L10a74

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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_18,11,19,12 X_20,13,7,14 X_12,19,13,20 X_14,17,15,18 X₆₇₁₈ X_4,15,5,16
Gauss code	{1, -2, 3, -10, 4, -9}, {9, -1, 2, -3, 5, -7, 6, -8, 10, -4, 8, -5, 7, -6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}v^{4}-2u^{2}v^{3}+2u^{2}v^{2}-u^{2}v-uv^{4}+3uv^{3}-3uv^{2}+3uv-u-v^{3}+2v^{2}-2v+2}{uv^{2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}-{\frac {1}{q^{27/2}}}+{\frac {3}{q^{25/2}}}-{\frac {5}{q^{23/2}}}+{\frac {7}{q^{21/2}}}-{\frac {8}{q^{19/2}}}+{\frac {8}{q^{17/2}}}-{\frac {7}{q^{15/2}}}+{\frac {5}{q^{13/2}}}-{\frac {4}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{11}z^{5}+3a^{11}z^{3}+a^{11}z-a^{11}z^{-1}-a^{9}z^{7}-4a^{9}z^{5}-2a^{9}z^{3}+5a^{9}z+3a^{9}z^{-1}-a^{7}z^{7}-6a^{7}z^{5}-12a^{7}z^{3}-9a^{7}z-2a^{7}z^{-1}$ (db)
Kauffman polynomial	$a^{17}z^{3}+3a^{16}z^{4}-a^{16}z^{2}+5a^{15}z^{5}-4a^{15}z^{3}+a^{15}z+6a^{14}z^{6}-8a^{14}z^{4}+3a^{14}z^{2}+5a^{13}z^{7}-7a^{13}z^{5}+a^{13}z+3a^{12}z^{8}-3a^{12}z^{6}-5a^{12}z^{4}+3a^{12}z^{2}-a^{12}+a^{11}z^{9}+2a^{11}z^{7}-10a^{11}z^{5}+4a^{11}z^{3}+a^{11}z^{-1}+4a^{10}z^{8}-12a^{10}z^{6}+5a^{10}z^{4}+6a^{10}z^{2}-3a^{10}+a^{9}z^{9}-2a^{9}z^{7}-4a^{9}z^{5}+11a^{9}z^{3}-9a^{9}z+3a^{9}z^{-1}+a^{8}z^{8}-3a^{8}z^{6}-a^{8}z^{4}+7a^{8}z^{2}-3a^{8}+a^{7}z^{7}-6a^{7}z^{5}+12a^{7}z^{3}-9a^{7}z+2a^{7}z^{-1}$ (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

-6

1

-8

1

0

-10

3

-12

2

1

-1

-14

5

3

2

-16

3

2

-1

-18

5

0

-20

3

4

1

-22

2

4

-2

-24

1

3

2

-26

2

-2

-28

1

Integral Khovanov Homology

(db, data source)

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