L11a520

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,8,19,7 X_16,6,17,5 X_10,16,11,15 X_14,3,15,4 X_4,11,5,12 X_2,20,3,19 X_20,9,21,10 X_12,13,7,14 X_22,18,13,17 X_6,21,1,22
Gauss code	{1, -7, 5, -6, 3, -11}, {2, -1, 8, -4, 6, -9}, {9, -5, 4, -3, 10, -2, 7, -8, 11, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)\left(uvw^{2}-uvw+uv-uw^{2}+2uw-u-vw^{2}+2vw-v+w^{2}-w+1\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+5q^{4}-13q^{3}+23q^{2}-31q+37-36q^{-1}+33q^{-2}-23q^{-3}+15q^{-4}-6q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+a^{4}z^{-2}-2a^{2}z^{6}-z^{6}a^{-2}-4a^{2}z^{4}-2z^{4}a^{-2}-2a^{2}z^{2}-2z^{2}a^{-2}-2a^{2}z^{-2}-a^{2}+z^{8}+4z^{6}+7z^{4}+4z^{2}+z^{-2}+1$ (db)
Kauffman polynomial	$7a^{2}z^{10}+7z^{10}+17a^{3}z^{9}+36az^{9}+19z^{9}a^{-1}+15a^{4}z^{8}+22a^{2}z^{8}+21z^{8}a^{-2}+28z^{8}+6a^{5}z^{7}-27a^{3}z^{7}-67az^{7}-21z^{7}a^{-1}+13z^{7}a^{-3}+a^{6}z^{6}-27a^{4}z^{6}-72a^{2}z^{6}-32z^{6}a^{-2}+5z^{6}a^{-4}-81z^{6}-6a^{5}z^{5}+6a^{3}z^{5}+26az^{5}-13z^{5}a^{-3}+z^{5}a^{-5}+11a^{4}z^{4}+41a^{2}z^{4}+21z^{4}a^{-2}-2z^{4}a^{-4}+53z^{4}+a^{3}z^{3}-az^{3}+3z^{3}a^{-1}+5z^{3}a^{-3}+2a^{4}z^{2}-2a^{2}z^{2}-6z^{2}a^{-2}-10z^{2}-a^{3}z-az+a^{4}+a^{2}+1+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

4

7

9

1

-8

5

14

4

10

3

17

9

-8

1

20

14

6

-1

18

19

1

-3

15

18

-3

-5

10

20

10

-7

5

13

-8

-9

1

10

9

-11

5

-5

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{20}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{15}$
$r=-1$	${\mathbb {Z} }^{18}\oplus {\mathbb {Z} }_{2}^{18}$	${\mathbb {Z} }^{18}$
$r=0$	${\mathbb {Z} }^{19}\oplus {\mathbb {Z} }_{2}^{18}$	${\mathbb {Z} }^{20}$
$r=1$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{17}$	${\mathbb {Z} }^{17}$
$r=2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$