L11a482

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-2)(t(3)-1)(2t(3)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{2}+5q-11+17q^{-1}-20q^{-2}+25q^{-3}-22q^{-4}+19q^{-5}-13q^{-6}+7q^{-7}-3q^{-8}+q^{-9}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{2}a^{8}+a^{8}-2z^{4}a^{6}-3z^{2}a^{6}+a^{6}z^{-2}-2a^{6}+z^{6}a^{4}+z^{4}a^{4}+z^{2}a^{4}-2a^{4}z^{-2}-a^{4}+z^{6}a^{2}+z^{4}a^{2}+z^{2}a^{2}+a^{2}z^{-2}+2a^{2}-z^{4}$ (db)
Kauffman polynomial	$z^{6}a^{10}-3z^{4}a^{10}+3z^{2}a^{10}-a^{10}+3z^{7}a^{9}-8z^{5}a^{9}+7z^{3}a^{9}-2za^{9}+4z^{8}a^{8}-5z^{6}a^{8}-4z^{4}a^{8}+6z^{2}a^{8}-a^{8}+4z^{9}a^{7}-z^{7}a^{7}-11z^{5}a^{7}+13z^{3}a^{7}-6za^{7}+2z^{10}a^{6}+8z^{8}a^{6}-20z^{6}a^{6}+11z^{4}a^{6}+z^{2}a^{6}+a^{6}z^{-2}-a^{6}+12z^{9}a^{5}-17z^{7}a^{5}+z^{5}a^{5}+8z^{3}a^{5}-2za^{5}-2a^{5}z^{-1}+2z^{10}a^{4}+17z^{8}a^{4}-39z^{6}a^{4}+23z^{4}a^{4}-z^{2}a^{4}+2a^{4}z^{-2}-2a^{4}+8z^{9}a^{3}-2z^{7}a^{3}-12z^{5}a^{3}+4z^{3}a^{3}+2za^{3}-2a^{3}z^{-1}+13z^{8}a^{2}-20z^{6}a^{2}+7z^{4}a^{2}+z^{2}a^{2}+a^{2}z^{-2}-2a^{2}+11z^{7}a-15z^{5}a+2z^{3}a+5z^{6}-4z^{4}+z^{5}a^{-1}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

4

1

7

1

-6

-1

10

4

6

-3

12

9

-3

-5

13

8

5

-7

9

12

3

-9

10

13

-3

-11

5

11

6

-13

2

8

-6

-15

1

5

4

-17

2

-2

-19

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-5$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{10}$
$r=-3$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$