L11a521

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{2}w^{3}-2u^{2}v^{2}w^{2}-2u^{2}vw^{3}+4u^{2}vw^{2}-2u^{2}vw+u^{2}w^{3}-2u^{2}w^{2}+u^{2}w-uv^{2}w^{3}+3uv^{2}w^{2}-2uv^{2}w+2uvw^{3}-5uvw^{2}+5uvw-2uv+2uw^{2}-3uw+u-v^{2}w^{2}+2v^{2}w-v^{2}+2vw^{2}-4vw+2v+2w-1}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{10}-3q^{9}+7q^{8}-11q^{7}+16q^{6}-17q^{5}+18q^{4}-15q^{3}+12q^{2}-7q+4-q^{-1}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{4}a^{-8}+3z^{2}a^{-8}+a^{-8}z^{-2}+2a^{-8}-2z^{6}a^{-6}-8z^{4}a^{-6}-10z^{2}a^{-6}-2a^{-6}z^{-2}-6a^{-6}+z^{8}a^{-4}+5z^{6}a^{-4}+9z^{4}a^{-4}+8z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}-z^{6}a^{-2}-3z^{4}a^{-2}-z^{2}a^{-2}+a^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-12}-z^{2}a^{-12}+3z^{5}a^{-11}-2z^{3}a^{-11}+6z^{6}a^{-10}-7z^{4}a^{-10}+5z^{2}a^{-10}-a^{-10}+8z^{7}a^{-9}-10z^{5}a^{-9}+5z^{3}a^{-9}+9z^{8}a^{-8}-17z^{6}a^{-8}+17z^{4}a^{-8}-12z^{2}a^{-8}-a^{-8}z^{-2}+5a^{-8}+6z^{9}a^{-7}-5z^{7}a^{-7}-12z^{5}a^{-7}+17z^{3}a^{-7}-9za^{-7}+2a^{-7}z^{-1}+2z^{10}a^{-6}+10z^{8}a^{-6}-43z^{6}a^{-6}+50z^{4}a^{-6}-31z^{2}a^{-6}-2a^{-6}z^{-2}+11a^{-6}+11z^{9}a^{-5}-30z^{7}a^{-5}+16z^{5}a^{-5}+7z^{3}a^{-5}-9za^{-5}+2a^{-5}z^{-1}+2z^{10}a^{-4}+5z^{8}a^{-4}-35z^{6}a^{-4}+41z^{4}a^{-4}-17z^{2}a^{-4}-a^{-4}z^{-2}+5a^{-4}+5z^{9}a^{-3}-16z^{7}a^{-3}+12z^{5}a^{-3}-z^{3}a^{-3}+4z^{8}a^{-2}-15z^{6}a^{-2}+16z^{4}a^{-2}-4z^{2}a^{-2}-a^{-2}+z^{7}a^{-1}-3z^{5}a^{-1}+2z^{3}a^{-1}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

7

8

χ

21

1

19

3

1

-2

17

4

15

7

3

-4

13

9

4

5

11

8

7

-1

9

10

9

1

7

10

3

5

8

-3

3

8

5

1

4

-3

-1

3

-3

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=3$	$i=5$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{5}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=7$	${\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=8$	${\mathbb {Z} }$	${\mathbb {Z} }$