L11a345

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Planar diagram presentation	X_12,1,13,2 X₈₄₉₃ X_18,14,19,13 X_22,20,11,19 X_20,15,21,16 X_14,21,15,22 X_6,17,7,18 X_16,7,17,8 X_10,6,1,5 X_4,10,5,9 X_2,11,3,12
Gauss code	{1, -11, 2, -10, 9, -7, 8, -2, 10, -9}, {11, -1, 3, -6, 5, -8, 7, -3, 4, -5, 6, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{3}v^{3}-4u^{3}v^{2}+6u^{3}v-2u^{3}-2u^{2}v^{3}+6u^{2}v^{2}-6u^{2}v+2u^{2}+2uv^{3}-6uv^{2}+6uv-2u-2v^{3}+6v^{2}-4v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+9q^{7/2}-14q^{5/2}+17q^{3/2}-19{\sqrt {q}}+{\frac {18}{\sqrt {q}}}-{\frac {15}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+10az^{3}-8z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-8a^{3}z+12az-9za^{-1}+2za^{-3}+2a^{5}z^{-1}-5a^{3}z^{-1}+6az^{-1}-4a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-5a^{5}z^{5}+4z^{5}a^{-5}+9a^{5}z^{3}-z^{3}a^{-5}-7a^{5}z+2a^{5}z^{-1}+2a^{4}z^{8}-7a^{4}z^{6}+9z^{6}a^{-4}+7a^{4}z^{4}-8z^{4}a^{-4}-a^{4}z^{2}+3z^{2}a^{-4}-a^{4}-a^{-4}+2a^{3}z^{9}-a^{3}z^{7}+13z^{7}a^{-3}-16a^{3}z^{5}-19z^{5}a^{-3}+31a^{3}z^{3}+11z^{3}a^{-3}-21a^{3}z-4za^{-3}+5a^{3}z^{-1}+a^{-3}z^{-1}+a^{2}z^{10}+6a^{2}z^{8}+11z^{8}a^{-2}-24a^{2}z^{6}-12z^{6}a^{-2}+21a^{2}z^{4}-5z^{4}a^{-2}-3a^{2}z^{2}+9z^{2}a^{-2}-a^{2}-3a^{-2}+7az^{9}+5z^{9}a^{-1}-6az^{7}+9z^{7}a^{-1}-30az^{5}-42z^{5}a^{-1}+50az^{3}+40z^{3}a^{-1}-29az-19za^{-1}+6az^{-1}+4a^{-1}z^{-1}+z^{10}+15z^{8}-38z^{6}+18z^{4}+4z^{2}-3$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

6

1

-5

6

8

3

5

4

9

6

-3

2

10

8

2

0

9

10

1

-2

6

9

-3

-4

4

9

5

-6

2

6

-4

-8

1

5

4

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

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