L11n385

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,12,6,13 X₃₈₄₉ X_9,16,10,17 X_17,19,18,22 X_11,20,12,21 X_19,10,20,11 X_21,5,22,18 X_13,2,14,3
Gauss code	{1, 11, -5, -3}, {-9, 8, -10, 7}, {-4, -1, 2, 5, -6, 9, -8, 4, -11, -2, 3, 6, -7, 10}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(v-1)(w-1)\left(uw^{4}-1\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{-1}-q^{-2}+3q^{-3}-q^{-4}+3q^{-5}-2q^{-6}+2q^{-7}-q^{-8}+q^{-9}-q^{-10}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-z^{2}a^{10}-a^{10}z^{-2}-2a^{10}+z^{6}a^{8}+6z^{4}a^{8}+11z^{2}a^{8}+4a^{8}z^{-2}+10a^{8}-z^{8}a^{6}-7z^{6}a^{6}-17z^{4}a^{6}-21z^{2}a^{6}-5a^{6}z^{-2}-16a^{6}+z^{6}a^{4}+6z^{4}a^{4}+11z^{2}a^{4}+2a^{4}z^{-2}+8a^{4}$ (db)
Kauffman polynomial	$a^{13}z+a^{12}z^{2}+a^{11}z^{3}-2a^{11}z+a^{11}z^{-1}+2a^{10}z^{4}-7a^{10}z^{2}-a^{10}z^{-2}+4a^{10}+2a^{9}z^{7}-11a^{9}z^{5}+19a^{9}z^{3}-17a^{9}z+5a^{9}z^{-1}+3a^{8}z^{8}-19a^{8}z^{6}+39a^{8}z^{4}-36a^{8}z^{2}-4a^{8}z^{-2}+17a^{8}+a^{7}z^{9}-3a^{7}z^{7}-8a^{7}z^{5}+30a^{7}z^{3}-29a^{7}z+9a^{7}z^{-1}+4a^{6}z^{8}-26a^{6}z^{6}+54a^{6}z^{4}-47a^{6}z^{2}-5a^{6}z^{-2}+22a^{6}+a^{5}z^{9}-5a^{5}z^{7}+3a^{5}z^{5}+12a^{5}z^{3}-15a^{5}z+5a^{5}z^{-1}+a^{4}z^{8}-7a^{4}z^{6}+17a^{4}z^{4}-19a^{4}z^{2}-2a^{4}z^{-2}+10a^{4}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

-1

1

-3

0

-5

3

1

2

-7

1

2

-9

3

1

2

-11

2

1

-13

1

4

3

0

-15

1

-17

2

0

-19

1

0

-21

1

-1

Integral Khovanov Homology

(db, data source)

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