L11a270

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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_18,8,19,7 X_20,15,21,16 X_22,17,9,18 X_16,21,17,22 X_6,20,7,19 X_8,13,1,14
Gauss code	{1, -2, 3, -4, 5, -10, 6, -11}, {4, -1, 2, -3, 11, -5, 7, -9, 8, -6, 10, -7, 9, -8}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}v^{4}-u^{3}v^{3}+u^{3}v^{2}-u^{3}v+u^{2}v^{5}-4u^{2}v^{4}+5u^{2}v^{3}-5u^{2}v^{2}+3u^{2}v-u^{2}-uv^{5}+3uv^{4}-5uv^{3}+5uv^{2}-4uv+u-v^{4}+v^{3}-v^{2}+v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {5}{q^{5/2}}}+{\frac {8}{q^{7/2}}}-{\frac {12}{q^{9/2}}}+{\frac {14}{q^{11/2}}}-{\frac {15}{q^{13/2}}}+{\frac {13}{q^{15/2}}}-{\frac {11}{q^{17/2}}}+{\frac {7}{q^{19/2}}}-{\frac {3}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{5}\right)-3a^{9}z^{3}-3a^{9}z-a^{9}z^{-1}+a^{7}z^{7}+4a^{7}z^{5}+6a^{7}z^{3}+6a^{7}z+3a^{7}z^{-1}+a^{5}z^{7}+4a^{5}z^{5}+4a^{5}z^{3}-a^{5}z-2a^{5}z^{-1}-a^{3}z^{5}-4a^{3}z^{3}-4a^{3}z$ (db)
Kauffman polynomial	$a^{14}z^{4}-a^{14}z^{2}+3a^{13}z^{5}-2a^{13}z^{3}+6a^{12}z^{6}-7a^{12}z^{4}+4a^{12}z^{2}+8a^{11}z^{7}-13a^{11}z^{5}+11a^{11}z^{3}-3a^{11}z+7a^{10}z^{8}-11a^{10}z^{6}+8a^{10}z^{4}-3a^{10}z^{2}+a^{10}+4a^{9}z^{9}-3a^{9}z^{7}-a^{9}z^{5}-3a^{9}z^{3}+2a^{9}z-a^{9}z^{-1}+a^{8}z^{10}+7a^{8}z^{8}-22a^{8}z^{6}+21a^{8}z^{4}-14a^{8}z^{2}+3a^{8}+6a^{7}z^{9}-17a^{7}z^{7}+21a^{7}z^{5}-24a^{7}z^{3}+14a^{7}z-3a^{7}z^{-1}+a^{6}z^{10}+2a^{6}z^{8}-13a^{6}z^{6}+14a^{6}z^{4}-8a^{6}z^{2}+3a^{6}+2a^{5}z^{9}-5a^{5}z^{7}+a^{5}z^{5}+5a^{5}z-2a^{5}z^{-1}+2a^{4}z^{8}-8a^{4}z^{6}+9a^{4}z^{4}-2a^{4}z^{2}+a^{3}z^{7}-5a^{3}z^{5}+8a^{3}z^{3}-4a^{3}z$ (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

1

2

χ

0

1

-2

1

-1

-4

4

1

3

-6

5

2

-3

-8

7

3

4

-10

7

5

-2

-12

8

7

1

-14

6

8

2

-16

5

7

-2

-18

2

6

4

-20

1

5

-4

-22

2

-24

1

-1

Integral Khovanov Homology

(db, data source)

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