L10a100

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Planar diagram presentation	X_10,1,11,2 X_12,3,13,4 X_14,19,15,20 X_18,7,19,8 X_16,5,17,6 X_4,15,5,16 X_6,17,7,18 X_20,13,9,14 X_2,9,3,10 X_8,11,1,12
Gauss code	{1, -9, 2, -6, 5, -7, 4, -10}, {9, -1, 10, -2, 8, -3, 6, -5, 7, -4, 3, -8}

A Braid Representative

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Polynomial invariants

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

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

2

1

-1

-8

2

-10

3

2

-1

-12

5

2

3

-14

3

0

-16

4

5

-1

-18

2

3

1

-20

2

4

-2

-22

1

3

2

-24

1

-1

-26

1

Integral Khovanov Homology

(db, data source)

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