L10a58

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_20,16,7,15 X_14,5,15,6 X_4,13,5,14 X_12,17,13,18 X_18,11,19,12 X_16,20,17,19 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -9, 2, -5, 4, -10}, {9, -1, 10, -2, 7, -6, 5, -4, 3, -8, 6, -7, 8, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{3}-4u^{2}v^{2}+2u^{2}v+uv^{4}-5uv^{3}+7uv^{2}-5uv+u+2v^{3}-4v^{2}+v}{uv^{2}}}$ (db)
Jones polynomial	${\frac {11}{q^{9/2}}}-{\frac {11}{q^{7/2}}}+{\frac {8}{q^{5/2}}}-{\frac {6}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {2}{q^{17/2}}}-{\frac {5}{q^{15/2}}}+{\frac {8}{q^{13/2}}}-{\frac {10}{q^{11/2}}}-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z+a^{9}z^{-1}-2a^{7}z^{3}-3a^{7}z-2a^{7}z^{-1}+a^{5}z^{5}+a^{5}z^{3}+a^{5}z+2a^{5}z^{-1}+a^{3}z^{5}+a^{3}z^{3}-a^{3}z-a^{3}z^{-1}-az^{3}-az$ (db)
Kauffman polynomial	$a^{11}z^{5}-3a^{11}z^{3}+2a^{11}z+2a^{10}z^{6}-4a^{10}z^{4}+a^{10}z^{2}+3a^{9}z^{7}-6a^{9}z^{5}+5a^{9}z^{3}-4a^{9}z+a^{9}z^{-1}+3a^{8}z^{8}-6a^{8}z^{6}+8a^{8}z^{4}-5a^{8}z^{2}+a^{7}z^{9}+5a^{7}z^{7}-17a^{7}z^{5}+23a^{7}z^{3}-11a^{7}z+2a^{7}z^{-1}+6a^{6}z^{8}-13a^{6}z^{6}+15a^{6}z^{4}-7a^{6}z^{2}+a^{6}+a^{5}z^{9}+6a^{5}z^{7}-18a^{5}z^{5}+20a^{5}z^{3}-9a^{5}z+2a^{5}z^{-1}+3a^{4}z^{8}-2a^{4}z^{6}-3a^{4}z^{4}+a^{4}z^{2}+4a^{3}z^{7}-7a^{3}z^{5}+3a^{3}z^{3}-3a^{3}z+a^{3}z^{-1}+3a^{2}z^{6}-6a^{2}z^{4}+2a^{2}z^{2}+az^{5}-2az^{3}+az$ (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

2

-2

4

1

3

-4

5

3

-2

-6

6

3

-8

5

0

-10

5

6

-1

-12

3

5

2

-14

2

5

-3

-16

1

4

3

-18

1

-1

-20

1

Integral Khovanov Homology

(db, data source)

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