L11n318

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,5,15,6 X_20,11,21,12 X_22,17,11,18 X_16,21,17,22 X_10,13,5,14 X_19,8,20,9 X_7,18,8,19 X_2,9,3,10 X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, -9, 8, 10, -7}, {4, -2, 7, -3, 11, -6, 5, 9, -8, -4, 6, -5}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{2}w^{3}-2uv^{2}w^{2}+uv^{2}w+uvw^{4}-3uvw^{3}+5uvw^{2}-3uvw+uw^{3}-2uw^{2}+2uw-2v^{2}w^{3}+2v^{2}w^{2}-v^{2}w+3vw^{3}-5vw^{2}+3vw-v-w^{3}+2w^{2}-w}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{-11}+4q^{-10}-8q^{-9}+11q^{-8}-14q^{-7}+15q^{-6}-12q^{-5}+11q^{-4}-5q^{-3}+3q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-z^{2}a^{10}-a^{10}z^{-2}-a^{10}+3z^{4}a^{8}+8z^{2}a^{8}+4a^{8}z^{-2}+8a^{8}-2z^{6}a^{6}-9z^{4}a^{6}-17z^{2}a^{6}-5a^{6}z^{-2}-15a^{6}+3z^{4}a^{4}+9z^{2}a^{4}+2a^{4}z^{-2}+8a^{4}$ (db)
Kauffman polynomial	$a^{13}z^{5}-a^{13}z^{3}+4a^{12}z^{6}-6a^{12}z^{4}+a^{12}z^{2}+7a^{11}z^{7}-13a^{11}z^{5}+6a^{11}z^{3}-2a^{11}z+a^{11}z^{-1}+6a^{10}z^{8}-8a^{10}z^{6}+2a^{10}z^{4}-3a^{10}z^{2}-a^{10}z^{-2}+2a^{10}+2a^{9}z^{9}+9a^{9}z^{7}-28a^{9}z^{5}+27a^{9}z^{3}-13a^{9}z+5a^{9}z^{-1}+11a^{8}z^{8}-27a^{8}z^{6}+35a^{8}z^{4}-27a^{8}z^{2}-4a^{8}z^{-2}+13a^{8}+2a^{7}z^{9}+5a^{7}z^{7}-20a^{7}z^{5}+33a^{7}z^{3}-27a^{7}z+9a^{7}z^{-1}+5a^{6}z^{8}-15a^{6}z^{6}+33a^{6}z^{4}-37a^{6}z^{2}-5a^{6}z^{-2}+20a^{6}+3a^{5}z^{7}-6a^{5}z^{5}+13a^{5}z^{3}-16a^{5}z+5a^{5}z^{-1}+6a^{4}z^{4}-14a^{4}z^{2}-2a^{4}z^{-2}+10a^{4}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

3

-5

4

2

-2

-7

7

1

6

-9

6

5

-1

-11

9

6

3

-13

5

6

1

-15

6

9

-3

-17

3

6

3

-19

1

5

-4

-21

3

-23

1

-1

Integral Khovanov Homology

(db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	$i=-5$	$i=-3$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-6$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-5$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$