L11a519

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,6,17,5 X_22,20,13,19 X_18,10,19,9 X_10,21,11,22 X_20,11,21,12 X_2,13,3,14 X_14,3,15,4 X₄₇₅₈ X_12,16,7,15 X_6,18,1,17
Gauss code	{1, -7, 8, -9, 2, -11}, {9, -1, 4, -5, 6, -10}, {7, -8, 10, -2, 11, -4, 3, -6, 5, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(2)-1)^{2}(t(3)-1)(t(3)t(1)-t(1)+1)(t(1)t(3)-t(3)+1)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-4q^{-5}+4q^{4}+9q^{-4}-9q^{3}-14q^{-3}+15q^{2}+21q^{-2}-20q-22q^{-1}+24$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+2a^{4}z^{2}+a^{4}z^{-2}+a^{4}-2a^{2}z^{6}-z^{6}a^{-2}-7a^{2}z^{4}-3z^{4}a^{-2}-8a^{2}z^{2}-3z^{2}a^{-2}-2a^{2}z^{-2}-4a^{2}-a^{-2}+z^{8}+5z^{6}+10z^{4}+9z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$2a^{2}z^{10}+2z^{10}+6a^{3}z^{9}+13az^{9}+7z^{9}a^{-1}+7a^{4}z^{8}+14a^{2}z^{8}+10z^{8}a^{-2}+17z^{8}+4a^{5}z^{7}-6a^{3}z^{7}-20az^{7}-2z^{7}a^{-1}+8z^{7}a^{-3}+a^{6}z^{6}-16a^{4}z^{6}-44a^{2}z^{6}-16z^{6}a^{-2}+4z^{6}a^{-4}-47z^{6}-9a^{5}z^{5}-7a^{3}z^{5}+3az^{5}-12z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-2a^{6}z^{4}+12a^{4}z^{4}+45a^{2}z^{4}+12z^{4}a^{-2}-5z^{4}a^{-4}+48z^{4}+5a^{5}z^{3}+6a^{3}z^{3}+6az^{3}+12z^{3}a^{-1}+6z^{3}a^{-3}-z^{3}a^{-5}+a^{6}z^{2}-6a^{4}z^{2}-25a^{2}z^{2}-6z^{2}a^{-2}+z^{2}a^{-4}-25z^{2}-2a^{3}z-4az-3za^{-1}-za^{-3}+3a^{4}+7a^{2}+2a^{-2}+7+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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

3

7

6

1

-5

5

9

3

6

3

11

6

-5

1

13

9

4

-1

11

13

2

-3

10

11

-1

-5

6

13

7

-7

3

8

-5

-9

1

6

5

-11

3

-3

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$