L10a111

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_16,7,17,8 X₈₃₉₄ X_2,18,3,17 X_14,6,15,5 X_6,12,7,11 X_18,10,19,9 X_20,15,11,16 X_10,13,1,14 X_4,19,5,20
Gauss code	{1, -4, 3, -10, 5, -6, 2, -3, 7, -9}, {6, -1, 9, -5, 8, -2, 4, -7, 10, -8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}+t(1)^{2}-2t(2)^{2}t(1)+3t(2)t(1)-2t(1)+t(2)^{2}-2t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {9}{q^{9/2}}}-q^{7/2}+{\frac {15}{q^{7/2}}}+5q^{5/2}-{\frac {19}{q^{5/2}}}-11q^{3/2}+{\frac {20}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+15{\sqrt {q}}-{\frac {20}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+3az^{5}-z^{5}a^{-1}+a^{5}z^{3}-4a^{3}z^{3}+3az^{3}-z^{3}a^{-1}+a^{5}z-2a^{3}z+az+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-4a^{3}z^{9}-4az^{9}-9a^{4}z^{8}-20a^{2}z^{8}-11z^{8}-8a^{5}z^{7}-12a^{3}z^{7}-15az^{7}-11z^{7}a^{-1}-4a^{6}z^{6}+11a^{4}z^{6}+34a^{2}z^{6}-5z^{6}a^{-2}+14z^{6}-a^{7}z^{5}+12a^{5}z^{5}+34a^{3}z^{5}+39az^{5}+17z^{5}a^{-1}-z^{5}a^{-3}+5a^{6}z^{4}-3a^{4}z^{4}-14a^{2}z^{4}+4z^{4}a^{-2}-2z^{4}+a^{7}z^{3}-8a^{5}z^{3}-22a^{3}z^{3}-19az^{3}-6z^{3}a^{-1}-2a^{6}z^{2}-a^{4}z^{2}+2a^{2}z^{2}+z^{2}+2a^{5}z+4a^{3}z+2az+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

4

-4

4

7

1

6

2

8

4

-4

0

12

7

5

-2

10

0

-4

9

10

-1

-6

6

10

4

-8

3

9

-6

-10

1

6

5

-12

3

-3

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$