L10a141

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,6,15,5 X_10,4,11,3 X_2,16,3,15 X_18,7,19,8 X_16,9,17,10 X_8,17,9,18 X_20,12,13,11 X_12,14,5,13 X_4,19,1,20
Gauss code	{1, -4, 3, -10}, {2, -1, 5, -7, 6, -3, 8, -9}, {9, -2, 4, -6, 7, -5, 10, -8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)(v+w)(vw+1)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{6}+3q^{5}-6q^{4}+9q^{3}-9q^{2}+11q-9+8q^{-1}-4q^{-2}+3q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-4}-2z^{2}a^{-4}-a^{-4}+z^{6}a^{-2}-a^{2}z^{4}+3z^{4}a^{-2}-2a^{2}z^{2}+3z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+a^{2}+3a^{-2}+z^{6}+3z^{4}+z^{2}-2z^{-2}-3$ (db)
Kauffman polynomial	$2az^{9}+2z^{9}a^{-1}+3a^{2}z^{8}+5z^{8}a^{-2}+8z^{8}+a^{3}z^{7}-5az^{7}+2z^{7}a^{-1}+8z^{7}a^{-3}-14a^{2}z^{6}-5z^{6}a^{-2}+9z^{6}a^{-4}-28z^{6}-4a^{3}z^{5}-az^{5}-16z^{5}a^{-1}-13z^{5}a^{-3}+6z^{5}a^{-5}+20a^{2}z^{4}-12z^{4}a^{-2}-15z^{4}a^{-4}+3z^{4}a^{-6}+26z^{4}+4a^{3}z^{3}+5az^{3}+7z^{3}a^{-1}+z^{3}a^{-3}-4z^{3}a^{-5}+z^{3}a^{-7}-7a^{2}z^{2}+13z^{2}a^{-2}+9z^{2}a^{-4}-3z^{2}+az+3za^{-1}+3za^{-3}+za^{-5}-2a^{2}-6a^{-2}-2a^{-4}-5-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

13

1

-1

11

2

9

4

1

-3

7

5

2

3

5

4

0

3

7

5

2

1

6

8

2

-1

2

3

-1

-3

2

6

4

-5

1

2

-1

-7

2

-9

1

-1

Integral Khovanov Homology

(db, data source)

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