L10a22

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,9,15,10 X_4,19,1,20 X_12,6,13,5 X_10,4,11,3 X_20,12,5,11 X_2,14,3,13 X_18,16,19,15 X_16,8,17,7 X_8,18,9,17
Gauss code	{1, -7, 5, -3}, {4, -1, 9, -10, 2, -5, 6, -4, 7, -2, 8, -9, 10, -8, 3, -6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)^{3}\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{17/2}-4q^{15/2}+9q^{13/2}-12q^{11/2}+15q^{9/2}-17q^{7/2}+14q^{5/2}-12q^{3/2}+7{\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+za^{-7}+a^{-7}z^{-1}-2z^{5}a^{-5}-5z^{3}a^{-5}-4za^{-5}-2a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+6z^{3}a^{-3}+3za^{-3}-z^{5}a^{-1}-2z^{3}a^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}+4z^{5}a^{-9}-z^{3}a^{-9}+9z^{6}a^{-8}-10z^{4}a^{-8}+5z^{2}a^{-8}-2a^{-8}+11z^{7}a^{-7}-15z^{5}a^{-7}+8z^{3}a^{-7}-2za^{-7}+a^{-7}z^{-1}+7z^{8}a^{-6}+z^{6}a^{-6}-19z^{4}a^{-6}+14z^{2}a^{-6}-5a^{-6}+2z^{9}a^{-5}+16z^{7}a^{-5}-41z^{5}a^{-5}+26z^{3}a^{-5}-7za^{-5}+2a^{-5}z^{-1}+12z^{8}a^{-4}-19z^{6}a^{-4}-4z^{4}a^{-4}+11z^{2}a^{-4}-3a^{-4}+2z^{9}a^{-3}+9z^{7}a^{-3}-33z^{5}a^{-3}+26z^{3}a^{-3}-6za^{-3}+5z^{8}a^{-2}-10z^{6}a^{-2}+2z^{4}a^{-2}+3z^{2}a^{-2}+a^{-2}+4z^{7}a^{-1}-11z^{5}a^{-1}+9z^{3}a^{-1}-za^{-1}-a^{-1}z^{-1}+z^{6}-2z^{4}+z^{2}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

7

χ

18

1

-1

16

3

14

6

1

-5

12

6

3

10

9

6

-3

8

6

2

6

9

3

4

6

8

-2

2

3

8

5

0

1

4

-3

-2

3

-4

1

-1

Integral Khovanov Homology

(db, data source)

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