L10a6

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Link Presentations

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

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	$-{\frac {8}{q^{9/2}}}-q^{7/2}+{\frac {13}{q^{7/2}}}+4q^{5/2}-{\frac {17}{q^{5/2}}}-9q^{3/2}+{\frac {17}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+13{\sqrt {q}}-{\frac {17}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z-2a^{3}z^{5}-5a^{3}z^{3}-4a^{3}z+az^{7}+4az^{5}-z^{5}a^{-1}+7az^{3}-2z^{3}a^{-1}+5az-2za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-a^{7}z^{3}+4a^{6}z^{6}-6a^{6}z^{4}+3a^{6}z^{2}+7a^{5}z^{7}-11a^{5}z^{5}+7a^{5}z^{3}-2a^{5}z+6a^{4}z^{8}-a^{4}z^{6}-12a^{4}z^{4}+8a^{4}z^{2}+2a^{3}z^{9}+16a^{3}z^{7}-39a^{3}z^{5}+z^{5}a^{-3}+28a^{3}z^{3}-z^{3}a^{-3}-8a^{3}z+13a^{2}z^{8}-14a^{2}z^{6}+4z^{6}a^{-2}-7a^{2}z^{4}-5z^{4}a^{-2}+7a^{2}z^{2}+2z^{2}a^{-2}+2az^{9}+17az^{7}+8z^{7}a^{-1}-42az^{5}-14z^{5}a^{-1}+31az^{3}+10z^{3}a^{-1}-10az-4za^{-1}+az^{-1}+a^{-1}z^{-1}+7z^{8}-5z^{6}-6z^{4}+4z^{2}-1$ (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

6

1

5

2

7

3

-4

0

10

6

4

-2

9

0

-4

8

0

-6

5

9

4

-8

3

8

-5

-10

1

5

4

-12

3

-3

-14

1

Integral Khovanov Homology

(db, data source)

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