L10a113

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Planar diagram presentation	X_12,1,13,2 X₈₄₉₃ X_18,14,19,13 X_20,8,11,7 X_6,20,7,19 X_4,15,5,16 X_14,5,15,6 X_16,10,17,9 X_2,11,3,12 X_10,18,1,17
Gauss code	{1, -9, 2, -6, 7, -5, 4, -2, 8, -10}, {9, -1, 3, -7, 6, -8, 10, -3, 5, -4}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)\left(t(1)^{2}-t(1)+1\right)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$10q^{9/2}-11q^{7/2}+11q^{5/2}-{\frac {1}{q^{5/2}}}-12q^{3/2}+{\frac {3}{q^{3/2}}}-q^{15/2}+3q^{13/2}-6q^{11/2}+8{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}+az^{3}-7z^{3}a^{-1}+10z^{3}a^{-3}-3z^{3}a^{-5}+2az-7za^{-1}+8za^{-3}-3za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{9}a^{-1}-2z^{9}a^{-3}-10z^{8}a^{-2}-7z^{8}a^{-4}-3z^{8}-az^{7}+z^{7}a^{-1}-9z^{7}a^{-3}-11z^{7}a^{-5}+32z^{6}a^{-2}+10z^{6}a^{-4}-10z^{6}a^{-6}+12z^{6}+4az^{5}+18z^{5}a^{-1}+44z^{5}a^{-3}+24z^{5}a^{-5}-6z^{5}a^{-7}-23z^{4}a^{-2}+10z^{4}a^{-4}+16z^{4}a^{-6}-3z^{4}a^{-8}-14z^{4}-6az^{3}-30z^{3}a^{-1}-44z^{3}a^{-3}-16z^{3}a^{-5}+3z^{3}a^{-7}-z^{3}a^{-9}-11z^{2}a^{-4}-8z^{2}a^{-6}+3z^{2}+4az+14za^{-1}+16za^{-3}+6za^{-5}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

16

1

14

2

-2

12

4

1

3

10

6

2

-4

8

5

4

1

6

0

4

6

5

1

2

4

8

4

0

2

4

-2

1

4

3

-4

2

-2

-6

1

Integral Khovanov Homology

(db, data source)

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