L11a230

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}v^{4}-5u^{2}v^{3}+6u^{2}v^{2}-2u^{2}v-uv^{4}+7uv^{3}-11uv^{2}+7uv-u-2v^{3}+6v^{2}-5v+2}{uv^{2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {7}{\sqrt {q}}}-{\frac {12}{q^{3/2}}}+{\frac {15}{q^{5/2}}}-{\frac {19}{q^{7/2}}}+{\frac {18}{q^{9/2}}}-{\frac {16}{q^{11/2}}}+{\frac {12}{q^{13/2}}}-{\frac {7}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+3z^{3}a^{7}+3za^{7}-z^{7}a^{5}-4z^{5}a^{5}-6z^{3}a^{5}-2za^{5}+2a^{5}z^{-1}-z^{7}a^{3}-4z^{5}a^{3}-7z^{3}a^{3}-7za^{3}-3a^{3}z^{-1}+z^{5}a+3z^{3}a+3za+az^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+5z^{4}a^{10}-2z^{2}a^{10}-5z^{7}a^{9}+7z^{5}a^{9}-3z^{3}a^{9}+za^{9}-6z^{8}a^{8}+9z^{6}a^{8}-8z^{4}a^{8}+4z^{2}a^{8}-4z^{9}a^{7}+6z^{5}a^{7}-4z^{3}a^{7}-za^{7}-z^{10}a^{6}-11z^{8}a^{6}+26z^{6}a^{6}-24z^{4}a^{6}+9z^{2}a^{6}-7z^{9}a^{5}+6z^{7}a^{5}+8z^{5}a^{5}-10z^{3}a^{5}+4za^{5}-2a^{5}z^{-1}-z^{10}a^{4}-9z^{8}a^{4}+21z^{6}a^{4}-10z^{4}a^{4}-2z^{2}a^{4}+3a^{4}-3z^{9}a^{3}-2z^{7}a^{3}+18z^{5}a^{3}-18z^{3}a^{3}+10za^{3}-3a^{3}z^{-1}-4z^{8}a^{2}+6z^{6}a^{2}+4z^{4}a^{2}-8z^{2}a^{2}+3a^{2}-3z^{7}a+8z^{5}a-7z^{3}a+3za-az^{-1}-z^{6}+3z^{4}-3z^{2}+1$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

0

5

1

-4

-2

7

2

5

-4

9

6

-3

-6

10

6

4

-8

8

9

1

-10

8

10

-2

-12

4

8

4

-14

3

8

-5

-16

1

5

4

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

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