L10a119

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Link Presentations

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

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

Polynomial invariants

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

-2

1

-4

3

1

-2

-6

3

-8

4

3

-1

-10

5

3

2

-12

4

0

-14

3

5

-2

-16

3

4

1

-18

1

3

-2

-20

3

-22

1

0

-24

1

Integral Khovanov Homology

(db, data source)

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