L11a509

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Link Presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(3)-1)(-t(2)t(1)+t(2)t(3)t(1)-t(3)t(1)+t(1)+t(2)+t(3)-1)(-t(1)t(2)+t(1)t(3)t(2)-t(3)t(2)+t(2)-t(1)t(3)+t(3)-1)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{6}-5q^{5}+12q^{4}-20q^{3}+28q^{2}-31q+33-27q^{-1}+21q^{-2}-12q^{-3}+5q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+4z^{6}-2a^{2}z^{4}-5z^{4}a^{-2}+z^{4}a^{-4}+6z^{4}-a^{2}z^{2}-3z^{2}a^{-2}+z^{2}a^{-4}+2z^{2}+a^{2}+a^{-2}-2+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-6}-z^{4}a^{-6}+5z^{7}a^{-5}+a^{5}z^{5}-8z^{5}a^{-5}+3z^{3}a^{-5}+11z^{8}a^{-4}+5a^{4}z^{6}-22z^{6}a^{-4}-3a^{4}z^{4}+14z^{4}a^{-4}-4z^{2}a^{-4}+12z^{9}a^{-3}+12a^{3}z^{7}-18z^{7}a^{-3}-13a^{3}z^{5}+z^{5}a^{-3}+3a^{3}z^{3}+4z^{3}a^{-3}+5z^{10}a^{-2}+18a^{2}z^{8}+18z^{8}a^{-2}-28a^{2}z^{6}-62z^{6}a^{-2}+16a^{2}z^{4}+49z^{4}a^{-2}-3a^{2}z^{2}-11z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{2}-2a^{-2}+15az^{9}+27z^{9}a^{-1}-15az^{7}-50z^{7}a^{-1}-4az^{5}+19z^{5}a^{-1}+4az^{3}+2z^{3}a^{-1}+2az+2za^{-1}-2az^{-1}-2a^{-1}z^{-1}+5z^{10}+25z^{8}-72z^{6}+53z^{4}-10z^{2}+2z^{-2}-3$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

13

1

11

4

-4

9

8

1

7

12

4

-8

5

16

8

3

16

13

-3

1

17

15

2

-1

12

18

6

-3

9

15

-6

-5

4

13

9

-7

1

8

-7

-9

4

-11

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=-1$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{18}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{17}$
$r=1$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{16}$	${\mathbb {Z} }^{16}$
$r=2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{16}$
$r=3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$