L11a344

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_18,7,19,8 X_10,5,1,6 X₆₃₇₄ X_4,9,5,10 X_20,14,21,13 X_22,16,11,15 X_14,22,15,21 X_16,20,17,19 X_2,11,3,12 X_8,17,9,18
Gauss code	{1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, {10, -1, 6, -8, 7, -9, 11, -2, 9, -6, 8, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{3}t(1)^{3}-2t(2)^{2}t(1)^{3}+2t(2)t(1)^{3}-2t(1)^{3}-4t(2)^{3}t(1)^{2}+6t(2)^{2}t(1)^{2}-6t(2)t(1)^{2}+4t(1)^{2}+4t(2)^{3}t(1)-6t(2)^{2}t(1)+6t(2)t(1)-4t(1)-2t(2)^{3}+2t(2)^{2}-2t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {10}{q^{9/2}}}-q^{7/2}+{\frac {13}{q^{7/2}}}+4q^{5/2}-{\frac {18}{q^{5/2}}}-8q^{3/2}+{\frac {17}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {5}{q^{11/2}}}+13{\sqrt {q}}-{\frac {16}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-2a^{7}z^{-1}+3z^{3}a^{5}+9za^{5}+7a^{5}z^{-1}-3z^{5}a^{3}-11z^{3}a^{3}-15za^{3}-7a^{3}z^{-1}+z^{7}a+4z^{5}a+7z^{3}a+6za+2az^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}-za^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-4a^{8}z^{4}+5a^{8}z^{2}-2a^{8}+2a^{7}z^{7}-6a^{7}z^{5}+6a^{7}z^{3}-4a^{7}z+2a^{7}z^{-1}+2a^{6}z^{8}-13a^{6}z^{4}+18a^{6}z^{2}-8a^{6}+2a^{5}z^{9}+a^{5}z^{7}-11a^{5}z^{5}+17a^{5}z^{3}-16a^{5}z+7a^{5}z^{-1}+a^{4}z^{10}+4a^{4}z^{8}-5a^{4}z^{6}-11a^{4}z^{4}+24a^{4}z^{2}-13a^{4}+6a^{3}z^{9}-6a^{3}z^{7}-5a^{3}z^{5}+z^{5}a^{-3}+15a^{3}z^{3}-z^{3}a^{-3}-15a^{3}z+7a^{3}z^{-1}+a^{2}z^{10}+9a^{2}z^{8}-16a^{2}z^{6}+4z^{6}a^{-2}+2a^{2}z^{4}-6z^{4}a^{-2}+13a^{2}z^{2}+2z^{2}a^{-2}-8a^{2}+4az^{9}+2az^{7}+7z^{7}a^{-1}-12az^{5}-11z^{5}a^{-1}+9az^{3}+4z^{3}a^{-1}-4az-za^{-1}+2az^{-1}+7z^{8}-8z^{6}-2z^{4}+4z^{2}-2$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

5

1

4

2

8

3

-5

0

8

5

3

-2

10

9

-1

-4

8

7

1

-6

5

10

5

-8

5

8

-3

-10

1

6

5

-12

1

4

-3

-14

1

-16

1

-1

Integral Khovanov Homology

(db, data source)

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