L10a15

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{5}+2t(1)t(2)^{4}-4t(2)^{4}-5t(1)t(2)^{3}+5t(2)^{3}+5t(1)t(2)^{2}-5t(2)^{2}-4t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {2}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {11}{q^{7/2}}}+{\frac {11}{q^{9/2}}}-{\frac {11}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z+a^{9}z^{-1}-3a^{7}z^{3}-6a^{7}z-3a^{7}z^{-1}+2a^{5}z^{5}+6a^{5}z^{3}+7a^{5}z+4a^{5}z^{-1}+a^{3}z^{5}+a^{3}z^{3}-2a^{3}z-2a^{3}z^{-1}-az^{3}-2az$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+6z^{4}a^{10}-3z^{2}a^{10}+a^{10}-4z^{7}a^{9}+6z^{5}a^{9}-z^{3}a^{9}+2za^{9}-a^{9}z^{-1}-3z^{8}a^{8}+10z^{4}a^{8}-9z^{2}a^{8}+3a^{8}-z^{9}a^{7}-7z^{7}a^{7}+21z^{5}a^{7}-25z^{3}a^{7}+14za^{7}-3a^{7}z^{-1}-6z^{8}a^{6}+10z^{6}a^{6}-4z^{4}a^{6}-6z^{2}a^{6}+3a^{6}-z^{9}a^{5}-6z^{7}a^{5}+21z^{5}a^{5}-31z^{3}a^{5}+19za^{5}-4a^{5}z^{-1}-3z^{8}a^{4}+5z^{6}a^{4}-4z^{4}a^{4}-z^{2}a^{4}+2a^{4}-3z^{7}a^{3}+6z^{5}a^{3}-6z^{3}a^{3}+6za^{3}-2a^{3}z^{-1}-2z^{6}a^{2}+4z^{4}a^{2}-z^{2}a^{2}-z^{5}a+3z^{3}a-2za$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

1

-1

-2

4

1

3

-4

5

2

-3

-6

6

3

-8

5

0

-10

6

0

-12

4

6

2

-14

2

5

-3

-16

1

4

3

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

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