L10a20

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)^{3}\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+9q^{7/2}-13q^{5/2}+15q^{3/2}-17{\sqrt {q}}+{\frac {14}{\sqrt {q}}}-{\frac {12}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-a^{3}z^{3}+2z^{3}a^{-3}-2a^{3}z+2za^{-3}-a^{3}z^{-1}+a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+6az^{3}-7z^{3}a^{-1}+7az-7za^{-1}+4az^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+4z^{5}a^{-5}-z^{3}a^{-5}+a^{4}z^{6}+9z^{6}a^{-4}-3a^{4}z^{4}-9z^{4}a^{-4}+3a^{4}z^{2}+4z^{2}a^{-4}-a^{4}-a^{-4}+3a^{3}z^{7}+12z^{7}a^{-3}-8a^{3}z^{5}-18z^{5}a^{-3}+8a^{3}z^{3}+11z^{3}a^{-3}-4a^{3}z-4za^{-3}+a^{3}z^{-1}+a^{-3}z^{-1}+4a^{2}z^{8}+8z^{8}a^{-2}-6a^{2}z^{6}-3z^{6}a^{-2}-4a^{2}z^{4}-14z^{4}a^{-2}+10a^{2}z^{2}+12z^{2}a^{-2}-4a^{2}-4a^{-2}+2az^{9}+2z^{9}a^{-1}+8az^{7}+17z^{7}a^{-1}-31az^{5}-45z^{5}a^{-1}+29az^{3}+33z^{3}a^{-1}-13az-13za^{-1}+4az^{-1}+4a^{-1}z^{-1}+12z^{8}-19z^{6}-5z^{4}+15z^{2}-7$ (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

3

8

6

1

-5

6

7

3

4

8

6

-2

2

9

7

2

0

7

10

3

-2

5

7

-2

-4

2

7

5

-6

1

5

-4

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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