L11n185

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_9,21,10,20 X_21,1,22,6 X_18,8,19,7 X_3,10,4,11 X_15,12,16,13 X_5,14,6,15 X_13,4,14,5 X_11,16,12,17 X_22,18,7,17 X_2,20,3,19
Gauss code	{1, -11, -5, 8, -7, 3}, {4, -1, -2, 5, -9, 6, -8, 7, -6, 9, 10, -4, 11, 2, -3, -10}

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

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

1

6

3

1

-2

4

3

1

2

3

0

5

3

2

-2

2

4

2

-4

3

4

-1

-6

1

3

2

-8

2

-2

-10

1

Integral Khovanov Homology

(db, data source)

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