L11a212

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,8,15,7 X_6,13,1,14 X_20,17,21,18 X_16,5,17,6 X_18,11,19,12 X_12,19,13,20 X_22,16,7,15 X_4,21,5,22
Gauss code	{1, -2, 3, -11, 7, -5}, {4, -1, 2, -3, 8, -9, 5, -4, 10, -7, 6, -8, 9, -6, 11, -10}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{4}-6u^{2}v^{3}+5u^{2}v^{2}-2u^{2}v-2uv^{4}+7uv^{3}-11uv^{2}+7uv-2u-2v^{3}+5v^{2}-6v+2}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {16}{q^{9/2}}}+{\frac {11}{q^{7/2}}}-{\frac {7}{q^{5/2}}}+{\frac {3}{q^{3/2}}}+{\frac {1}{q^{23/2}}}-{\frac {4}{q^{21/2}}}+{\frac {8}{q^{19/2}}}-{\frac {13}{q^{17/2}}}+{\frac {17}{q^{15/2}}}-{\frac {19}{q^{13/2}}}+{\frac {18}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{5}\right)-2a^{9}z^{3}+a^{7}z^{7}+3a^{7}z^{5}+2a^{7}z^{3}+a^{7}z+a^{7}z^{-1}+a^{5}z^{7}+3a^{5}z^{5}+a^{5}z^{3}-2a^{5}z-a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-2a^{3}z$ (db)
Kauffman polynomial	$-z^{4}a^{14}-4z^{5}a^{13}+2z^{3}a^{13}-8z^{6}a^{12}+7z^{4}a^{12}-z^{2}a^{12}-11z^{7}a^{11}+14z^{5}a^{11}-5z^{3}a^{11}-11z^{8}a^{10}+17z^{6}a^{10}-7z^{4}a^{10}-7z^{9}a^{9}+6z^{7}a^{9}+9z^{5}a^{9}-8z^{3}a^{9}+2za^{9}-2z^{10}a^{8}-11z^{8}a^{8}+39z^{6}a^{8}-31z^{4}a^{8}+7z^{2}a^{8}-11z^{9}a^{7}+30z^{7}a^{7}-22z^{5}a^{7}+7z^{3}a^{7}-3za^{7}+a^{7}z^{-1}-2z^{10}a^{6}-3z^{8}a^{6}+25z^{6}a^{6}-28z^{4}a^{6}+10z^{2}a^{6}-a^{6}-4z^{9}a^{5}+12z^{7}a^{5}-9z^{5}a^{5}+3z^{3}a^{5}-3za^{5}+a^{5}z^{-1}-3z^{8}a^{4}+11z^{6}a^{4}-12z^{4}a^{4}+4z^{2}a^{4}-z^{7}a^{3}+4z^{5}a^{3}-5z^{3}a^{3}+2za^{3}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

2

-2

-4

5

1

4

-6

7

3

-4

-8

9

4

5

-10

9

7

-2

-12

10

9

1

-14

7

9

2

-16

6

10

-4

-18

3

8

5

-20

1

5

-4

-22

3

-24

1

-1

Integral Khovanov Homology

(db, data source)

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