L11n152

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X₆₇₁₈ X_19,7,20,22 X_12,5,13,6 X_10,4,11,3 X_4,15,5,16 X_16,12,17,11 X_13,21,14,20 X_21,15,22,14 X_2,18,3,17
Gauss code	{1, -11, 6, -7, 5, -3}, {3, -1, 2, -6, 8, -5, -9, 10, 7, -8, 11, -2, -4, 9, -10, 4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{4}-t(2)^{4}-3t(1)t(2)^{3}+2t(2)^{3}-t(1)^{2}t(2)^{2}+3t(1)t(2)^{2}-t(2)^{2}+2t(1)^{2}t(2)-3t(1)t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$2q^{7/2}-4q^{5/2}+5q^{3/2}-7{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$az^{5}-a^{3}z^{3}+3az^{3}-3z^{3}a^{-1}-a^{3}z+4az-6za^{-1}+2za^{-3}+2az^{-1}-3a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$a^{4}z^{6}-3a^{4}z^{4}+a^{4}z^{2}+2z^{2}a^{-4}-a^{-4}+3a^{3}z^{7}-11a^{3}z^{5}+z^{5}a^{-3}+9a^{3}z^{3}+2z^{3}a^{-3}-2a^{3}z-3za^{-3}+a^{-3}z^{-1}+3a^{2}z^{8}+z^{8}a^{-2}-10a^{2}z^{6}-3z^{6}a^{-2}+6a^{2}z^{4}+4z^{4}a^{-2}+3z^{2}a^{-2}-3a^{-2}+az^{9}+z^{9}a^{-1}+az^{7}-2z^{7}a^{-1}-14az^{5}-2z^{5}a^{-1}+18az^{3}+11z^{3}a^{-1}-9az-10za^{-1}+2az^{-1}+3a^{-1}z^{-1}+4z^{8}-14z^{6}+13z^{4}-3$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

χ

8

2

-2

6

2

4

3

2

-1

2

4

2

0

3

4

1

-2

3

0

-4

2

4

2

-6

1

2

-1

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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