L11n232

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_18,11,19,12 X_8,9,1,10 X_22,19,9,20 X_20,6,21,5 X_4,22,5,21 X_7,15,8,14 X_12,4,13,3 X_13,16,14,17 X_15,7,16,6 X_2,18,3,17
Gauss code	{1, -11, 8, -6, 5, 10, -7, -3}, {3, -1, 2, -8, -9, 7, -10, 9, 11, -2, 4, -5, 6, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}\left(-v^{2}\right)+2u^{3}v-u^{3}+u^{2}v^{2}-u^{2}v+u^{2}+uv^{3}-uv^{2}+uv-v^{3}+2v^{2}-v}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$2q^{9/2}-4q^{7/2}+4q^{5/2}-{\frac {1}{q^{5/2}}}-5q^{3/2}+{\frac {3}{q^{3/2}}}-q^{11/2}+4{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-1}+az^{3}-3z^{3}a^{-1}+2z^{3}a^{-3}+az-2za^{-1}+4za^{-3}-za^{-5}+a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{7}a^{-5}-5z^{5}a^{-5}+8z^{3}a^{-5}-5za^{-5}+a^{-5}z^{-1}+2z^{8}a^{-4}-9z^{6}a^{-4}+10z^{4}a^{-4}-2z^{2}a^{-4}-a^{-4}+z^{9}a^{-3}-z^{7}a^{-3}-11z^{5}a^{-3}+18z^{3}a^{-3}-8za^{-3}+a^{-3}z^{-1}+4z^{8}a^{-2}-17z^{6}a^{-2}+17z^{4}a^{-2}+a^{2}z^{2}-4z^{2}a^{-2}+z^{9}a^{-1}+az^{7}-z^{7}a^{-1}-4az^{5}-10z^{5}a^{-1}+5az^{3}+15z^{3}a^{-1}-2az-5za^{-1}+2z^{8}-8z^{6}+7z^{4}-z^{2}$ (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		\

-2

-1

0

1

2

3

4

5

6

χ

12

1

10

1

-1

8

3

1

2

6

2

0

4

3

2

1

2

1

2

1

0

3

0

-2

1

2

1

-4

2

-2

-6

1

Integral Khovanov Homology

(db, data source)

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