L11a12

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Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_14,12,15,11 X₈₄₉₃ X_12,5,13,6 X_22,13,5,14 X_18,15,19,16 X_16,9,17,10 X_10,17,11,18 X_2,20,3,19
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 9, -10, 4, -6, 7, -4, 8, -9, 10, -8, 11, -2, 3, -7}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{4}-5v^{3}+7v^{2}-5v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-5{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {17}{q^{3/2}}}+{\frac {21}{q^{5/2}}}-{\frac {25}{q^{7/2}}}+{\frac {24}{q^{9/2}}}-{\frac {21}{q^{11/2}}}+{\frac {15}{q^{13/2}}}-{\frac {8}{q^{15/2}}}+{\frac {4}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z-3a^{7}z^{3}-4a^{7}z-a^{7}z^{-1}+3a^{5}z^{5}+7a^{5}z^{3}+6a^{5}z+2a^{5}z^{-1}-a^{3}z^{7}-3a^{3}z^{5}-4a^{3}z^{3}-2a^{3}z+az^{5}+az^{3}-az-az^{-1}$ (db)
Kauffman polynomial	$a^{11}z^{5}-a^{11}z^{3}+4a^{10}z^{6}-6a^{10}z^{4}+3a^{10}z^{2}+7a^{9}z^{7}-9a^{9}z^{5}+5a^{9}z^{3}-2a^{9}z+8a^{8}z^{8}-5a^{8}z^{6}-7a^{8}z^{4}+9a^{8}z^{2}-2a^{8}+6a^{7}z^{9}+5a^{7}z^{7}-25a^{7}z^{5}+24a^{7}z^{3}-8a^{7}z+a^{7}z^{-1}+2a^{6}z^{10}+17a^{6}z^{8}-36a^{6}z^{6}+15a^{6}z^{4}+8a^{6}z^{2}-5a^{6}+13a^{5}z^{9}-10a^{5}z^{7}-26a^{5}z^{5}+31a^{5}z^{3}-11a^{5}z+2a^{5}z^{-1}+2a^{4}z^{10}+18a^{4}z^{8}-47a^{4}z^{6}+27a^{4}z^{4}-3a^{4}+7a^{3}z^{9}-3a^{3}z^{7}-21a^{3}z^{5}+18a^{3}z^{3}-4a^{3}z+9a^{2}z^{8}-19a^{2}z^{6}+10a^{2}z^{4}-2a^{2}z^{2}+a^{2}+5az^{7}-10az^{5}+5az^{3}+az-az^{-1}+z^{6}-z^{4}$ (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

4

0

6

1

-5

-2

11

4

7

-4

12

8

-4

-6

13

9

4

-8

11

12

1

-10

10

13

-3

-12

5

11

6

-14

3

10

-7

-16

1

5

4

-18

3

-3

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-3$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$