L11a439

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Planar diagram presentation	X₆₁₇₂ X_14,6,15,5 X_10,4,11,3 X_2,16,3,15 X_20,18,21,17 X_16,9,17,10 X_8,20,9,19 X_18,8,19,7 X_22,12,13,11 X_12,14,5,13 X_4,21,1,22
Gauss code	{1, -4, 3, -11}, {2, -1, 8, -7, 6, -3, 9, -10}, {10, -2, 4, -6, 5, -8, 7, -5, 11, -9}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)^{2}(t(3)-1)^{2}(t(2)t(3)+1)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{10}-4q^{9}+9q^{8}-14q^{7}+18q^{6}-20q^{5}+21q^{4}-16q^{3}+13q^{2}-7q+4-q^{-1}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{4}a^{-8}+2z^{2}a^{-8}+a^{-8}-2z^{6}a^{-6}-7z^{4}a^{-6}-7z^{2}a^{-6}+a^{-6}z^{-2}-2a^{-6}+z^{8}a^{-4}+5z^{6}a^{-4}+9z^{4}a^{-4}+6z^{2}a^{-4}-2a^{-4}z^{-2}-a^{-4}-z^{6}a^{-2}-3z^{4}a^{-2}-z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-12}+4z^{5}a^{-11}-z^{3}a^{-11}+9z^{6}a^{-10}-8z^{4}a^{-10}+3z^{2}a^{-10}-a^{-10}+13z^{7}a^{-9}-18z^{5}a^{-9}+9z^{3}a^{-9}-2za^{-9}+12z^{8}a^{-8}-15z^{6}a^{-8}+4z^{2}a^{-8}-a^{-8}+7z^{9}a^{-7}+z^{7}a^{-7}-27z^{5}a^{-7}+21z^{3}a^{-7}-6za^{-7}+2z^{10}a^{-6}+15z^{8}a^{-6}-49z^{6}a^{-6}+38z^{4}a^{-6}-9z^{2}a^{-6}+a^{-6}z^{-2}-a^{-6}+12z^{9}a^{-5}-28z^{7}a^{-5}+8z^{5}a^{-5}+10z^{3}a^{-5}-2za^{-5}-2a^{-5}z^{-1}+2z^{10}a^{-4}+7z^{8}a^{-4}-40z^{6}a^{-4}+46z^{4}a^{-4}-15z^{2}a^{-4}+2a^{-4}z^{-2}-2a^{-4}+5z^{9}a^{-3}-15z^{7}a^{-3}+10z^{5}a^{-3}+z^{3}a^{-3}+2za^{-3}-2a^{-3}z^{-1}+4z^{8}a^{-2}-15z^{6}a^{-2}+17z^{4}a^{-2}-5z^{2}a^{-2}+a^{-2}z^{-2}-2a^{-2}+z^{7}a^{-1}-3z^{5}a^{-1}+2z^{3}a^{-1}$ (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

-3

17

6

1

5

15

8

3

-5

13

10

6

4

11

12

10

-2

9

8

1

7

12

5

6

9

-3

3

9

6

1

4

-3

-1

3

-3

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=3$	$i=5$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{6}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{10}$
$r=5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=8$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$