L10a106

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{3}t(1)^{3}-2t(2)^{2}t(1)^{3}+t(2)t(1)^{3}-3t(2)^{3}t(1)^{2}+7t(2)^{2}t(1)^{2}-5t(2)t(1)^{2}+2t(1)^{2}+2t(2)^{3}t(1)-5t(2)^{2}t(1)+7t(2)t(1)-3t(1)+t(2)^{2}-2t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {13}{q^{9/2}}}-{\frac {14}{q^{7/2}}}+{\frac {13}{q^{5/2}}}+q^{3/2}-{\frac {11}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {11}{q^{11/2}}}-4{\sqrt {q}}+{\frac {7}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-2a^{7}z+2a^{5}z^{5}+6a^{5}z^{3}+5a^{5}z+a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-6a^{3}z^{3}-5a^{3}z-a^{3}z^{-1}+az^{5}+2az^{3}$ (db)
Kauffman polynomial	$-z^{4}a^{10}+z^{2}a^{10}-3z^{5}a^{9}+3z^{3}a^{9}-za^{9}-5z^{6}a^{8}+4z^{4}a^{8}-z^{2}a^{8}-7z^{7}a^{7}+10z^{5}a^{7}-8z^{3}a^{7}+za^{7}-6z^{8}a^{6}+7z^{6}a^{6}-2z^{4}a^{6}-z^{2}a^{6}-2z^{9}a^{5}-10z^{7}a^{5}+33z^{5}a^{5}-30z^{3}a^{5}+10za^{5}-a^{5}z^{-1}-11z^{8}a^{4}+24z^{6}a^{4}-12z^{4}a^{4}+z^{2}a^{4}+a^{4}-2z^{9}a^{3}-7z^{7}a^{3}+31z^{5}a^{3}-27z^{3}a^{3}+9za^{3}-a^{3}z^{-1}-5z^{8}a^{2}+11z^{6}a^{2}-3z^{4}a^{2}-z^{2}a^{2}-4z^{7}a+11z^{5}a-8z^{3}a+za-z^{6}+2z^{4}-z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

3

0

4

1

-3

-2

7

3

4

-4

7

5

-2

-6

7

6

1

-8

6

7

1

-10

5

7

-2

-12

2

7

5

-14

1

4

-3

-16

2

-18

1

-1

Integral Khovanov Homology

(db, data source)

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