L10a105

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{3}t(1)^{3}-2t(2)^{2}t(1)^{3}-2t(2)^{3}t(1)^{2}+4t(2)^{2}t(1)^{2}-3t(2)t(1)^{2}-3t(2)^{2}t(1)+4t(2)t(1)-2t(1)-2t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {4}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {8}{q^{13/2}}}+{\frac {6}{q^{15/2}}}-{\frac {4}{q^{17/2}}}+{\frac {2}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{9}+3za^{9}+a^{9}z^{-1}-2z^{5}a^{7}-8z^{3}a^{7}-8za^{7}-a^{7}z^{-1}+z^{7}a^{5}+5z^{5}a^{5}+8z^{3}a^{5}+5za^{5}-z^{5}a^{3}-4z^{3}a^{3}-4za^{3}$ (db)
Kauffman polynomial	$a^{13}z^{3}-a^{13}z+2a^{12}z^{4}-a^{12}z^{2}+3a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+4a^{10}z^{6}-5a^{10}z^{4}+2a^{10}z^{2}+5a^{9}z^{7}-13a^{9}z^{5}+14a^{9}z^{3}-7a^{9}z+a^{9}z^{-1}+3a^{8}z^{8}-4a^{8}z^{6}-6a^{8}z^{4}+6a^{8}z^{2}-a^{8}+a^{7}z^{9}+4a^{7}z^{7}-23a^{7}z^{5}+25a^{7}z^{3}-9a^{7}z+a^{7}z^{-1}+5a^{6}z^{8}-17a^{6}z^{6}+13a^{6}z^{4}-2a^{6}z^{2}+a^{5}z^{9}-12a^{5}z^{5}+16a^{5}z^{3}-4a^{5}z+2a^{4}z^{8}-9a^{4}z^{6}+12a^{4}z^{4}-5a^{4}z^{2}+a^{3}z^{7}-5a^{3}z^{5}+8a^{3}z^{3}-4a^{3}z$ (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

0

1

-2

1

-1

-4

3

1

2

-6

4

2

-2

-8

3

2

1

-10

4

0

-12

4

3

1

-14

2

4

2

-16

2

4

-2

-18

2

-20

1

2

-1

-22

1

Integral Khovanov Homology

(db, data source)

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