L10a146

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_20,12,13,11 X_18,8,19,7 X_16,10,17,9 X_8,18,9,17 X_10,14,11,13 X_12,20,5,19 X₂₅₃₆ X_4,16,1,15
Gauss code	{1, -9, 2, -10}, {9, -1, 4, -6, 5, -7, 3, -8}, {7, -2, 10, -5, 6, -4, 8, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{3}w+2uv^{2}w^{2}-3uv^{2}w+uv^{2}+uvw^{3}-3uvw^{2}+2uvw-uw^{3}+uw^{2}-v^{3}w+v^{3}-2v^{2}w^{2}+3v^{2}w-v^{2}-vw^{3}+3vw^{2}-2vw-w^{2}}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{9}-2q^{8}+5q^{7}-7q^{6}+10q^{5}-9q^{4}+10q^{3}-8q^{2}+5q+q^{-1}-2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-2}-2z^{4}a^{-4}-z^{4}a^{-6}-2z^{2}a^{-4}-z^{2}a^{-6}+z^{2}a^{-8}+z^{2}+a^{-4}-3a^{-6}+a^{-8}+1+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-10}-4z^{4}a^{-10}+4z^{2}a^{-10}+2z^{7}a^{-9}-6z^{5}a^{-9}+4z^{3}a^{-9}+3z^{8}a^{-8}-11z^{6}a^{-8}+18z^{4}a^{-8}-19z^{2}a^{-8}-a^{-8}z^{-2}+8a^{-8}+z^{9}a^{-7}+3z^{7}a^{-7}-15z^{5}a^{-7}+17z^{3}a^{-7}-11za^{-7}+2a^{-7}z^{-1}+6z^{8}a^{-6}-18z^{6}a^{-6}+28z^{4}a^{-6}-30z^{2}a^{-6}-2a^{-6}z^{-2}+13a^{-6}+z^{9}a^{-5}+5z^{7}a^{-5}-17z^{5}a^{-5}+21z^{3}a^{-5}-11za^{-5}+2a^{-5}z^{-1}+3z^{8}a^{-4}-3z^{6}a^{-4}+3z^{4}a^{-4}-5z^{2}a^{-4}-a^{-4}z^{-2}+5a^{-4}+4z^{7}a^{-3}-6z^{5}a^{-3}+6z^{3}a^{-3}+3z^{6}a^{-2}-2z^{4}a^{-2}+2z^{5}a^{-1}-2z^{3}a^{-1}+z^{4}-2z^{2}+1$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

χ

19

1

17

2

1

-1

15

3

13

4

2

-2

11

6

3

9

4

5

1

7

6

5

1

5

2

4

2

3

6

-3

1

4

3

-1

1

-1

-3

1

Integral Khovanov Homology

(db, data source)

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