}a^{-6}-8z^{2}a^{-6}+a^{-6}+2z^{9}a^{-5}-7z^{7}a^{-5}+5z^{5}a^{-5}-2z^{3}a^{-5}+3za^{-5}-a^{-5}z^{-1}+2z^{8}a^{-4}-9z^{6}a^{-4}+11z^{4}a^{-4}-4z^{2}a^{-4}+z^{7}a^{-3}-5z^{5}a^{-3}+7z^{3}a^{-3}-3za^{-3}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

24

1

-1

22

2

20

3

1

-2

18

5

2

3

16

6

4

-2

14

5

4

1

12

5

6

1

10

4

5

-1

8

2

5

3

6

2

4

-2

4

1

3

2

1

-1

0

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=4$	$i=6$
$r=-2$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{5}$
$r=7$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=8$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=9$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$