L10a42

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2uv^{3}-9uv^{2}+8uv-2u-2v^{3}+8v^{2}-9v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {14}{q^{7/2}}}+{\frac {14}{q^{9/2}}}-{\frac {13}{q^{11/2}}}+{\frac {10}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z-2a^{7}z^{3}-a^{7}z+a^{5}z^{5}+a^{5}z^{-1}+a^{3}z^{5}-2a^{3}z-a^{3}z^{-1}-az^{3}$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-6a^{10}z^{4}+3a^{10}z^{2}+4a^{9}z^{7}-5a^{9}z^{5}+4a^{8}z^{8}-4a^{8}z^{6}+a^{8}z^{4}-a^{8}z^{2}+2a^{7}z^{9}+4a^{7}z^{7}-11a^{7}z^{5}+8a^{7}z^{3}-a^{7}z+10a^{6}z^{8}-21a^{6}z^{6}+19a^{6}z^{4}-6a^{6}z^{2}+2a^{5}z^{9}+7a^{5}z^{7}-20a^{5}z^{5}+14a^{5}z^{3}-2a^{5}z+a^{5}z^{-1}+6a^{4}z^{8}-10a^{4}z^{6}+6a^{4}z^{4}-2a^{4}z^{2}-a^{4}+7a^{3}z^{7}-14a^{3}z^{5}+7a^{3}z^{3}-2a^{3}z+a^{3}z^{-1}+4a^{2}z^{6}-6a^{2}z^{4}+az^{5}-az^{3}$ (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

χ

2

1

0

3

-3

-2

5

1

4

-4

6

4

-2

-6

8

4

-8

6

0

-10

7

8

-1

-12

4

7

3

-14

2

6

-4

-16

1

4

3

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

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