L11a463

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{3}w^{2}-2uv^{3}w+uv^{2}w^{3}-3uv^{2}w^{2}+4uv^{2}w-2uv^{2}-2uvw^{3}+4uvw^{2}-3uvw+uv+uw^{3}-2uw^{2}+uw-v^{3}w^{2}+2v^{3}w-v^{3}-v^{2}w^{3}+3v^{2}w^{2}-4v^{2}w+2v^{2}+2vw^{3}-4vw^{2}+3vw-v+2w^{2}-w}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{9}-3q^{8}+7q^{7}-9q^{6}+15q^{5}-17q^{4}+17q^{3}-15q^{2}+12q-7+4q^{-1}-q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}+2z^{4}a^{-4}-2z^{4}a^{-6}-z^{4}+z^{2}a^{-2}+2z^{2}a^{-4}-4z^{2}a^{-6}+z^{2}a^{-8}-z^{2}+a^{-4}-3a^{-6}+a^{-8}+1+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-10}-3z^{4}a^{-10}+z^{2}a^{-10}+3z^{7}a^{-9}-8z^{5}a^{-9}+2z^{3}a^{-9}+6z^{8}a^{-8}-22z^{6}a^{-8}+28z^{4}a^{-8}-20z^{2}a^{-8}-a^{-8}z^{-2}+8a^{-8}+5z^{9}a^{-7}-13z^{7}a^{-7}+5z^{5}a^{-7}+11z^{3}a^{-7}-11za^{-7}+2a^{-7}z^{-1}+2z^{10}a^{-6}+4z^{8}a^{-6}-30z^{6}a^{-6}+49z^{4}a^{-6}-31z^{2}a^{-6}-2a^{-6}z^{-2}+13a^{-6}+10z^{9}a^{-5}-30z^{7}a^{-5}+32z^{5}a^{-5}-11za^{-5}+2a^{-5}z^{-1}+2z^{10}a^{-4}+4z^{8}a^{-4}-18z^{6}a^{-4}+24z^{4}a^{-4}-12z^{2}a^{-4}-a^{-4}z^{-2}+5a^{-4}+5z^{9}a^{-3}-8z^{7}a^{-3}+9z^{5}a^{-3}-7z^{3}a^{-3}+6z^{8}a^{-2}-7z^{6}a^{-2}-z^{4}a^{-2}+6z^{7}a^{-1}+az^{5}-9z^{5}a^{-1}-az^{3}+z^{3}a^{-1}+4z^{6}-7z^{4}+2z^{2}+1$ (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

19

1

17

3

1

-2

15

4

13

5

3

-2

11

10

4

6

9

8

6

-2

7

9

0

5

6

8

2

3

6

9

-3

1

3

8

5

-1

1

4

-3

3

-5

1

-1

Integral Khovanov Homology

(db, data source)

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