L11a208

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X₆₇₁₈ X_20,13,21,14 X_10,4,11,3 X_16,6,17,5 X_4,12,5,11 X_22,15,7,16 X_12,19,13,20 X_14,21,15,22 X_2,18,3,17
Gauss code	{1, -11, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -9, 4, -10, 8, -6, 11, -2, 9, -4, 10, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {\left(v^{4}-v^{3}+v^{2}-v+1\right)(uv-u+1)(uv-v+1)}{uv^{3}}}$ (db)
Jones polynomial	${\frac {13}{q^{9/2}}}-{\frac {14}{q^{7/2}}}-q^{5/2}+{\frac {13}{q^{5/2}}}+3q^{3/2}-{\frac {13}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {9}{q^{11/2}}}-6{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{5}z^{7}-5a^{5}z^{5}-8a^{5}z^{3}-5a^{5}z-2a^{5}z^{-1}+a^{3}z^{9}+7a^{3}z^{7}+18a^{3}z^{5}+21a^{3}z^{3}+12a^{3}z+5a^{3}z^{-1}-az^{7}-5az^{5}-8az^{3}-6az-3az^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}-a^{10}z^{2}+3a^{9}z^{5}-3a^{9}z^{3}+5a^{8}z^{6}-6a^{8}z^{4}+3a^{8}z^{2}-a^{8}+6a^{7}z^{7}-8a^{7}z^{5}+4a^{7}z^{3}+6a^{6}z^{8}-10a^{6}z^{6}+5a^{6}z^{4}+2a^{6}z^{2}+5a^{5}z^{9}-12a^{5}z^{7}+15a^{5}z^{5}-13a^{5}z^{3}+6a^{5}z-2a^{5}z^{-1}+2a^{4}z^{10}+a^{4}z^{8}-15a^{4}z^{6}+19a^{4}z^{4}-12a^{4}z^{2}+5a^{4}+9a^{3}z^{9}-35a^{3}z^{7}+50a^{3}z^{5}-38a^{3}z^{3}+15a^{3}z-5a^{3}z^{-1}+2a^{2}z^{10}-2a^{2}z^{8}-12a^{2}z^{6}+19a^{2}z^{4}-12a^{2}z^{2}+5a^{2}+4az^{9}-16az^{7}+z^{7}a^{-1}+20az^{5}-4z^{5}a^{-1}-14az^{3}+4z^{3}a^{-1}+8az-3az^{-1}-za^{-1}+3z^{8}-12z^{6}+12z^{4}-2z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

2

-2

2

4

1

3

0

5

3

-2

8

3

5

-4

6

0

-6

8

7

1

-8

5

6

1

-10

4

8

-4

-12

2

5

3

-14

1

4

-3

-16

2

-18

1

-1

Integral Khovanov Homology

(db, data source)

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