L11n231

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_16,11,17,12 X_5,21,6,20 X_12,4,13,3 X_14,8,15,7 X_6,14,7,13 X_17,9,18,22 X_21,19,22,18 X_8,9,1,10 X_19,5,20,4 X_2,16,3,15
Gauss code	{1, -11, 4, 10, -3, -6, 5, -9}, {9, -1, 2, -4, 6, -5, 11, -2, -7, 8, -10, 3, -8, 7}

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

Polynomial invariants

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

-3

-2

-1

0

1

2

3

4

5

6

χ

16

1

14

2

-2

12

4

1

3

10

5

3

-2

8

4

3

1

6

4

5

1

4

0

2

5

3

0

1

3

-2

2

-4

1

-1

Integral Khovanov Homology

(db, data source)

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