L10a147

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(3)^{2}t(2)^{3}-t(3)^{2}t(2)^{3}-t(1)t(3)t(2)^{3}+t(3)t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-t(3)^{3}t(2)^{2}-3t(1)t(3)^{2}t(2)^{2}+3t(3)^{2}t(2)^{2}-t(1)t(2)^{2}+4t(1)t(3)t(2)^{2}-3t(3)t(2)^{2}+t(2)^{2}-t(1)t(3)^{3}t(2)+t(3)^{3}t(2)+3t(1)t(3)^{2}t(2)-4t(3)^{2}t(2)+t(1)t(2)-3t(1)t(3)t(2)+3t(3)t(2)-t(2)-t(1)t(3)^{2}+t(3)^{2}+t(1)t(3)-t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{6}-3q^{5}+6q^{4}+q^{-4}-9q^{3}-4q^{-3}+14q^{2}+8q^{-2}-13q-11q^{-1}+14$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-z^{6}+a^{2}z^{4}-3z^{4}a^{-2}+z^{4}a^{-4}-2z^{4}+a^{2}z^{2}-4z^{2}a^{-2}+2z^{2}a^{-4}-3a^{-2}+a^{-4}+2-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$2z^{9}a^{-1}+2z^{9}a^{-3}+10z^{8}a^{-2}+4z^{8}a^{-4}+6z^{8}+9az^{7}+8z^{7}a^{-1}+2z^{7}a^{-3}+3z^{7}a^{-5}+8a^{2}z^{6}-23z^{6}a^{-2}-11z^{6}a^{-4}+z^{6}a^{-6}-3z^{6}+4a^{3}z^{5}-11az^{5}-19z^{5}a^{-1}-13z^{5}a^{-3}-9z^{5}a^{-5}+a^{4}z^{4}-9a^{2}z^{4}+20z^{4}a^{-2}+11z^{4}a^{-4}-3z^{4}a^{-6}-4z^{4}-2a^{3}z^{3}+3az^{3}+8z^{3}a^{-1}+10z^{3}a^{-3}+7z^{3}a^{-5}+3a^{2}z^{2}-13z^{2}a^{-2}-7z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}-3za^{-1}-3za^{-3}+5a^{-2}+3a^{-4}+3+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

13

1

11

2

-2

9

4

1

3

7

6

3

-3

5

8

3

5

3

5

6

1

9

8

1

-1

5

8

3

-3

3

6

-3

-5

1

5

4

-7

3

-3

-9

1

Integral Khovanov Homology

(db, data source)

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