L11a497

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,8,15,7 X_20,11,21,12 X_18,10,5,9 X_10,19,11,20 X_8,16,9,15 X_16,21,17,22 X_22,17,19,18 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {6, -4, 8, -9}, {10, -1, 3, -7, 5, -6, 4, -2, 11, -3, 7, -8, 9, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(w-1)\left(vw^{4}-2vw^{3}+2vw^{2}-2vw+2w^{3}-2w^{2}+2w-1\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-3q^{-5}+3q^{4}+8q^{-4}-7q^{3}-11q^{-3}+11q^{2}+16q^{-2}-15q-17q^{-1}+19$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+3a^{4}z^{2}+2a^{4}z^{-2}+4a^{4}-2a^{2}z^{6}-z^{6}a^{-2}-9a^{2}z^{4}-4z^{4}a^{-2}-16a^{2}z^{2}-6z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-14a^{2}-4a^{-2}+z^{8}+6z^{6}+15z^{4}+19z^{2}+4z^{-2}+14$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+4a^{3}z^{9}+8az^{9}+4z^{9}a^{-1}+5a^{4}z^{8}+13a^{2}z^{8}+6z^{8}a^{-2}+14z^{8}+3a^{5}z^{7}-3a^{3}z^{7}-10az^{7}+z^{7}a^{-1}+5z^{7}a^{-3}+a^{6}z^{6}-12a^{4}z^{6}-49a^{2}z^{6}-11z^{6}a^{-2}+3z^{6}a^{-4}-50z^{6}-7a^{5}z^{5}-14a^{3}z^{5}-19az^{5}-21z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-3a^{6}z^{4}+10a^{4}z^{4}+68a^{2}z^{4}+12z^{4}a^{-2}-5z^{4}a^{-4}+72z^{4}+3a^{5}z^{3}+22a^{3}z^{3}+50az^{3}+37z^{3}a^{-1}+4z^{3}a^{-3}-2z^{3}a^{-5}+3a^{6}z^{2}-13a^{4}z^{2}-50a^{2}z^{2}-11z^{2}a^{-2}+z^{2}a^{-4}-46z^{2}-17a^{3}z-35az-23za^{-1}-4za^{-3}+za^{-5}-a^{6}+8a^{4}+22a^{2}+5a^{-2}+19+5a^{3}z^{-1}+9az^{-1}+5a^{-1}z^{-1}+a^{-3}z^{-1}-2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-4z^{-2}$ (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

2

7

5

1

-4

5

6

2

4

3

9

5

-4

1

10

6

4

-1

9

11

2

-3

7

8

-1

-5

4

9

5

-7

4

7

-3

-9

1

6

5

-11

2

-2

-13

1

Integral Khovanov Homology

(db, data source)

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