L10a24

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_14,12,15,11 X_10,4,11,3 X_12,5,13,6 X_20,13,5,14 X_16,9,17,10 X_2,16,3,15 X_8,17,9,18
Gauss code	{1, -9, 5, -3}, {6, -1, 2, -10, 8, -5, 4, -6, 7, -4, 9, -8, 10, -2, 3, -7}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2(t(1)-1)(t(2)-1)\left(t(2)^{2}-3t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+7{\sqrt {q}}-{\frac {11}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {14}{q^{5/2}}}+{\frac {12}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {6}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+2a^{5}z^{3}+a^{5}z-a^{5}z^{-1}-a^{3}z^{5}+2a^{3}z+3a^{3}z^{-1}-az^{5}-az^{3}+z^{3}a^{-1}-2az-2az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+2a^{8}z^{2}+3a^{7}z^{7}-9a^{7}z^{5}+7a^{7}z^{3}-2a^{7}z+4a^{6}z^{8}-11a^{6}z^{6}+9a^{6}z^{4}-4a^{6}z^{2}+a^{6}+2a^{5}z^{9}+2a^{5}z^{7}-16a^{5}z^{5}+14a^{5}z^{3}-2a^{5}z-a^{5}z^{-1}+10a^{4}z^{8}-26a^{4}z^{6}+24a^{4}z^{4}-11a^{4}z^{2}+3a^{4}+2a^{3}z^{9}+7a^{3}z^{7}-21a^{3}z^{5}+12a^{3}z^{3}+3a^{3}z-3a^{3}z^{-1}+6a^{2}z^{8}-7a^{2}z^{6}+4a^{2}z^{4}+z^{4}a^{-2}-5a^{2}z^{2}+3a^{2}+8az^{7}-10az^{5}+4z^{5}a^{-1}+2az^{3}-3z^{3}a^{-1}+3az-2az^{-1}+7z^{6}-7z^{4}$ (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

6

1

-1

4

3

2

4

1

-3

0

7

3

4

-2

7

6

-1

-4

7

5

2

-6

5

7

2

-8

4

7

-3

-10

2

5

3

-12

1

4

-3

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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