L10a3

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_14,10,15,9 X₈₄₉₃ X_10,5,11,6 X_20,11,5,12 X_18,13,19,14 X_12,19,13,20 X_2,16,3,15
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, 4, -6, 7, -9, 8, -4, 10, -2, 3, -8, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-2)(t(2)-1)(2t(2)-1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+6{\sqrt {q}}-{\frac {10}{\sqrt {q}}}+{\frac {11}{q^{3/2}}}-{\frac {12}{q^{5/2}}}+{\frac {11}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+2z^{3}a^{5}+2za^{5}-z^{5}a^{3}-z^{3}a^{3}-za^{3}-z^{5}a-z^{3}a+az^{-1}+z^{3}a^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{5}z^{9}-2a^{3}z^{9}-4a^{6}z^{8}-9a^{4}z^{8}-5a^{2}z^{8}-3a^{7}z^{7}-3a^{3}z^{7}-6az^{7}-a^{8}z^{6}+13a^{6}z^{6}+27a^{4}z^{6}+7a^{2}z^{6}-6z^{6}+10a^{7}z^{5}+14a^{5}z^{5}+13a^{3}z^{5}+5az^{5}-4z^{5}a^{-1}+3a^{8}z^{4}-12a^{6}z^{4}-28a^{4}z^{4}-6a^{2}z^{4}-z^{4}a^{-2}+6z^{4}-8a^{7}z^{3}-17a^{5}z^{3}-11a^{3}z^{3}+2az^{3}+4z^{3}a^{-1}-a^{8}z^{2}+4a^{6}z^{2}+9a^{4}z^{2}+4a^{2}z^{2}+2a^{7}z+4a^{5}z+2a^{3}z+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

3

2

3

1

-2

0

7

3

4

-2

6

5

-1

-4

6

5

1

-6

5

6

1

-8

3

6

-3

-10

2

5

3

-12

1

3

-2

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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