L11a63

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,8,15,7 X_22,16,5,15 X_16,11,17,12 X_20,18,21,17 X_18,10,19,9 X_10,20,11,19 X_8,21,9,22 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 7, -8, 5, -2, 11, -3, 4, -5, 6, -7, 8, -6, 9, -4}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-3t(2)+1\right)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{17/2}-4q^{15/2}+9q^{13/2}-13q^{11/2}+17q^{9/2}-20q^{7/2}+18q^{5/2}-16q^{3/2}+11{\sqrt {q}}-{\frac {7}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+za^{-7}+a^{-7}z^{-1}-2z^{5}a^{-5}-5z^{3}a^{-5}-5za^{-5}-3a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+8z^{3}a^{-3}+8za^{-3}+3a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+2az-6za^{-1}+az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}+4z^{5}a^{-9}-z^{3}a^{-9}+9z^{6}a^{-8}-9z^{4}a^{-8}+4z^{2}a^{-8}-a^{-8}+12z^{7}a^{-7}-16z^{5}a^{-7}+9z^{3}a^{-7}-3za^{-7}+a^{-7}z^{-1}+10z^{8}a^{-6}-8z^{6}a^{-6}-7z^{4}a^{-6}+5z^{2}a^{-6}-2a^{-6}+5z^{9}a^{-5}+9z^{7}a^{-5}-39z^{5}a^{-5}+31z^{3}a^{-5}-12za^{-5}+3a^{-5}z^{-1}+z^{10}a^{-4}+17z^{8}a^{-4}-46z^{6}a^{-4}+31z^{4}a^{-4}-7z^{2}a^{-4}+8z^{9}a^{-3}-8z^{7}a^{-3}-28z^{5}a^{-3}+41z^{3}a^{-3}-18za^{-3}+3a^{-3}z^{-1}+z^{10}a^{-2}+10z^{8}a^{-2}-40z^{6}a^{-2}+41z^{4}a^{-2}-13z^{2}a^{-2}+2a^{-2}+3z^{9}a^{-1}+az^{7}-4z^{7}a^{-1}-4az^{5}-13z^{5}a^{-1}+6az^{3}+26z^{3}a^{-1}-4az-13za^{-1}+az^{-1}+2a^{-1}z^{-1}+3z^{8}-11z^{6}+13z^{4}-5z^{2}$ (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

7

χ

18

1

-1

16

3

14

6

1

-5

12

7

3

4

10

6

-4

8

10

7

3

6

8

10

2

4

8

10

-2

2

5

10

5

0

2

6

-4

-2

1

5

4

-4

2

-2

-6

1

Integral Khovanov Homology

(db, data source)

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