L10a19

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2(t(1)-1)(t(2)-1)^{3}}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{5/2}-3q^{3/2}+5{\sqrt {q}}-{\frac {9}{\sqrt {q}}}+{\frac {10}{q^{3/2}}}-{\frac {11}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {2}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+2a^{5}z^{3}+3a^{5}z+2a^{5}z^{-1}-a^{3}z^{5}-a^{3}z^{3}-az^{5}-2az^{3}+z^{3}a^{-1}-3az-az^{-1}+za^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-4a^{8}z^{4}+5a^{8}z^{2}-2a^{8}+2a^{7}z^{7}-6a^{7}z^{5}+5a^{7}z^{3}-2a^{7}z+a^{7}z^{-1}+2a^{6}z^{8}-2a^{6}z^{6}-8a^{6}z^{4}+12a^{6}z^{2}-5a^{6}+a^{5}z^{9}+3a^{5}z^{7}-12a^{5}z^{5}+9a^{5}z^{3}-5a^{5}z+2a^{5}z^{-1}+5a^{4}z^{8}-7a^{4}z^{6}-4a^{4}z^{4}+8a^{4}z^{2}-3a^{4}+a^{3}z^{9}+5a^{3}z^{7}-10a^{3}z^{5}+3a^{3}z^{3}+3a^{2}z^{8}-4a^{2}z^{4}+z^{4}a^{-2}+2a^{2}z^{2}-z^{2}a^{-2}+a^{2}+4az^{7}-az^{5}+3z^{5}a^{-1}-5az^{3}-4z^{3}a^{-1}+5az+2za^{-1}-az^{-1}+4z^{6}-3z^{4}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

2

3

1

-2

0

6

2

4

-2

6

5

-1

-4

5

4

1

-6

4

6

2

-8

4

5

-1

-10

1

4

3

-12

1

4

-3

-14

1

-16

1

-1

Integral Khovanov Homology

(db, data source)

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