L10a114

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(2)^{3}t(1)^{4}-t(2)^{4}t(1)^{3}+t(2)^{3}t(1)^{3}-t(2)^{2}t(1)^{3}-t(2)^{3}t(1)^{2}+t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(1)+t(2)t(1)-t(1)-t(2)}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}-{\frac {1}{q^{27/2}}}+{\frac {1}{q^{25/2}}}-{\frac {2}{q^{23/2}}}+{\frac {3}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {2}{q^{13/2}}}-{\frac {2}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$z^{5}a^{11}+5z^{3}a^{11}+6za^{11}+a^{11}z^{-1}-z^{7}a^{9}-6z^{5}a^{9}-11z^{3}a^{9}-7za^{9}-a^{9}z^{-1}-z^{7}a^{7}-6z^{5}a^{7}-10z^{3}a^{7}-4za^{7}$ (db)
Kauffman polynomial	$-z^{3}a^{17}+2za^{17}-z^{4}a^{16}+z^{2}a^{16}-z^{5}a^{15}+z^{3}a^{15}-za^{15}-z^{6}a^{14}+2z^{4}a^{14}-3z^{2}a^{14}-z^{7}a^{13}+3z^{5}a^{13}-4z^{3}a^{13}+za^{13}-z^{8}a^{12}+4z^{6}a^{12}-6z^{4}a^{12}+4z^{2}a^{12}-z^{9}a^{11}+6z^{7}a^{11}-15z^{5}a^{11}+19z^{3}a^{11}-8za^{11}+a^{11}z^{-1}-2z^{8}a^{10}+10z^{6}a^{10}-15z^{4}a^{10}+9z^{2}a^{10}-a^{10}-z^{9}a^{9}+6z^{7}a^{9}-13z^{5}a^{9}+15z^{3}a^{9}-8za^{9}+a^{9}z^{-1}-z^{8}a^{8}+5z^{6}a^{8}-6z^{4}a^{8}+z^{2}a^{8}-z^{7}a^{7}+6z^{5}a^{7}-10z^{3}a^{7}+4za^{7}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-6

1

-8

1

0

-10

1

-12

1

0

-14

2

1

-16

1

0

-18

2

0

-20

1

0

-22

1

2

-1

-24

1

-26

1

0

-28

1

Integral Khovanov Homology

(db, data source)

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