L11a297

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Link Presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{3}v^{2}-3u^{3}v+u^{3}+2u^{2}v^{3}-7u^{2}v^{2}+7u^{2}v-3u^{2}-3uv^{3}+7uv^{2}-7uv+2u+v^{3}-3v^{2}+2v}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	${\frac {16}{q^{9/2}}}-{\frac {14}{q^{7/2}}}+{\frac {10}{q^{5/2}}}-{\frac {7}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {6}{q^{17/2}}}-{\frac {10}{q^{15/2}}}+{\frac {13}{q^{13/2}}}-{\frac {16}{q^{11/2}}}-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{9}-za^{9}+z^{5}a^{7}+z^{3}a^{7}+2z^{5}a^{5}+4z^{3}a^{5}+3za^{5}+a^{5}z^{-1}+z^{5}a^{3}-3za^{3}-a^{3}z^{-1}-z^{3}a-za$ (db)
Kauffman polynomial	$a^{12}z^{6}-3a^{12}z^{4}+2a^{12}z^{2}+3a^{11}z^{7}-9a^{11}z^{5}+7a^{11}z^{3}-2a^{11}z+4a^{10}z^{8}-10a^{10}z^{6}+6a^{10}z^{4}-2a^{10}z^{2}+3a^{9}z^{9}-3a^{9}z^{7}-6a^{9}z^{5}+7a^{9}z^{3}-3a^{9}z+a^{8}z^{10}+6a^{8}z^{8}-19a^{8}z^{6}+17a^{8}z^{4}-5a^{8}z^{2}+6a^{7}z^{9}-9a^{7}z^{7}-a^{7}z^{5}+10a^{7}z^{3}-3a^{7}z+a^{6}z^{10}+7a^{6}z^{8}-18a^{6}z^{6}+16a^{6}z^{4}-2a^{6}z^{2}+3a^{5}z^{9}+2a^{5}z^{7}-14a^{5}z^{5}+18a^{5}z^{3}-8a^{5}z+a^{5}z^{-1}+5a^{4}z^{8}-7a^{4}z^{6}+3a^{4}z^{4}-a^{4}+5a^{3}z^{7}-9a^{3}z^{5}+6a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}+3a^{2}z^{6}-5a^{2}z^{4}+a^{2}z^{2}+az^{5}-2az^{3}+az$ (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

2

1

0

2

-2

5

1

4

-4

6

3

-3

-6

8

4

-8

8

6

-2

-10

8

0

-12

6

9

3

-14

4

7

-3

-16

2

6

4

-18

1

4

-3

-20

2

-22

1

-1

Integral Khovanov Homology

(db, data source)

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