L11n175

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Planar diagram presentation	X₈₁₉₂ X_9,21,10,20 X_14,5,15,6 X_18,12,19,11 X_3,10,4,11 X_12,7,13,8 X_16,13,17,14 X_22,17,7,18 X_6,15,1,16 X_4,21,5,22 X_19,2,20,3
Gauss code	{1, 11, -5, -10, 3, -9}, {6, -1, -2, 5, 4, -6, 7, -3, 9, -7, 8, -4, -11, 2, 10, -8}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-3u^{2}v^{2}+3u^{2}v-u^{2}+2uv^{4}-5uv^{3}+5uv^{2}-5uv+2u-v^{4}+3v^{3}-3v^{2}}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {10}{q^{9/2}}}-{\frac {11}{q^{11/2}}}+{\frac {11}{q^{13/2}}}-{\frac {9}{q^{15/2}}}+{\frac {6}{q^{17/2}}}-{\frac {4}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)+a^{9}z^{-1}+a^{7}z^{5}+a^{7}z^{3}-a^{7}z-2a^{7}z^{-1}+a^{5}z^{5}+a^{5}z^{3}+a^{5}z+2a^{5}z^{-1}-2a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{6}-2a^{12}z^{4}+4a^{11}z^{7}-12a^{11}z^{5}+7a^{11}z^{3}+a^{11}z+5a^{10}z^{8}-15a^{10}z^{6}+11a^{10}z^{4}-3a^{10}z^{2}+2a^{9}z^{9}+2a^{9}z^{7}-19a^{9}z^{5}+17a^{9}z^{3}-5a^{9}z+a^{9}z^{-1}+9a^{8}z^{8}-27a^{8}z^{6}+27a^{8}z^{4}-11a^{8}z^{2}+2a^{7}z^{9}+a^{7}z^{7}-12a^{7}z^{5}+17a^{7}z^{3}-9a^{7}z+2a^{7}z^{-1}+4a^{6}z^{8}-10a^{6}z^{6}+16a^{6}z^{4}-9a^{6}z^{2}+a^{6}+3a^{5}z^{7}-5a^{5}z^{5}+10a^{5}z^{3}-7a^{5}z+2a^{5}z^{-1}+a^{4}z^{6}+2a^{4}z^{4}-a^{4}z^{2}+3a^{3}z^{3}-4a^{3}z+a^{3}z^{-1}$ (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

3

1

-2

-6

5

1

4

-8

5

3

-2

-10

6

5

1

-12

5

0

-14

4

6

-2

-16

3

6

3

-18

1

3

-2

-20

3

-22

1

-1

Integral Khovanov Homology

(db, data source)

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