L10a159

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{2}t(3)^{3}+t(1)^{2}t(2)t(3)^{3}-t(1)t(2)t(3)^{3}+t(1)^{2}t(3)^{2}+t(1)^{2}t(2)^{2}t(3)^{2}-2t(1)t(2)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}-t(1)t(3)^{2}-2t(1)^{2}t(2)t(3)^{2}+4t(1)t(2)t(3)^{2}-t(2)t(3)^{2}-t(1)^{2}t(3)+t(1)t(2)^{2}t(3)-t(2)^{2}t(3)+2t(1)t(3)+t(1)^{2}t(2)t(3)-4t(1)t(2)t(3)+2t(2)t(3)-t(3)-t(1)+t(1)t(2)-t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-13}-3q^{-12}+6q^{-11}-8q^{-10}+11q^{-9}-10q^{-8}+10q^{-7}-7q^{-6}+5q^{-5}-2q^{-4}+q^{-3}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$z^{2}a^{12}+a^{12}z^{-2}+2a^{12}-3z^{4}a^{10}-10z^{2}a^{10}-2a^{10}z^{-2}-9a^{10}+2z^{6}a^{8}+9z^{4}a^{8}+13z^{2}a^{8}+a^{8}z^{-2}+7a^{8}+z^{6}a^{6}+4z^{4}a^{6}+4z^{2}a^{6}$ (db)
Kauffman polynomial	$a^{16}z^{4}-a^{16}z^{2}+3a^{15}z^{5}-3a^{15}z^{3}+5a^{14}z^{6}-7a^{14}z^{4}+5a^{14}z^{2}-2a^{14}+5a^{13}z^{7}-6a^{13}z^{5}+3a^{13}z^{3}+3a^{12}z^{8}+a^{12}z^{6}-8a^{12}z^{4}+5a^{12}z^{2}-a^{12}z^{-2}+3a^{12}+a^{11}z^{9}+6a^{11}z^{7}-17a^{11}z^{5}+17a^{11}z^{3}-9a^{11}z+2a^{11}z^{-1}+6a^{10}z^{8}-16a^{10}z^{6}+22a^{10}z^{4}-23a^{10}z^{2}-2a^{10}z^{-2}+11a^{10}+a^{9}z^{9}+3a^{9}z^{7}-14a^{9}z^{5}+15a^{9}z^{3}-9a^{9}z+2a^{9}z^{-1}+3a^{8}z^{8}-11a^{8}z^{6}+18a^{8}z^{4}-18a^{8}z^{2}-a^{8}z^{-2}+7a^{8}+2a^{7}z^{7}-6a^{7}z^{5}+4a^{7}z^{3}+a^{6}z^{6}-4a^{6}z^{4}+4a^{6}z^{2}$ (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

-5

1

-7

2

1

-1

-9

3

-11

4

2

-2

-13

6

3

-15

4

0

-17

7

6

1

-19

3

6

3

-21

3

5

-2

-23

1

4

3

-25

2

-2

-27

1

Integral Khovanov Homology

(db, data source)

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