L11n11

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,1,18,4 X_5,10,6,11 X₃₈₄₉ X_13,22,14,5 X_11,20,12,21 X_21,12,22,13 X_19,14,20,15 X_9,18,10,19 X_2,16,3,15
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -10, 4, -7, 8, -6, 9, 11, -2, -3, 10, -9, 7, -8, 6}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {6}{q^{13/2}}}-{\frac {5}{q^{15/2}}}+{\frac {4}{q^{17/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)-2a^{9}z-a^{9}z^{-1}+a^{7}z^{5}+3a^{7}z^{3}+3a^{7}z+a^{7}z^{-1}+a^{5}z^{5}+3a^{5}z^{3}+4a^{5}z+2a^{5}z^{-1}-2a^{3}z^{3}-5a^{3}z-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{6}-4a^{12}z^{4}+4a^{12}z^{2}-a^{12}+2a^{11}z^{7}-7a^{11}z^{5}+5a^{11}z^{3}-a^{11}z+2a^{10}z^{8}-6a^{10}z^{6}+3a^{10}z^{4}-a^{10}z^{2}+a^{9}z^{9}-a^{9}z^{7}-4a^{9}z^{5}+3a^{9}z^{3}-a^{9}z^{-1}+4a^{8}z^{8}-15a^{8}z^{6}+20a^{8}z^{4}-12a^{8}z^{2}+3a^{8}+a^{7}z^{9}-a^{7}z^{7}-4a^{7}z^{5}+8a^{7}z^{3}-a^{7}z^{-1}+2a^{6}z^{8}-7a^{6}z^{6}+11a^{6}z^{4}-3a^{6}z^{2}+2a^{5}z^{7}-7a^{5}z^{5}+13a^{5}z^{3}-8a^{5}z+2a^{5}z^{-1}+a^{4}z^{6}-2a^{4}z^{4}+4a^{4}z^{2}-3a^{4}+3a^{3}z^{3}-7a^{3}z+2a^{3}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

2

-4

2

0

-6

3

-8

3

2

-1

-10

4

3

1

-12

2

3

1

-14

3

4

-1

-16

1

2

1

-18

1

3

-2

-20

1

-22

1

-1

Integral Khovanov Homology

(db, data source)

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