L10a116

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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_8,15,9,16 X_16,5,17,6 X_18,7,19,8 X_6,17,7,18 X_4,19,5,20 X_2,11,3,12 X_10,13,1,14
Gauss code	{1, -9, 3, -8, 5, -7, 6, -4, 2, -10}, {9, -1, 10, -3, 4, -5, 7, -6, 8, -2}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-4

1

-6

2

1

-1

-8

3

-10

4

2

-2

-12

5

3

2

-14

5

4

-1

-16

4

5

-1

-18

3

5

2

-20

2

4

-2

-22

3

-24

1

2

-1

-26

1

Integral Khovanov Homology

(db, data source)

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