L10a1

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{4}-3v^{3}+5v^{2}-3v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {9}{q^{9/2}}}-q^{7/2}+{\frac {13}{q^{7/2}}}+4q^{5/2}-{\frac {17}{q^{5/2}}}-8q^{3/2}+{\frac {17}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {5}{q^{11/2}}}+12{\sqrt {q}}-{\frac {17}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+4az^{5}-z^{5}a^{-1}+a^{5}z^{3}-4a^{3}z^{3}+6az^{3}-2z^{3}a^{-1}+az-za^{-1}-a^{5}z^{-1}+3a^{3}z^{-1}-2az^{-1}$ (db)
Kauffman polynomial	$-2a^{3}z^{9}-2az^{9}-7a^{4}z^{8}-13a^{2}z^{8}-6z^{8}-9a^{5}z^{7}-17a^{3}z^{7}-15az^{7}-7z^{7}a^{-1}-5a^{6}z^{6}+3a^{4}z^{6}+15a^{2}z^{6}-4z^{6}a^{-2}+3z^{6}-a^{7}z^{5}+15a^{5}z^{5}+40a^{3}z^{5}+37az^{5}+12z^{5}a^{-1}-z^{5}a^{-3}+6a^{6}z^{4}+10a^{4}z^{4}+5a^{2}z^{4}+6z^{4}a^{-2}+7z^{4}-6a^{5}z^{3}-23a^{3}z^{3}-26az^{3}-8z^{3}a^{-1}+z^{3}a^{-3}-2a^{4}z^{2}-5a^{2}z^{2}-2z^{2}a^{-2}-5z^{2}+a^{3}z+3az+2za^{-1}-a^{6}-3a^{4}-3a^{2}+a^{5}z^{-1}+3a^{3}z^{-1}+2az^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

5

1

4

2

7

3

-4

0

10

5

-2

9

0

-4

8

0

-6

5

9

4

-8

4

8

-4

-10

1

5

4

-12

4

-4

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$