L11n159

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_7,17,8,16 X_10,4,11,3 X_2,15,3,16 X_14,10,15,9 X_11,19,12,18 X_5,13,6,12 X_21,1,22,6 X_20,14,21,13 X_17,7,18,22 X_4,20,5,19
Gauss code	{1, -4, 3, -11, -7, 8}, {-2, -1, 5, -3, -6, 7, 9, -5, 4, 2, -10, 6, 11, -9, -8, 10}

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

Polynomial invariants

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

-2

-1

0

1

2

3

4

5

χ

16

2

-2

14

1

0

12

2

0

10

2

1

2

8

1

3

2

6

2

1

4

1

2

1

2

1

-1

0

1

Integral Khovanov Homology

(db, data source)

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