L10a5

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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_14,10,15,9 X₈₄₉₃ X_12,5,13,6 X_20,13,5,14 X_16,12,17,11 X_10,16,11,15 X_2,18,3,17
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, 4, -9, 8, -6, 7, -4, 9, -8, 10, -2, 3, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{4}-3v^{3}+3v^{2}-3v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {1}{q^{9/2}}}+8q^{7/2}-{\frac {3}{q^{7/2}}}-12q^{5/2}+{\frac {6}{q^{5/2}}}+14q^{3/2}-{\frac {11}{q^{3/2}}}+q^{11/2}-15{\sqrt {q}}+{\frac {13}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-a^{3}z^{3}+6az^{3}-6z^{3}a^{-1}+2z^{3}a^{-3}-2a^{3}z+5az-4za^{-1}+za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2az^{9}-2z^{9}a^{-1}-4a^{2}z^{8}-7z^{8}a^{-2}-11z^{8}-3a^{3}z^{7}-5az^{7}-12z^{7}a^{-1}-10z^{7}a^{-3}-a^{4}z^{6}+9a^{2}z^{6}+5z^{6}a^{-2}-8z^{6}a^{-4}+23z^{6}+9a^{3}z^{5}+25az^{5}+34z^{5}a^{-1}+14z^{5}a^{-3}-4z^{5}a^{-5}+3a^{4}z^{4}-4a^{2}z^{4}+5z^{4}a^{-2}+8z^{4}a^{-4}-z^{4}a^{-6}-11z^{4}-9a^{3}z^{3}-26az^{3}-26z^{3}a^{-1}-7z^{3}a^{-3}+2z^{3}a^{-5}-2a^{4}z^{2}-a^{2}z^{2}-3z^{2}a^{-2}-2z^{2}a^{-4}+4a^{3}z+10az+8za^{-1}+2za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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

3

8

5

1

-4

6

7

3

4

7

5

-2

2

8

7

1

0

7

9

2

-2

4

6

-2

-4

2

7

5

-6

1

4

-3

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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