L11a249

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(u^{2}v^{4}-2u^{2}v^{3}+u^{2}v^{2}-uv^{4}+3uv^{3}-6uv^{2}+3uv-u+v^{2}-2v+1\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+5q^{11/2}-11q^{9/2}+18q^{7/2}-25q^{5/2}+28q^{3/2}-29{\sqrt {q}}+{\frac {24}{\sqrt {q}}}-{\frac {18}{q^{3/2}}}+{\frac {11}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{9}a^{-1}-az^{7}+5z^{7}a^{-1}-z^{7}a^{-3}-3az^{5}+7z^{5}a^{-1}-3z^{5}a^{-3}-az^{3}-z^{3}a^{-3}+2az-4za^{-1}+2za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}+5z^{6}a^{-6}-4z^{4}a^{-6}+11z^{7}a^{-5}-14z^{5}a^{-5}+4z^{3}a^{-5}+14z^{8}a^{-4}+a^{4}z^{6}-18z^{6}a^{-4}-a^{4}z^{4}+4z^{4}a^{-4}+z^{2}a^{-4}+11z^{9}a^{-3}+5a^{3}z^{7}-6z^{7}a^{-3}-9a^{3}z^{5}-14z^{5}a^{-3}+3a^{3}z^{3}+13z^{3}a^{-3}-4za^{-3}+4z^{10}a^{-2}+10a^{2}z^{8}+18z^{8}a^{-2}-21a^{2}z^{6}-48z^{6}a^{-2}+11a^{2}z^{4}+30z^{4}a^{-2}-a^{2}z^{2}-3z^{2}a^{-2}+10az^{9}+21z^{9}a^{-1}-15az^{7}-37z^{7}a^{-1}-2az^{5}+8z^{5}a^{-1}+9az^{3}+15z^{3}a^{-1}-4az-8za^{-1}+az^{-1}+a^{-1}z^{-1}+4z^{10}+14z^{8}-47z^{6}+34z^{4}-5z^{2}-1$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

4

-4

10

7

1

6

8

11

4

-7

6

14

7

4

14

11

-3

2

15

14

1

0

11

16

5

-2

7

13

-6

-4

4

11

7

-6

1

7

-6

-8

4

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{15}$
$r=1$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$