L10a44

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_16,8,17,7 X_18,10,19,9 X_20,12,5,11 X_8,18,9,17 X_10,20,11,19 X_12,16,13,15 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -6, 4, -7, 5, -8, 10, -2, 8, -3, 6, -4, 7, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

16

1

14

2

-2

12

2

1

10

4

2

-2

8

3

2

1

6

4

0

4

2

3

-1

2

3

5

2

0

1

0

-2

3

-4

1

0

-6

1

Integral Khovanov Homology

(db, data source)

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