L11n221

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_8,9,1,10 X_12,4,13,3 X_22,16,9,15 X_2,17,3,18 X_21,4,22,5 X_5,15,6,14 X_13,21,14,20 X_16,12,17,11 X_6,19,7,20 X_18,7,19,8
Gauss code	{1, -5, 3, 6, -7, -10, 11, -2}, {2, -1, 9, -3, -8, 7, 4, -9, 5, -11, 10, 8, -6, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{2}t(1)^{3}-t(2)t(1)^{3}+2t(2)^{3}t(1)^{2}-6t(2)^{2}t(1)^{2}+6t(2)t(1)^{2}-t(1)^{2}-t(2)^{3}t(1)+6t(2)^{2}t(1)-6t(2)t(1)+2t(1)-t(2)^{2}+t(2)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {2}{q^{9/2}}}-3q^{7/2}+{\frac {5}{q^{7/2}}}+6q^{5/2}-{\frac {9}{q^{5/2}}}-9q^{3/2}+{\frac {10}{q^{3/2}}}+11{\sqrt {q}}-{\frac {12}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$2a^{3}z^{3}+z^{3}a^{-3}+3a^{3}z+za^{-3}+2a^{3}z^{-1}-2az^{5}-z^{5}a^{-1}-6az^{3}-z^{3}a^{-1}-7az+za^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2az^{9}-2z^{9}a^{-1}-4a^{2}z^{8}-4z^{8}a^{-2}-8z^{8}-3a^{3}z^{7}-3z^{7}a^{-3}-a^{4}z^{6}+7a^{2}z^{6}+11z^{6}a^{-2}-z^{6}a^{-4}+20z^{6}-az^{5}+8z^{5}a^{-1}+9z^{5}a^{-3}-4a^{4}z^{4}-11a^{2}z^{4}-8z^{4}a^{-2}+3z^{4}a^{-4}-18z^{4}-3a^{5}z^{3}+3a^{3}z^{3}+11az^{3}-2z^{3}a^{-1}-7z^{3}a^{-3}+2a^{4}z^{2}+9a^{2}z^{2}+3z^{2}a^{-2}-2z^{2}a^{-4}+12z^{2}+2a^{5}z-5a^{3}z-10az-2za^{-1}+za^{-3}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

4

1

-3

4

5

2

3

2

6

4

-2

0

6

5

1

-2

5

7

2

-4

4

5

-1

-6

1

5

4

-8

1

4

-3

-10

2

Integral Khovanov Homology

(db, data source)

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