L10a107

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_20,5,9,6 X_14,3,15,4 X_4,15,5,16 X_16,7,17,8 X_18,12,19,11 X_12,18,13,17 X_2,9,3,10 X_8,13,1,14 X_6,19,7,20
Gauss code	{1, -8, 3, -4, 2, -10, 5, -9}, {8, -1, 6, -7, 9, -3, 4, -5, 7, -6, 10, -2}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{2}t(1)^{3}-2t(2)t(1)^{3}+2t(2)^{3}t(1)^{2}-7t(2)^{2}t(1)^{2}+7t(2)t(1)^{2}-3t(1)^{2}-3t(2)^{3}t(1)+7t(2)^{2}t(1)-7t(2)t(1)+2t(1)-2t(2)^{2}+t(2)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {15}{q^{9/2}}}-{\frac {14}{q^{7/2}}}+{\frac {11}{q^{5/2}}}-{\frac {7}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {8}{q^{15/2}}}+{\frac {11}{q^{13/2}}}-{\frac {14}{q^{11/2}}}-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}-3z^{3}a^{7}-4za^{7}-a^{7}z^{-1}+2z^{5}a^{5}+4z^{3}a^{5}+2za^{5}+z^{5}a^{3}-2za^{3}-z^{3}a-za$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-4a^{10}z^{4}+a^{10}z^{2}+6a^{9}z^{7}-12a^{9}z^{5}+12a^{9}z^{3}-8a^{9}z+a^{9}z^{-1}+5a^{8}z^{8}-3a^{8}z^{6}-7a^{8}z^{4}+6a^{8}z^{2}-a^{8}+2a^{7}z^{9}+8a^{7}z^{7}-21a^{7}z^{5}+17a^{7}z^{3}-7a^{7}z+a^{7}z^{-1}+10a^{6}z^{8}-15a^{6}z^{6}+5a^{6}z^{4}+a^{6}z^{2}+2a^{5}z^{9}+7a^{5}z^{7}-17a^{5}z^{5}+10a^{5}z^{3}-a^{5}z+5a^{4}z^{8}-6a^{4}z^{6}+3a^{4}z^{4}-2a^{4}z^{2}+5a^{3}z^{7}-8a^{3}z^{5}+5a^{3}z^{3}-2a^{3}z+3a^{2}z^{6}-5a^{2}z^{4}+2a^{2}z^{2}+az^{5}-2az^{3}+az$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

2

-2

5

1

4

-4

7

3

-4

-6

7

4

3

-8

8

7

-1

-10

6

7

-1

-12

5

8

3

-14

3

6

-3

-16

5

-18

1

3

-2

-20

1

Integral Khovanov Homology

(db, data source)

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