L11a338

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{3}v^{3}-3u^{3}v^{2}+u^{3}v-4u^{2}v^{3}+12u^{2}v^{2}-9u^{2}v+2u^{2}+2uv^{3}-9uv^{2}+12uv-4u+v^{2}-3v+2}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	${\frac {21}{q^{9/2}}}-{\frac {22}{q^{7/2}}}+{\frac {18}{q^{5/2}}}+q^{3/2}-{\frac {15}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {7}{q^{15/2}}}+{\frac {13}{q^{13/2}}}-{\frac {18}{q^{11/2}}}-4{\sqrt {q}}+{\frac {9}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+3z^{3}a^{7}+3za^{7}-z^{7}a^{5}-4z^{5}a^{5}-7z^{3}a^{5}-4za^{5}+a^{5}z^{-1}-z^{7}a^{3}-3z^{5}a^{3}-3z^{3}a^{3}-2za^{3}-a^{3}z^{-1}+z^{5}a+2z^{3}a+za$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-5a^{10}z^{4}+2a^{10}z^{2}+5a^{9}z^{7}-6a^{9}z^{5}+a^{9}z^{3}+7a^{8}z^{8}-12a^{8}z^{6}+13a^{8}z^{4}-7a^{8}z^{2}+6a^{7}z^{9}-7a^{7}z^{7}+4a^{7}z^{5}+a^{7}z+2a^{6}z^{10}+12a^{6}z^{8}-38a^{6}z^{6}+42a^{6}z^{4}-15a^{6}z^{2}+12a^{5}z^{9}-22a^{5}z^{7}+9a^{5}z^{5}+2a^{5}z^{3}+a^{5}z^{-1}+2a^{4}z^{10}+12a^{4}z^{8}-39a^{4}z^{6}+33a^{4}z^{4}-9a^{4}z^{2}-a^{4}+6a^{3}z^{9}-6a^{3}z^{7}-11a^{3}z^{5}+11a^{3}z^{3}-4a^{3}z+a^{3}z^{-1}+7a^{2}z^{8}-15a^{2}z^{6}+7a^{2}z^{4}-2a^{2}z^{2}+4az^{7}-9az^{5}+6az^{3}-2az+z^{6}-2z^{4}+z^{2}$ (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

3

0

6

1

-5

-2

9

3

6

-4

10

7

-3

-6

12

8

4

-8

9

10

1

-10

9

12

-3

-12

5

10

5

-14

2

8

-6

-16

1

5

4

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-5$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=-3$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$