L11n169

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,5,15,6 X_15,20,16,21 X_11,18,12,19 X_19,12,20,13 X_17,22,18,7 X_21,16,22,17 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, -6, 7, 11, -4, -5, 9, -8, 6, -7, 5, -9, 8}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{6}-u^{2}v^{4}+u^{2}v^{3}+2uv^{4}-3uv^{3}+2uv^{2}+v^{3}-v^{2}+1}{uv^{3}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}+{\frac {1}{q^{29/2}}}-{\frac {2}{q^{27/2}}}+{\frac {3}{q^{25/2}}}-{\frac {4}{q^{23/2}}}+{\frac {4}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{15/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-2z^{3}a^{13}-5za^{13}-2a^{13}z^{-1}+z^{7}a^{11}+9z^{5}a^{11}+24z^{3}a^{11}+22za^{11}+5a^{11}z^{-1}-z^{9}a^{9}-9z^{7}a^{9}-28z^{5}a^{9}-37z^{3}a^{9}-20za^{9}-3a^{9}z^{-1}$ (db)
Kauffman polynomial	$a^{18}z^{4}-2a^{18}z^{2}+2a^{17}z^{5}-4a^{17}z^{3}+a^{17}z+2a^{16}z^{6}-4a^{16}z^{4}+3a^{16}z^{2}-a^{16}+a^{15}z^{7}-a^{15}z^{5}+a^{15}z^{3}+2a^{14}z^{6}-3a^{14}z^{4}+3a^{14}z^{2}+2a^{13}z^{5}-3a^{13}z^{3}+5a^{13}z-2a^{13}z^{-1}+a^{12}z^{8}-9a^{12}z^{6}+26a^{12}z^{4}-24a^{12}z^{2}+5a^{12}+a^{11}z^{9}-10a^{11}z^{7}+33a^{11}z^{5}-45a^{11}z^{3}+26a^{11}z-5a^{11}z^{-1}+a^{10}z^{8}-9a^{10}z^{6}+24a^{10}z^{4}-22a^{10}z^{2}+5a^{10}+a^{9}z^{9}-9a^{9}z^{7}+28a^{9}z^{5}-37a^{9}z^{3}+20a^{9}z-3a^{9}z^{-1}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-8

1

-10

1

-12

1

-14

2

-16

1

-1

-18

3

2

1

-20

2

1

-1

-22

2

0

-24

1

2

1

-26

1

2

-1

-28

1

-30

1

-1

Integral Khovanov Homology

(db, data source)

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