L10a54

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_18,11,19,12 X_12,6,13,5 X_4,18,5,17 X_14,7,15,8 X_16,14,17,13 X_20,15,7,16 X_6,19,1,20
Gauss code	{1, -2, 3, -6, 5, -10}, {7, -1, 2, -3, 4, -5, 8, -7, 9, -8, 6, -4, 10, -9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-4t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-t(2)^{3}+4t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+4t(2)^{2}-t(1)^{2}t(2)+5t(1)t(2)-4t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {12}{q^{5/2}}}-{\frac {13}{q^{7/2}}}+{\frac {12}{q^{9/2}}}-{\frac {10}{q^{11/2}}}+{\frac {6}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-2a^{7}z+2a^{5}z^{5}+6a^{5}z^{3}+4a^{5}z-a^{3}z^{7}-4a^{3}z^{5}-5a^{3}z^{3}-2a^{3}z+a^{3}z^{-1}+az^{5}+2az^{3}-az-az^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{10}+z^{2}a^{10}-3z^{5}a^{9}+3z^{3}a^{9}-za^{9}-5z^{6}a^{8}+5z^{4}a^{8}-2z^{2}a^{8}-6z^{7}a^{7}+7z^{5}a^{7}-4z^{3}a^{7}+za^{7}-5z^{8}a^{6}+5z^{6}a^{6}-z^{4}a^{6}-2z^{9}a^{5}-6z^{7}a^{5}+21z^{5}a^{5}-18z^{3}a^{5}+5za^{5}-10z^{8}a^{4}+25z^{6}a^{4}-19z^{4}a^{4}+6z^{2}a^{4}-2z^{9}a^{3}-4z^{7}a^{3}+23z^{5}a^{3}-19z^{3}a^{3}+2za^{3}+a^{3}z^{-1}-5z^{8}a^{2}+14z^{6}a^{2}-10z^{4}a^{2}+3z^{2}a^{2}-a^{2}-4z^{7}a+12z^{5}a-8z^{3}a-za+az^{-1}-z^{6}+2z^{4}$ (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

3

1

-2

7

3

4

-4

6

4

-2

-6

7

6

1

-8

6

7

1

-10

4

6

-2

-12

2

6

4

-14

1

4

-3

-16

2

-18

1

-1

Integral Khovanov Homology

(db, data source)

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