L11a260

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Link Presentations

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

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

Polynomial invariants

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

1

-1

-2

4

1

3

-4

5

2

-3

-6

6

3

-8

6

5

-1

-10

6

0

-12

5

7

2

-14

3

5

-2

-16

1

5

4

-18

1

3

-2

-20

1

-22

1

-1

Integral Khovanov Homology

(db, data source)

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