L10a2

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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_20,13,5,14 X_18,11,19,12 X_12,19,13,20 X_2,16,3,15
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, 4, -6, 8, -9, 7, -4, 10, -2, 3, -8, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-3t(2)^{3}+3t(2)^{2}-3t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {14}{q^{9/2}}}-{\frac {15}{q^{7/2}}}+{\frac {13}{q^{5/2}}}+q^{3/2}-{\frac {11}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {4}{q^{15/2}}}+{\frac {8}{q^{13/2}}}-{\frac {11}{q^{11/2}}}-4{\sqrt {q}}+{\frac {6}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-a^{7}z+2a^{5}z^{5}+5a^{5}z^{3}+2a^{5}z-a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-5a^{3}z^{3}+3a^{3}z^{-1}+az^{5}+2az^{3}-az-2az^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}+4a^{9}z^{5}-2a^{9}z^{3}+8a^{8}z^{6}-9a^{8}z^{4}+3a^{8}z^{2}+9a^{7}z^{7}-11a^{7}z^{5}+5a^{7}z^{3}-2a^{7}z+6a^{6}z^{8}-a^{6}z^{6}-9a^{6}z^{4}+2a^{6}z^{2}+a^{6}+2a^{5}z^{9}+11a^{5}z^{7}-30a^{5}z^{5}+20a^{5}z^{3}-4a^{5}z-a^{5}z^{-1}+11a^{4}z^{8}-23a^{4}z^{6}+12a^{4}z^{4}-4a^{4}z^{2}+3a^{4}+2a^{3}z^{9}+6a^{3}z^{7}-27a^{3}z^{5}+22a^{3}z^{3}-a^{3}z-3a^{3}z^{-1}+5a^{2}z^{8}-13a^{2}z^{6}+9a^{2}z^{4}-3a^{2}z^{2}+3a^{2}+4az^{7}-12az^{5}+9az^{3}+az-2az^{-1}+z^{6}-2z^{4}$ (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

4

1

-1

2

3

0

3

1

-2

8

3

5

-4

7

5

-2

-6

8

6

2

-8

6

7

1

-10

5

8

-3

-12

3

6

3

-14

1

5

-4

-16

3

-18

1

-1

Integral Khovanov Homology

(db, data source)

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