L11n154

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_7,16,8,17 X_13,20,14,21 X_15,22,16,7 X_19,1,20,6 X_18,11,19,12 X_5,12,6,13 X_21,14,22,15 X_4,18,5,17
Gauss code	{1, -2, 3, -11, -9, 7}, {-4, -1, 2, -3, 8, 9, -5, 10, -6, 4, 11, -8, -7, 5, -10, 6}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{4}-u^{2}v^{3}+u^{2}v^{2}-u^{2}v-2uv^{4}+3uv^{3}-3uv^{2}+3uv-2u-v^{3}+v^{2}-v+1}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {5}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {2}{q^{5/2}}}+{\frac {1}{q^{23/2}}}-{\frac {2}{q^{21/2}}}+{\frac {4}{q^{19/2}}}-{\frac {6}{q^{17/2}}}+{\frac {7}{q^{15/2}}}-{\frac {7}{q^{13/2}}}+{\frac {6}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{5}\right)-4a^{9}z^{3}-5a^{9}z-2a^{9}z^{-1}+a^{7}z^{7}+6a^{7}z^{5}+14a^{7}z^{3}+15a^{7}z+5a^{7}z^{-1}-2a^{5}z^{5}-9a^{5}z^{3}-11a^{5}z-3a^{5}z^{-1}$ (db)
Kauffman polynomial	$a^{14}z^{4}-2a^{14}z^{2}+2a^{13}z^{5}-3a^{13}z^{3}+3a^{12}z^{6}-6a^{12}z^{4}+5a^{12}z^{2}-a^{12}+3a^{11}z^{7}-7a^{11}z^{5}+9a^{11}z^{3}-2a^{11}z+2a^{10}z^{8}-4a^{10}z^{6}+5a^{10}z^{4}+a^{9}z^{9}-2a^{9}z^{7}+5a^{9}z^{5}-6a^{9}z^{3}+5a^{9}z-2a^{9}z^{-1}+3a^{8}z^{8}-11a^{8}z^{6}+21a^{8}z^{4}-20a^{8}z^{2}+5a^{8}+a^{7}z^{9}-5a^{7}z^{7}+17a^{7}z^{5}-30a^{7}z^{3}+20a^{7}z-5a^{7}z^{-1}+a^{6}z^{8}-4a^{6}z^{6}+9a^{6}z^{4}-13a^{6}z^{2}+5a^{6}+3a^{5}z^{5}-12a^{5}z^{3}+13a^{5}z-3a^{5}z^{-1}$ (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

-4

2

-6

1

0

-8

4

1

3

-10

3

2

-1

-12

4

3

1

-14

3

0

-16

3

4

-1

-18

1

3

2

-20

1

3

-2

-22

1

-24

1

-1

Integral Khovanov Homology

(db, data source)

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