L10a77

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_16,5,17,6 X_14,7,15,8 X_18,11,19,12 X_20,15,7,16 X_12,17,13,18 X_4,13,5,14 X_6,19,1,20
Gauss code	{1, -2, 3, -9, 4, -10}, {5, -1, 2, -3, 6, -8, 9, -5, 7, -4, 8, -6, 10, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {4u^{2}v^{3}-4u^{2}v^{2}+u^{2}v+uv^{4}-5uv^{3}+9uv^{2}-5uv+u+v^{3}-4v^{2}+4v}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{25/2}}}+{\frac {3}{q^{23/2}}}-{\frac {6}{q^{21/2}}}+{\frac {10}{q^{19/2}}}-{\frac {12}{q^{17/2}}}+{\frac {13}{q^{15/2}}}-{\frac {13}{q^{13/2}}}+{\frac {9}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{11}z^{3}+a^{11}z-a^{11}z^{-1}-a^{9}z^{5}+5a^{9}z+3a^{9}z^{-1}-3a^{7}z^{5}-10a^{7}z^{3}-9a^{7}z-2a^{7}z^{-1}-a^{5}z^{5}-2a^{5}z^{3}$ (db)
Kauffman polynomial	$a^{15}z^{5}-2a^{15}z^{3}+a^{15}z+3a^{14}z^{6}-6a^{14}z^{4}+3a^{14}z^{2}+4a^{13}z^{7}-5a^{13}z^{5}-a^{13}z^{3}+a^{13}z+4a^{12}z^{8}-5a^{12}z^{6}+3a^{12}z^{4}-3a^{12}z^{2}-a^{12}+2a^{11}z^{9}+2a^{11}z^{7}-5a^{11}z^{5}+a^{11}z^{3}+a^{11}z^{-1}+9a^{10}z^{8}-20a^{10}z^{6}+18a^{10}z^{4}-2a^{10}z^{2}-3a^{10}+2a^{9}z^{9}+4a^{9}z^{7}-16a^{9}z^{5}+19a^{9}z^{3}-9a^{9}z+3a^{9}z^{-1}+5a^{8}z^{8}-9a^{8}z^{6}+4a^{8}z^{4}+4a^{8}z^{2}-3a^{8}+6a^{7}z^{7}-16a^{7}z^{5}+17a^{7}z^{3}-9a^{7}z+2a^{7}z^{-1}+3a^{6}z^{6}-5a^{6}z^{4}+a^{5}z^{5}-2a^{5}z^{3}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

3

1

-2

-8

4

-10

5

3

-2

-12

8

4

-14

5

0

-16

7

8

-1

-18

4

6

2

-20

2

6

-4

-22

1

4

3

-24

2

-2

-26

1

Integral Khovanov Homology

(db, data source)

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