L11n182

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_9,21,10,20 X_14,5,15,6 X_16,8,17,7 X_3,10,4,11 X_22,14,7,13 X_18,12,19,11 X_12,18,13,17 X_6,15,1,16 X_4,21,5,22 X_19,2,20,3
Gauss code	{1, 11, -5, -10, 3, -9}, {4, -1, -2, 5, 7, -8, 6, -3, 9, -4, 8, -7, -11, 2, 10, -6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{3}-3u^{2}v^{2}+3u^{2}v+2uv^{4}-5uv^{3}+7uv^{2}-5uv+2u+3v^{3}-3v^{2}+v}{uv^{2}}}$ (db)
Jones polynomial	$2q^{5/2}-5q^{3/2}+8{\sqrt {q}}-{\frac {11}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {12}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-a^{3}z^{3}-2az^{5}-6az^{3}+2z^{3}a^{-1}-7az-2az^{-1}+3za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-2a^{7}z^{3}+3a^{6}z^{6}-5a^{6}z^{4}+6a^{5}z^{7}-16a^{5}z^{5}+16a^{5}z^{3}-8a^{5}z+a^{5}z^{-1}+5a^{4}z^{8}-10a^{4}z^{6}+6a^{4}z^{4}+a^{4}z^{2}-a^{4}+2a^{3}z^{9}+2a^{3}z^{7}-10a^{3}z^{5}+10a^{3}z^{3}-2a^{3}z+8a^{2}z^{8}-21a^{2}z^{6}+27a^{2}z^{4}+3z^{4}a^{-2}-14a^{2}z^{2}-5z^{2}a^{-2}+3a^{2}+2a^{-2}+2az^{9}-3az^{7}+z^{7}a^{-1}+10az^{5}+3z^{5}a^{-1}-15az^{3}-7z^{3}a^{-1}+9az+3za^{-1}-2az^{-1}-a^{-1}z^{-1}+3z^{8}-8z^{6}+19z^{4}-20z^{2}+5$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

2

-2

4

3

2

5

2

-3

0

6

3

-2

7

6

-1

-4

5

0

-6

4

7

3

-8

3

5

-2

-10

4

-12

1

3

-2

-14

1

Integral Khovanov Homology

(db, data source)

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