L11n227

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_22,12,9,11 X_2,9,3,10 X_17,21,18,20 X_6,14,7,13 X_14,8,15,7 X_8,16,1,15 X_19,4,20,5 X_5,18,6,19 X_21,17,22,16
Gauss code	{1, -4, 2, 9, -10, -6, 7, -8}, {4, -1, 3, -2, 6, -7, 8, 11, -5, 10, -9, 5, -11, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(u^{2}v^{2}-u^{2}v-uv^{2}-uv-u-v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$5q^{9/2}-6q^{7/2}+6q^{5/2}-{\frac {1}{q^{5/2}}}-7q^{3/2}+{\frac {2}{q^{3/2}}}+q^{13/2}-3q^{11/2}+5{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}+az^{3}-8z^{3}a^{-1}+8z^{3}a^{-3}-4z^{3}a^{-5}+3az-7za^{-1}+6za^{-3}-3za^{-5}+za^{-7}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-5z^{8}a^{-2}-3z^{8}a^{-4}-2z^{8}-az^{7}-2z^{7}a^{-3}-3z^{7}a^{-5}+19z^{6}a^{-2}+9z^{6}a^{-4}-z^{6}a^{-6}+9z^{6}+5az^{5}+14z^{5}a^{-1}+17z^{5}a^{-3}+8z^{5}a^{-5}-17z^{4}a^{-2}-7z^{4}a^{-4}-z^{4}a^{-6}-11z^{4}-8az^{3}-22z^{3}a^{-1}-18z^{3}a^{-3}-7z^{3}a^{-5}-3z^{3}a^{-7}+3z^{2}a^{-2}+2z^{2}a^{-4}+z^{2}a^{-6}-z^{2}a^{-8}+3z^{2}+5az+10za^{-1}+6za^{-3}+2za^{-5}+za^{-7}+1-az^{-1}-a^{-1}z^{-1}$ (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

14

1

-1

12

2

10

3

1

-2

8

3

2

1

6

3

0

4

3

1

2

3

5

2

0

1

2

-1

-2

1

3

2

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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