L11a269

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_4,9,5,10 X_14,5,15,6 X_22,20,9,19 X_18,7,19,8 X_6,17,7,18 X_16,22,17,21 X_20,16,21,15 X_8,13,1,14
Gauss code	{1, -2, 3, -4, 5, -8, 7, -11}, {4, -1, 2, -3, 11, -5, 10, -9, 8, -7, 6, -10, 9, -6}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{5}+t(1)^{3}t(2)^{4}-4t(1)^{2}t(2)^{4}+3t(1)t(2)^{4}-2t(1)^{3}t(2)^{3}+8t(1)^{2}t(2)^{3}-7t(1)t(2)^{3}+2t(2)^{3}+2t(1)^{3}t(2)^{2}-7t(1)^{2}t(2)^{2}+8t(1)t(2)^{2}-2t(2)^{2}+3t(1)^{2}t(2)-4t(1)t(2)+t(2)+t(1)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {11}{q^{3/2}}}+{\frac {15}{q^{5/2}}}-{\frac {18}{q^{7/2}}}+{\frac {18}{q^{9/2}}}-{\frac {16}{q^{11/2}}}+{\frac {12}{q^{13/2}}}-{\frac {8}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{5}+3a^{7}z^{3}+4a^{7}z+2a^{7}z^{-1}-a^{5}z^{7}-4a^{5}z^{5}-7a^{5}z^{3}-7a^{5}z-3a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-6a^{3}z^{3}-3a^{3}z+a^{3}z^{-1}+az^{5}+3az^{3}+2az$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-4a^{10}z^{4}+a^{10}z^{2}+6a^{9}z^{7}-11a^{9}z^{5}+10a^{9}z^{3}-5a^{9}z+6a^{8}z^{8}-6a^{8}z^{6}-a^{8}z^{4}+2a^{8}z^{2}+4a^{7}z^{9}-3a^{7}z^{5}-4a^{7}z^{3}+6a^{7}z-2a^{7}z^{-1}+a^{6}z^{10}+10a^{6}z^{8}-21a^{6}z^{6}+17a^{6}z^{4}-9a^{6}z^{2}+3a^{6}+7a^{5}z^{9}-10a^{5}z^{7}+9a^{5}z^{5}-15a^{5}z^{3}+12a^{5}z-3a^{5}z^{-1}+a^{4}z^{10}+8a^{4}z^{8}-22a^{4}z^{6}+22a^{4}z^{4}-13a^{4}z^{2}+3a^{4}+3a^{3}z^{9}-a^{3}z^{7}-9a^{3}z^{5}+9a^{3}z^{3}-2a^{3}z-a^{3}z^{-1}+4a^{2}z^{8}-9a^{2}z^{6}+5a^{2}z^{4}-a^{2}z^{2}+a^{2}+3az^{7}-9az^{5}+8az^{3}-2az+z^{6}-3z^{4}+2z^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

0

4

1

-3

-2

7

2

5

-4

9

5

-4

-6

9

6

3

-8

9

0

-10

7

9

-2

-12

5

9

4

-14

3

7

-4

-16

5

-18

1

3

-2

-20

1

Integral Khovanov Homology

(db, data source)

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