L10a48

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Link Presentations

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

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

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

12

1

10

1

-1

8

1

0

6

2

1

-1

4

2

1

2

0

2

0

-2

2

3

1

-4

1

0

-6

2

-8

1

0

-10

1

Integral Khovanov Homology

(db, data source)

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