L11a284

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Link Presentations

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

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

Polynomial invariants

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

20

1

18

2

-2

16

4

1

3

14

6

2

-4

12

8

4

10

9

7

-2

8

7

0

6

9

3

4

5

7

-2

2

7

5

0

1

4

-3

-2

2

-4

1

-1

Integral Khovanov Homology

(db, data source)

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