L11n216

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,3,13,4 X_18,8,19,7 X_14,6,15,5 X_17,9,18,22 X_21,17,22,16 X_20,13,21,14 X_6,16,7,15 X_4,20,5,19 X_2,9,3,10 X_8,11,1,12
Gauss code	{1, -10, 2, -9, 4, -8, 3, -11}, {10, -1, 11, -2, 7, -4, 8, 6, -5, -3, 9, -7, -6, 5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

2

-2

10

3

1

2

8

4

2

-2

6

4

3

1

4

0

2

4

0

2

6

4

-2

1

2

-1

-4

2

-6

1

-1

Integral Khovanov Homology

(db, data source)

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