L10a161

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{2}t(3)^{3}-t(1)t(2)^{2}t(3)^{3}-t(1)^{2}t(2)t(3)^{3}-t(1)^{2}t(3)^{2}+t(1)t(2)^{2}t(3)^{2}-t(2)^{2}t(3)^{2}+t(1)^{2}t(2)t(3)^{2}-t(1)t(2)t(3)^{2}+t(1)^{2}t(3)+t(2)^{2}t(3)-t(1)t(3)+t(1)t(2)t(3)-t(2)t(3)+t(1)+t(2)-1}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{11}-2q^{10}+3q^{9}-4q^{8}+5q^{7}-4q^{6}+5q^{5}-3q^{4}+3q^{3}-q^{2}+q$ (db)
Signature	6 (db)
HOMFLY-PT polynomial	$z^{6}a^{-8}+5z^{4}a^{-8}+7z^{2}a^{-8}+a^{-8}z^{-2}+3a^{-8}-z^{8}a^{-6}-7z^{6}a^{-6}-17z^{4}a^{-6}-18z^{2}a^{-6}-2a^{-6}z^{-2}-9a^{-6}+z^{6}a^{-4}+6z^{4}a^{-4}+11z^{2}a^{-4}+a^{-4}z^{-2}+6a^{-4}$ (db)
Kauffman polynomial	$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+4z^{8}a^{-6}+3z^{8}a^{-8}-5z^{7}a^{-5}-z^{7}a^{-7}+4z^{7}a^{-9}-7z^{6}a^{-4}-23z^{6}a^{-6}-11z^{6}a^{-8}+5z^{6}a^{-10}+5z^{5}a^{-5}-11z^{5}a^{-7}-12z^{5}a^{-9}+4z^{5}a^{-11}+17z^{4}a^{-4}+43z^{4}a^{-6}+9z^{4}a^{-8}-14z^{4}a^{-10}+3z^{4}a^{-12}+6z^{3}a^{-5}+20z^{3}a^{-7}+6z^{3}a^{-9}-6z^{3}a^{-11}+2z^{3}a^{-13}-17z^{2}a^{-4}-32z^{2}a^{-6}-3z^{2}a^{-8}+9z^{2}a^{-10}-2z^{2}a^{-12}+z^{2}a^{-14}-9za^{-5}-9za^{-7}+7a^{-4}+11a^{-6}+3a^{-8}-2a^{-10}+2a^{-5}z^{-1}+2a^{-7}z^{-1}-a^{-4}z^{-2}-2a^{-6}z^{-2}-a^{-8}z^{-2}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

χ

23

1

21

2

1

-1

19

1

17

3

2

-1

15

2

1

13

2

3

1

11

3

2

1

9

2

4

2

7

1

0

5

1

3

2

3

0

1

Integral Khovanov Homology

(db, data source)

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