L11a268

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}v^{3}-3u^{3}v^{2}+3u^{3}v-u^{3}-3u^{2}v^{3}+9u^{2}v^{2}-10u^{2}v+3u^{2}+3uv^{3}-10uv^{2}+9uv-3u-v^{3}+3v^{2}-3v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {14}{q^{9/2}}}-q^{7/2}+{\frac {18}{q^{7/2}}}+4q^{5/2}-{\frac {21}{q^{5/2}}}-8q^{3/2}+{\frac {21}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {8}{q^{11/2}}}+13{\sqrt {q}}-{\frac {19}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+3z^{3}a^{5}+4za^{5}+a^{5}z^{-1}-3z^{5}a^{3}-8z^{3}a^{3}-7za^{3}-a^{3}z^{-1}+z^{7}a+4z^{5}a+7z^{3}a+3za-z^{5}a^{-1}-2z^{3}a^{-1}-za^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-4a^{5}z^{9}-8a^{3}z^{9}-4az^{9}-6a^{6}z^{8}-15a^{4}z^{8}-16a^{2}z^{8}-7z^{8}-4a^{7}z^{7}-4a^{5}z^{7}-3a^{3}z^{7}-10az^{7}-7z^{7}a^{-1}-a^{8}z^{6}+12a^{6}z^{6}+36a^{4}z^{6}+32a^{2}z^{6}-4z^{6}a^{-2}+5z^{6}+10a^{7}z^{5}+29a^{5}z^{5}+40a^{3}z^{5}+33az^{5}+11z^{5}a^{-1}-z^{5}a^{-3}+2a^{8}z^{4}-5a^{6}z^{4}-22a^{4}z^{4}-17a^{2}z^{4}+6z^{4}a^{-2}+4z^{4}-8a^{7}z^{3}-29a^{5}z^{3}-42a^{3}z^{3}-29az^{3}-7z^{3}a^{-1}+z^{3}a^{-3}-a^{8}z^{2}+3a^{4}z^{2}+a^{2}z^{2}-2z^{2}a^{-2}-3z^{2}+2a^{7}z+11a^{5}z+14a^{3}z+7az+2za^{-1}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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

5

1

4

2

8

3

-5

0

11

5

6

-2

11

9

-2

-4

10

0

-6

8

11

3

-8

6

10

-4

-10

3

9

6

-12

1

5

-4

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

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