L10a8

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-2t(2)^{3}+4t(2)^{2}-2t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-3q^{9/2}+7q^{7/2}-11q^{5/2}+12q^{3/2}-14{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-5z^{5}a^{-1}+z^{5}a^{-3}-a^{3}z^{3}+7az^{3}-11z^{3}a^{-1}+3z^{3}a^{-3}-2a^{3}z+9az-11za^{-1}+4za^{-3}-a^{3}z^{-1}+4az^{-1}-4a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-3a^{2}z^{8}-5z^{8}a^{-2}-8z^{8}-3a^{3}z^{7}-10az^{7}-15z^{7}a^{-1}-8z^{7}a^{-3}-a^{4}z^{6}+3a^{2}z^{6}-6z^{6}a^{-4}+10z^{6}+9a^{3}z^{5}+35az^{5}+43z^{5}a^{-1}+14z^{5}a^{-3}-3z^{5}a^{-5}+3a^{4}z^{4}+9a^{2}z^{4}+14z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}+12z^{4}-9a^{3}z^{3}-35az^{3}-43z^{3}a^{-1}-15z^{3}a^{-3}+2z^{3}a^{-5}-3a^{4}z^{2}-12a^{2}z^{2}-15z^{2}a^{-2}-5z^{2}a^{-4}+z^{2}a^{-6}-18z^{2}+4a^{3}z+17az+21za^{-1}+8za^{-3}+a^{4}+4a^{2}+4a^{-2}+a^{-4}+7-a^{3}z^{-1}-4az^{-1}-4a^{-1}z^{-1}-a^{-3}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

2

8

5

1

-4

6

2

4

6

5

-1

2

8

6

2

0

6

8

2

-2

4

6

-2

-4

2

6

4

-6

1

4

-3

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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