L10a110

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X₈₄₉₃ X_18,14,19,13 X_20,16,11,15 X_14,20,15,19 X_6,17,7,18 X_16,7,17,8 X_10,6,1,5 X_4,10,5,9 X_2,11,3,12
Gauss code	{1, -10, 2, -9, 8, -6, 7, -2, 9, -8}, {10, -1, 3, -5, 4, -7, 6, -3, 5, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{3}t(1)^{3}-2t(2)^{2}t(1)^{3}+2t(2)t(1)^{3}-2t(1)^{3}-2t(2)^{3}t(1)^{2}+2t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}+2t(1)^{2}+2t(2)^{3}t(1)-2t(2)^{2}t(1)+2t(2)t(1)-2t(1)-2t(2)^{3}+2t(2)^{2}-2t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{15/2}+3q^{13/2}-6q^{11/2}+9q^{9/2}-9q^{7/2}+10q^{5/2}-9q^{3/2}+6{\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-5}-3z^{3}a^{-5}-3za^{-5}-a^{-5}z^{-1}+z^{7}a^{-3}+5z^{5}a^{-3}+10z^{3}a^{-3}+11za^{-3}+5a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-9z^{3}a^{-1}+4az-14za^{-1}+4az^{-1}-8a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-5z^{8}a^{-2}-4z^{8}a^{-4}-z^{8}-az^{7}-7z^{7}a^{-3}-8z^{7}a^{-5}+11z^{6}a^{-2}-9z^{6}a^{-6}+2z^{6}+6az^{5}+12z^{5}a^{-1}+26z^{5}a^{-3}+14z^{5}a^{-5}-6z^{5}a^{-7}+9z^{4}a^{-2}+22z^{4}a^{-4}+15z^{4}a^{-6}-3z^{4}a^{-8}+5z^{4}-13az^{3}-24z^{3}a^{-1}-20z^{3}a^{-3}-4z^{3}a^{-5}+4z^{3}a^{-7}-z^{3}a^{-9}-28z^{2}a^{-2}-23z^{2}a^{-4}-9z^{2}a^{-6}-14z^{2}+12az+20za^{-1}+9za^{-3}-za^{-7}+14a^{-2}+9a^{-4}+2a^{-6}+8-4az^{-1}-8a^{-1}z^{-1}-5a^{-3}z^{-1}-a^{-5}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

16

1

14

2

-2

12

4

1

3

10

5

2

-3

8

4

0

6

5

-1

4

3

4

-1

2

4

7

3

0

1

2

-1

-2

4

-4

1

0

-6

1

Integral Khovanov Homology

(db, data source)

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