L10a27

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-2t(2)^{3}+t(2)^{2}-2t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{15/2}+3q^{13/2}-5q^{11/2}+7q^{9/2}-9q^{7/2}+9q^{5/2}-8q^{3/2}+6{\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-5}-3z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+z^{7}a^{-3}+5z^{5}a^{-3}+9z^{3}a^{-3}+8za^{-3}+3a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-8z^{3}a^{-1}+3az-9za^{-1}+2az^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-9}+3z^{4}a^{-8}-z^{2}a^{-8}+5z^{5}a^{-7}-4z^{3}a^{-7}+za^{-7}+6z^{6}a^{-6}-7z^{4}a^{-6}+2z^{2}a^{-6}-a^{-6}+6z^{7}a^{-5}-10z^{5}a^{-5}+3z^{3}a^{-5}-2za^{-5}+a^{-5}z^{-1}+4z^{8}a^{-4}-5z^{6}a^{-4}-8z^{4}a^{-4}+10z^{2}a^{-4}-3a^{-4}+z^{9}a^{-3}+7z^{7}a^{-3}-32z^{5}a^{-3}+34z^{3}a^{-3}-15za^{-3}+3a^{-3}z^{-1}+6z^{8}a^{-2}-19z^{6}a^{-2}+10z^{4}a^{-2}+7z^{2}a^{-2}-3a^{-2}+z^{9}a^{-1}+az^{7}+2z^{7}a^{-1}-5az^{5}-22z^{5}a^{-1}+9az^{3}+35z^{3}a^{-1}-7az-19za^{-1}+2az^{-1}+4a^{-1}z^{-1}+2z^{8}-8z^{6}+8z^{4}-2$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

16

1

14

2

-2

12

3

1

2

10

4

2

-2

8

5

3

2

6

4

0

4

5

-1

2

4

6

2

0

1

2

-1

-2

1

4

3

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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