L11a213

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_18,10,19,9 X_22,20,7,19 X_20,14,21,13 X_14,22,15,21 X_10,16,11,15 X_16,6,17,5 X₂₇₃₈ X_4,11,5,12 X_6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 3, -7, 10, -2, 5, -6, 7, -8, 11, -3, 4, -5, 6, -4}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+2t(2)^{4}-4t(1)^{2}t(2)^{3}+7t(1)t(2)^{3}-5t(2)^{3}+6t(1)^{2}t(2)^{2}-9t(1)t(2)^{2}+6t(2)^{2}-5t(1)^{2}t(2)+7t(1)t(2)-4t(2)+2t(1)^{2}-3t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{17/2}-4q^{15/2}+9q^{13/2}-14q^{11/2}+19q^{9/2}-21q^{7/2}+20q^{5/2}-18q^{3/2}+12{\sqrt {q}}-{\frac {8}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+za^{-7}-2z^{5}a^{-5}-4z^{3}a^{-5}-za^{-5}+a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}+3z^{3}a^{-3}-2a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-5z^{3}a^{-1}+2az-3za^{-1}+az^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}a^{-4}-4z^{9}a^{-1}-12z^{9}a^{-3}-8z^{9}a^{-5}-6z^{8}a^{-2}-16z^{8}a^{-4}-13z^{8}a^{-6}-3z^{8}-az^{7}+9z^{7}a^{-1}+27z^{7}a^{-3}+4z^{7}a^{-5}-13z^{7}a^{-7}+33z^{6}a^{-2}+51z^{6}a^{-4}+19z^{6}a^{-6}-9z^{6}a^{-8}+10z^{6}+4az^{5}+z^{5}a^{-1}-9z^{5}a^{-3}+15z^{5}a^{-5}+17z^{5}a^{-7}-4z^{5}a^{-9}-35z^{4}a^{-2}-42z^{4}a^{-4}-8z^{4}a^{-6}+8z^{4}a^{-8}-z^{4}a^{-10}-10z^{4}-6az^{3}-10z^{3}a^{-1}-3z^{3}a^{-3}-8z^{3}a^{-5}-8z^{3}a^{-7}+z^{3}a^{-9}+13z^{2}a^{-2}+18z^{2}a^{-4}+4z^{2}a^{-6}-3z^{2}a^{-8}+2z^{2}+4az+3za^{-1}-2za^{-3}+za^{-7}-3a^{-2}-5a^{-4}-2a^{-6}+1-az^{-1}+2a^{-3}z^{-1}+a^{-5}z^{-1}$ (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

3

14

6

1

-5

12

8

3

5

10

11

6

-5

8

10

8

2

6

10

11

1

4

8

10

-2

2

5

11

6

0

3

7

-4

-2

1

6

5

-4

2

-2

-6

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=2$	$i=4$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$