L11a252

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}v^{5}-3u^{3}v^{4}+3u^{3}v^{3}-u^{3}v^{2}-u^{2}v^{5}+5u^{2}v^{4}-9u^{2}v^{3}+6u^{2}v^{2}-2u^{2}v-2uv^{4}+6uv^{3}-9uv^{2}+5uv-u-v^{3}+3v^{2}-3v+1}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	${\frac {17}{q^{9/2}}}-{\frac {20}{q^{7/2}}}-q^{5/2}+{\frac {19}{q^{5/2}}}+4q^{3/2}-{\frac {17}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {4}{q^{15/2}}}+{\frac {8}{q^{13/2}}}-{\frac {13}{q^{11/2}}}-8{\sqrt {q}}+{\frac {12}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{3}z^{9}-a^{5}z^{7}+6a^{3}z^{7}-az^{7}-4a^{5}z^{5}+12a^{3}z^{5}-4az^{5}-4a^{5}z^{3}+8a^{3}z^{3}-4az^{3}-a^{3}z-az+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}+4a^{9}z^{5}-2a^{9}z^{3}+8a^{8}z^{6}-7a^{8}z^{4}+a^{8}z^{2}+11a^{7}z^{7}-14a^{7}z^{5}+5a^{7}z^{3}+11a^{6}z^{8}-16a^{6}z^{6}+5a^{6}z^{4}+8a^{5}z^{9}-10a^{5}z^{7}-4a^{5}z^{5}+6a^{5}z^{3}-3a^{5}z+a^{5}z^{-1}+3a^{4}z^{10}+7a^{4}z^{8}-34a^{4}z^{6}+28a^{4}z^{4}-5a^{4}z^{2}-a^{4}+14a^{3}z^{9}-43a^{3}z^{7}+38a^{3}z^{5}-10a^{3}z^{3}-2a^{3}z+a^{3}z^{-1}+3a^{2}z^{10}-24a^{2}z^{6}+28a^{2}z^{4}-7a^{2}z^{2}+6az^{9}-21az^{7}+z^{7}a^{-1}+21az^{5}-3z^{5}a^{-1}-7az^{3}+2z^{3}a^{-1}+az+4z^{8}-14z^{6}+13z^{4}-3z^{2}$ (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

3

-3

2

5

1

4

0

7

3

-4

-2

10

5

-4

10

8

-2

-6

10

9

1

-8

7

10

3

-10

6

10

-4

-12

3

8

5

-14

1

5

-4

-16

3

-18

1

-1

Integral Khovanov Homology

(db, data source)

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