L10a164

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{2}t(3)^{3}+t(1)t(2)t(3)^{3}-t(2)t(3)^{3}-t(1)^{2}t(3)^{2}+2t(1)t(2)^{2}t(3)^{2}-2t(2)^{2}t(3)^{2}+2t(1)t(3)^{2}+2t(1)^{2}t(2)t(3)^{2}-4t(1)t(2)t(3)^{2}+3t(2)t(3)^{2}-t(3)^{2}+2t(1)^{2}t(3)+t(1)^{2}t(2)^{2}t(3)-2t(1)t(2)^{2}t(3)+t(2)^{2}t(3)-2t(1)t(3)-3t(1)^{2}t(2)t(3)+4t(1)t(2)t(3)-2t(2)t(3)-t(1)^{2}+t(1)^{2}t(2)-t(1)t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-2}-3q^{-3}+7q^{-4}-10q^{-5}+13q^{-6}-13q^{-7}+13q^{-8}-9q^{-9}+7q^{-10}-3q^{-11}+q^{-12}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{12}z^{-2}+a^{12}-4a^{10}z^{2}-2a^{10}z^{-2}-6a^{10}+3a^{8}z^{4}+6a^{8}z^{2}+a^{8}z^{-2}+4a^{8}+3a^{6}z^{4}+5a^{6}z^{2}+a^{6}+a^{4}z^{4}+a^{4}z^{2}$ (db)
Kauffman polynomial	$z^{6}a^{14}-3z^{4}a^{14}+3z^{2}a^{14}-a^{14}+3z^{7}a^{13}-8z^{5}a^{13}+5z^{3}a^{13}+4z^{8}a^{12}-9z^{6}a^{12}+4z^{4}a^{12}-3z^{2}a^{12}-a^{12}z^{-2}+5a^{12}+2z^{9}a^{11}+3z^{7}a^{11}-18z^{5}a^{11}+17z^{3}a^{11}-9za^{11}+2a^{11}z^{-1}+10z^{8}a^{10}-25z^{6}a^{10}+26z^{4}a^{10}-23z^{2}a^{10}-2a^{10}z^{-2}+11a^{10}+2z^{9}a^{9}+7z^{7}a^{9}-22z^{5}a^{9}+19z^{3}a^{9}-9za^{9}+2a^{9}z^{-1}+6z^{8}a^{8}-9z^{6}a^{8}+10z^{4}a^{8}-10z^{2}a^{8}-a^{8}z^{-2}+5a^{8}+7z^{7}a^{7}-9z^{5}a^{7}+5z^{3}a^{7}+6z^{6}a^{6}-8z^{4}a^{6}+6z^{2}a^{6}-a^{6}+3z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-z^{2}a^{4}$ (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

-3

1

-5

3

1

-2

-7

4

-9

6

3

-3

-11

7

4

3

-13

6

0

-15

7

0

-17

4

8

4

-19

3

5

-2

-21

1

5

4

-23

2

-2

-25

1

Integral Khovanov Homology

(db, data source)

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