L10a149

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(3)-1)^{2}(2t(3)t(2)-t(2)-t(3)+2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+5q^{7}-9q^{6}+13q^{5}-15q^{4}+17q^{3}-14q^{2}+12q-6+3q^{-1}-q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}+a^{-6}z^{-2}+2a^{-6}+z^{6}a^{-4}+z^{4}a^{-4}-4z^{2}a^{-4}-2a^{-4}z^{-2}-7a^{-4}+z^{6}a^{-2}+3z^{4}a^{-2}+6z^{2}a^{-2}+a^{-2}z^{-2}+6a^{-2}-z^{4}-2z^{2}-1$ (db)
Kauffman polynomial	$z^{5}a^{-9}+5z^{6}a^{-8}-6z^{4}a^{-8}+a^{-8}+9z^{7}a^{-7}-15z^{5}a^{-7}+4z^{3}a^{-7}+7z^{8}a^{-6}-5z^{6}a^{-6}-8z^{4}a^{-6}+5z^{2}a^{-6}+a^{-6}z^{-2}-3a^{-6}+2z^{9}a^{-5}+13z^{7}a^{-5}-28z^{5}a^{-5}+9z^{3}a^{-5}+5za^{-5}-2a^{-5}z^{-1}+11z^{8}a^{-4}-12z^{6}a^{-4}-9z^{4}a^{-4}+18z^{2}a^{-4}+2a^{-4}z^{-2}-10a^{-4}+2z^{9}a^{-3}+8z^{7}a^{-3}-16z^{5}a^{-3}+4z^{3}a^{-3}+7za^{-3}-2a^{-3}z^{-1}+4z^{8}a^{-2}+z^{6}a^{-2}-13z^{4}a^{-2}+18z^{2}a^{-2}+a^{-2}z^{-2}-9a^{-2}+4z^{7}a^{-1}+az^{5}-3z^{5}a^{-1}-2az^{3}-3z^{3}a^{-1}+az+3za^{-1}+3z^{6}-6z^{4}+5z^{2}-2$ (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

17

1

-1

15

4

13

5

1

-4

11

8

4

9

7

-2

7

8

6

2

5

6

9

3

6

8

-2

1

2

8

6

-1

1

4

-3

2

-5

1

-1

Integral Khovanov Homology

(db, data source)

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