L10a120

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_2,13,3,14 X_14,3,15,4 X_16,5,17,6 X_6,11,7,12 X_20,7,11,8 X_18,9,19,10 X_10,17,1,18 X_8,19,9,20 X_4,15,5,16
Gauss code	{1, -2, 3, -10, 4, -5, 6, -9, 7, -8}, {5, -1, 2, -3, 10, -4, 8, -7, 9, -6}

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

Polynomial invariants

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

1

0

-8

1

-10

2

1

-1

-12

2

1

-14

2

0

-16

2

0

-18

1

2

1

-20

1

2

-1

-22

1

-24

1

0

-26

1

Integral Khovanov Homology

(db, data source)

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