L11a522

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{2}w^{3}-2u^{2}v^{2}w^{2}+u^{2}v^{2}w-2u^{2}vw^{3}+4u^{2}vw^{2}-3u^{2}vw+u^{2}v+u^{2}w^{3}-2u^{2}w^{2}+u^{2}w-uv^{2}w^{3}+3uv^{2}w^{2}-4uv^{2}w+uv^{2}+3uvw^{3}-8uvw^{2}+8uvw-3uv-uw^{3}+4uw^{2}-3uw+u-v^{2}w^{2}+2v^{2}w-v^{2}-vw^{3}+3vw^{2}-4vw+2v-w^{2}+2w-1}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{6}-4q^{5}+10q^{4}-15q^{3}+22q^{2}-24q+25-21q^{-1}+16q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}+2z^{2}a^{-4}+a^{-4}z^{-2}+2a^{-4}-a^{2}z^{6}-2z^{6}a^{-2}-3a^{2}z^{4}-7z^{4}a^{-2}-3a^{2}z^{2}-9z^{2}a^{-2}-2a^{-2}z^{-2}-5a^{-2}+z^{8}+5z^{6}+10z^{4}+8z^{2}+z^{-2}+3$ (db)
Kauffman polynomial	$3z^{10}a^{-2}+3z^{10}+9az^{9}+17z^{9}a^{-1}+8z^{9}a^{-3}+11a^{2}z^{8}+13z^{8}a^{-2}+8z^{8}a^{-4}+16z^{8}+8a^{3}z^{7}-10az^{7}-36z^{7}a^{-1}-14z^{7}a^{-3}+4z^{7}a^{-5}+4a^{4}z^{6}-19a^{2}z^{6}-52z^{6}a^{-2}-19z^{6}a^{-4}+z^{6}a^{-6}-55z^{6}+a^{5}z^{5}-11a^{3}z^{5}-2az^{5}+21z^{5}a^{-1}+3z^{5}a^{-3}-8z^{5}a^{-5}-5a^{4}z^{4}+15a^{2}z^{4}+62z^{4}a^{-2}+16z^{4}a^{-4}-2z^{4}a^{-6}+64z^{4}-a^{5}z^{3}+4a^{3}z^{3}+9az^{3}+5z^{3}a^{-1}+4z^{3}a^{-3}+3z^{3}a^{-5}+a^{4}z^{2}-7a^{2}z^{2}-37z^{2}a^{-2}-12z^{2}a^{-4}+z^{2}a^{-6}-32z^{2}-a^{3}z-3az-8za^{-1}-6za^{-3}+a^{2}+12a^{-2}+6a^{-4}+8+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

13

1

11

3

-3

9

7

1

6

7

9

4

-5

5

13

6

7

3

12

10

-2

1

13

12

1

-1

9

13

4

-3

7

12

-5

3

10

7

-7

1

6

-5

-9

3

-11

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{13}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$