L11a209

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {\left(2v^{2}-3v+2\right)(uv-u+1)(uv-v+1)}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {12}{q^{9/2}}}-q^{7/2}+{\frac {17}{q^{7/2}}}+4q^{5/2}-{\frac {20}{q^{5/2}}}-9q^{3/2}+{\frac {20}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {7}{q^{11/2}}}+13{\sqrt {q}}-{\frac {19}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+4a^{3}z^{5}+3az^{5}-z^{5}a^{-1}-3a^{5}z^{3}+7a^{3}z^{3}+2az^{3}-2z^{3}a^{-1}-3a^{5}z+6a^{3}z-az-za^{-1}-a^{5}z^{-1}+2a^{3}z^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+2a^{8}z^{2}+3a^{7}z^{7}-8a^{7}z^{5}+5a^{7}z^{3}+5a^{6}z^{8}-13a^{6}z^{6}+12a^{6}z^{4}-7a^{6}z^{2}+2a^{6}+5a^{5}z^{9}-11a^{5}z^{7}+12a^{5}z^{5}-11a^{5}z^{3}+4a^{5}z-a^{5}z^{-1}+2a^{4}z^{10}+6a^{4}z^{8}-24a^{4}z^{6}+31a^{4}z^{4}-21a^{4}z^{2}+5a^{4}+11a^{3}z^{9}-25a^{3}z^{7}+27a^{3}z^{5}+z^{5}a^{-3}-18a^{3}z^{3}-z^{3}a^{-3}+8a^{3}z-2a^{3}z^{-1}+2a^{2}z^{10}+10a^{2}z^{8}-29a^{2}z^{6}+4z^{6}a^{-2}+30a^{2}z^{4}-5z^{4}a^{-2}-14a^{2}z^{2}+3a^{2}+6az^{9}-3az^{7}+8z^{7}a^{-1}-8az^{5}-14z^{5}a^{-1}+7az^{3}+8z^{3}a^{-1}+az-3za^{-1}+a^{-1}z^{-1}+9z^{8}-15z^{6}+9z^{4}-2z^{2}-1$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

6

1

5

2

8

4

-4

0

11

5

6

-2

10

9

-1

-4

10

0

-6

7

10

3

-8

5

10

-5

-10

2

7

5

-12

1

5

-4

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$