L11a462

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_16,5,17,6 X_18,14,19,13 X_12,15,5,16 X_10,19,11,20 X_8,21,9,22 X_20,7,21,8 X_22,9,13,10 X_2,11,3,12 X_4,18,1,17
Gauss code	{1, -10, 2, -11}, {3, -1, 8, -7, 9, -6, 10, -5}, {4, -2, 5, -3, 11, -4, 6, -8, 7, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{3}t(2)^{3}-t(3)^{3}t(2)^{3}-2t(1)t(3)^{2}t(2)^{3}+2t(3)^{2}t(2)^{3}+t(1)t(3)t(2)^{3}-2t(1)t(3)^{3}t(2)^{2}+2t(3)^{3}t(2)^{2}+6t(1)t(3)^{2}t(2)^{2}-6t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-5t(1)t(3)t(2)^{2}+4t(3)t(2)^{2}-t(3)^{3}t(2)-4t(1)t(3)^{2}t(2)+5t(3)^{2}t(2)-2t(1)t(2)+6t(1)t(3)t(2)-6t(3)t(2)+2t(2)-t(3)^{2}+t(1)-2t(1)t(3)+2t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q+4-7q^{-1}+13q^{-2}-17q^{-3}+21q^{-4}-21q^{-5}+19q^{-6}-14q^{-7}+10q^{-8}-4q^{-9}+q^{-10}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{4}+2a^{8}z^{2}+a^{8}z^{-2}+2a^{8}-2a^{6}z^{6}-7a^{6}z^{4}-8a^{6}z^{2}-2a^{6}z^{-2}-5a^{6}+a^{4}z^{8}+5a^{4}z^{6}+9a^{4}z^{4}+6a^{4}z^{2}+a^{4}z^{-2}+a^{4}-a^{2}z^{6}-3a^{2}z^{4}-a^{2}z^{2}+2a^{2}$ (db)
Kauffman polynomial	$a^{12}z^{4}+4a^{11}z^{5}+10a^{10}z^{6}-10a^{10}z^{4}+6a^{10}z^{2}-2a^{10}+14a^{9}z^{7}-18a^{9}z^{5}+6a^{9}z^{3}+a^{9}z+13a^{8}z^{8}-17a^{8}z^{6}+6a^{8}z^{4}-6a^{8}z^{2}-a^{8}z^{-2}+3a^{8}+7a^{7}z^{9}+4a^{7}z^{7}-32a^{7}z^{5}+22a^{7}z^{3}-8a^{7}z+2a^{7}z^{-1}+2a^{6}z^{10}+16a^{6}z^{8}-50a^{6}z^{6}+42a^{6}z^{4}-22a^{6}z^{2}-2a^{6}z^{-2}+9a^{6}+12a^{5}z^{9}-26a^{5}z^{7}+4a^{5}z^{5}+14a^{5}z^{3}-8a^{5}z+2a^{5}z^{-1}+2a^{4}z^{10}+7a^{4}z^{8}-38a^{4}z^{6}+42a^{4}z^{4}-14a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+5a^{3}z^{9}-15a^{3}z^{7}+11a^{3}z^{5}+a^{3}z+4a^{2}z^{8}-15a^{2}z^{6}+17a^{2}z^{4}-4a^{2}z^{2}-2a^{2}+az^{7}-3az^{5}+2az^{3}$ (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

3

1

-1

1

3

-1

4

1

-3

9

3

6

-5

10

6

-4

-7

11

7

4

-9

10

0

-11

9

11

-2

-13

6

11

5

-15

4

8

-4

-17

6

-19

1

4

-3

-21

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-5$	$i=-3$
$r=-8$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{4}$
$r=-6$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=-3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$