L10a139

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(2)-1)\left(t(3)t(2)^{2}-2t(2)^{2}-3t(1)t(2)+2t(1)t(3)t(2)-3t(3)t(2)+2t(2)+t(1)-2t(1)t(3)\right)}{{\sqrt {t(1)}}t(2)^{3/2}{\sqrt {t(3)}}}}$ (db)
Jones polynomial	$q^{-6}-3q^{-5}+q^{4}+6q^{-4}-2q^{3}-8q^{-3}+5q^{2}+10q^{-2}-7q-10q^{-1}+11$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}-3z^{2}a^{4}-2a^{4}+2z^{4}a^{2}+3z^{2}a^{2}+a^{2}z^{-2}+3a^{2}+z^{4}-2z^{2}-2z^{-2}-3-2z^{2}a^{-2}+a^{-2}z^{-2}+a^{-4}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-9a^{5}z^{5}+7a^{5}z^{3}-2a^{5}z+3a^{4}z^{8}-5a^{4}z^{6}-5a^{4}z^{4}+z^{4}a^{-4}+6a^{4}z^{2}-2z^{2}a^{-4}-a^{4}+a^{-4}+a^{3}z^{9}+6a^{3}z^{7}-24a^{3}z^{5}+2z^{5}a^{-3}+21a^{3}z^{3}-2z^{3}a^{-3}-6a^{3}z+6a^{2}z^{8}-11a^{2}z^{6}+3z^{6}a^{-2}-a^{2}z^{4}-3z^{4}a^{-2}+8a^{2}z^{2}+3z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-4a^{2}-3a^{-2}+az^{9}+6az^{7}+3z^{7}a^{-1}-17az^{5}+11az^{3}-5z^{3}a^{-1}+2az+6za^{-1}-2az^{-1}-2a^{-1}z^{-1}+3z^{8}-2z^{6}-3z^{4}+10z^{2}+2z^{-2}-7$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

9

1

7

2

1

-1

5

3

4

2

-2

1

7

3

4

-1

5

6

1

-3

5

0

-5

3

5

2

-7

3

5

-2

-9

1

4

3

-11

2

-2

-13

1

Integral Khovanov Homology

(db, data source)

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