L11a487

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_22,10,19,9 X₈₄₉₃ X_16,21,17,22 X_18,11,5,12 X_20,5,21,6 X_10,17,11,18 X_12,20,13,19 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {10, -8, 6, -4}, {8, -1, 2, -5, 4, -9, 7, -10, 11, -2, 3, -6, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)^{3}\left(t(3)^{2}-t(3)+1\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{3}-5q^{2}+12q-18+28q^{-1}-30q^{-2}+32q^{-3}-27q^{-4}+20q^{-5}-13q^{-6}+5q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{4}-a^{6}z^{2}-a^{6}z^{-2}-a^{6}+2a^{4}z^{6}+5a^{4}z^{4}+4a^{4}z^{2}+4a^{4}z^{-2}+4a^{4}-a^{2}z^{8}-4a^{2}z^{6}-6a^{2}z^{4}-4a^{2}z^{2}-5a^{2}z^{-2}-5a^{2}+z^{6}+2z^{4}+z^{2}+2z^{-2}+2$ (db)
Kauffman polynomial	$a^{9}z^{5}+5a^{8}z^{6}-2a^{8}z^{4}+13a^{7}z^{7}-16a^{7}z^{5}+7a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+18a^{6}z^{8}-27a^{6}z^{6}+14a^{6}z^{4}-4a^{6}z^{2}-a^{6}z^{-2}+a^{6}+13a^{5}z^{9}-4a^{5}z^{7}-23a^{5}z^{5}+18a^{5}z^{3}-10a^{5}z+5a^{5}z^{-1}+4a^{4}z^{10}+28a^{4}z^{8}-72a^{4}z^{6}+49a^{4}z^{4}-13a^{4}z^{2}-4a^{4}z^{-2}+4a^{4}+24a^{3}z^{9}-38a^{3}z^{7}+3a^{3}z^{5}+14a^{3}z^{3}-12a^{3}z+9a^{3}z^{-1}+4a^{2}z^{10}+21a^{2}z^{8}-65a^{2}z^{6}+z^{6}a^{-2}+51a^{2}z^{4}-z^{4}a^{-2}-13a^{2}z^{2}-5a^{2}z^{-2}+5a^{2}+11az^{9}-16az^{7}+5z^{7}a^{-1}+az^{5}-8z^{5}a^{-1}+5az^{3}+2z^{3}a^{-1}-5az+5az^{-1}+11z^{8}-24z^{6}+17z^{4}-4z^{2}-2z^{-2}+3$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

4

-4

3

8

1

7

1

10

4

-6

-1

18

8

10

-3

16

14

-2

-5

16

14

2

-7

11

16

5

-9

9

16

-7

-11

4

11

7

-13

1

9

-8

-15

4

-17

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-3$	${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-2$	${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{16}$	${\mathbb {Z} }^{16}$
$r=-1$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{16}$	${\mathbb {Z} }^{16}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{18}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$