L11n384

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_10,6,11,5 X₈₄₉₃ X_22,18,19,17 X_11,20,12,21 X_19,12,20,13 X_18,22,5,21 X_16,10,17,9 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {-8, 7, 9, -6}, {4, -1, 2, -5, 10, -4, -7, 8, 11, -2, 3, -10, 6, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)^{3}}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{6}-3q^{5}+5q^{4}-8q^{3}+11q^{2}-10q+11-7q^{-1}+6q^{-2}-2q^{-3}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-4z^{4}a^{-2}+z^{4}a^{-4}+3z^{4}-2a^{2}z^{2}-7z^{2}a^{-2}+2z^{2}a^{-4}+7z^{2}-a^{2}-3a^{-2}+a^{-4}+3+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$2z^{9}a^{-1}+2z^{9}a^{-3}+9z^{8}a^{-2}+4z^{8}a^{-4}+5z^{8}+4az^{7}+z^{7}a^{-1}+3z^{7}a^{-5}+a^{2}z^{6}-31z^{6}a^{-2}-13z^{6}a^{-4}+z^{6}a^{-6}-16z^{6}-7az^{5}-13z^{5}a^{-1}-16z^{5}a^{-3}-10z^{5}a^{-5}+6a^{2}z^{4}+41z^{4}a^{-2}+12z^{4}a^{-4}-3z^{4}a^{-6}+32z^{4}+3a^{3}z^{3}+10az^{3}+20z^{3}a^{-1}+21z^{3}a^{-3}+8z^{3}a^{-5}-9a^{2}z^{2}-26z^{2}a^{-2}-6z^{2}a^{-4}+z^{2}a^{-6}-28z^{2}-a^{3}z-5az-9za^{-1}-7za^{-3}-2za^{-5}+2a^{2}+6a^{-2}+2a^{-4}+7-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

χ

13

1

11

2

-2

9

3

1

2

7

5

2

-3

5

6

3

4

5

1

7

6

1

-1

4

8

4

-3

2

3

-1

-5

4

-7

2

-2

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-3$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{7}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$