L11n416

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Planar diagram presentation	X₈₁₉₂ X_14,5,15,6 X_10,3,11,4 X_4,13,5,14 X₂₇₃₈ X_6,9,1,10 X_18,12,19,11 X_12,18,7,17 X_15,20,16,21 X_19,22,20,13 X_21,16,22,17
Gauss code	{1, -5, 3, -4, 2, -6}, {5, -1, 6, -3, 7, -8}, {4, -2, -9, 11, 8, -7, -10, 9, -11, 10}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}vw^{3}-u^{2}vw^{2}+u^{2}vw-u^{2}v-u^{2}w^{3}+uv^{2}w^{3}-uv^{2}w^{2}+uv^{2}w-uv^{2}-2uvw^{3}+2uvw^{2}-2uvw+2uv+uw^{3}-uw^{2}+uw-u+v^{2}+vw^{3}-vw^{2}+vw-v}{uvw^{3/2}}}$ (db)
Jones polynomial	$-2q^{-9}+4q^{-8}-6q^{-7}+9q^{-6}-8q^{-5}+9q^{-4}-6q^{-3}+5q^{-2}-2q^{-1}+1$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+a^{8}z^{4}+4a^{8}z^{2}+4a^{8}z^{-2}+6a^{8}-a^{6}z^{6}-4a^{6}z^{4}-7a^{6}z^{2}-5a^{6}z^{-2}-9a^{6}-a^{4}z^{6}-3a^{4}z^{4}-a^{4}z^{2}+2a^{4}z^{-2}+2a^{4}+a^{2}z^{4}+3a^{2}z^{2}+2a^{2}$ (db)
Kauffman polynomial	$3a^{11}z^{3}-5a^{11}z+a^{11}z^{-1}+a^{10}z^{6}+2a^{10}z^{4}-5a^{10}z^{2}-a^{10}z^{-2}+4a^{10}+4a^{9}z^{7}-15a^{9}z^{5}+30a^{9}z^{3}-21a^{9}z+5a^{9}z^{-1}+4a^{8}z^{8}-16a^{8}z^{6}+34a^{8}z^{4}-32a^{8}z^{2}-4a^{8}z^{-2}+17a^{8}+a^{7}z^{9}+4a^{7}z^{7}-23a^{7}z^{5}+43a^{7}z^{3}-33a^{7}z+9a^{7}z^{-1}+6a^{6}z^{8}-22a^{6}z^{6}+36a^{6}z^{4}-36a^{6}z^{2}-5a^{6}z^{-2}+20a^{6}+a^{5}z^{9}+2a^{5}z^{7}-14a^{5}z^{5}+19a^{5}z^{3}-16a^{5}z+5a^{5}z^{-1}+2a^{4}z^{8}-4a^{4}z^{6}-4a^{4}z^{2}-2a^{4}z^{-2}+6a^{4}+2a^{3}z^{7}-6a^{3}z^{5}+3a^{3}z^{3}+a^{3}z+a^{2}z^{6}-4a^{2}z^{4}+5a^{2}z^{2}-2a^{2}$ (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

χ

1

-1

1

-1

-3

4

1

3

-5

4

3

-1

-7

5

2

3

-9

4

5

1

-11

5

4

1

-13

1

4

3

-15

3

5

-2

-17

2

-19

2

-2

Integral Khovanov Homology

(db, data source)

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