L11n290

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₈₄₉₃ X_2,14,3,13 X_14,7,15,8 X_11,18,12,19 X_9,21,10,20 X_19,5,20,10 X_4,15,1,16 X_17,22,18,11 X_21,16,22,17
Gauss code	{1, -4, 3, -9}, {-2, -1, 5, -3, -7, 8}, {-6, 2, 4, -5, 9, 11, -10, 6, -8, 7, -11, 10}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)\left(w^{2}-w+1\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q-1+6q^{-1}-6q^{-2}+9q^{-3}-8q^{-4}+7q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{4}\right)-2a^{6}z^{2}-a^{6}z^{-2}-2a^{6}+a^{4}z^{6}+4a^{4}z^{4}+7a^{4}z^{2}+4a^{4}z^{-2}+7a^{4}-2a^{2}z^{4}-6a^{2}z^{2}-5a^{2}z^{-2}-8a^{2}+z^{2}+2z^{-2}+3$ (db)
Kauffman polynomial	$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-6z^{4}a^{8}+z^{2}a^{8}+4z^{7}a^{7}-8z^{5}a^{7}+2z^{3}a^{7}-2za^{7}+a^{7}z^{-1}+3z^{8}a^{6}-6z^{6}a^{6}+5z^{4}a^{6}-5z^{2}a^{6}-a^{6}z^{-2}+3a^{6}+z^{9}a^{5}+z^{7}a^{5}-6z^{5}a^{5}+13z^{3}a^{5}-13za^{5}+5a^{5}z^{-1}+4z^{8}a^{4}-14z^{6}a^{4}+26z^{4}a^{4}-20z^{2}a^{4}-4a^{4}z^{-2}+10a^{4}+z^{9}a^{3}-3z^{7}a^{3}+4z^{5}a^{3}+10z^{3}a^{3}-17za^{3}+9a^{3}z^{-1}+z^{8}a^{2}-5z^{6}a^{2}+16z^{4}a^{2}-18z^{2}a^{2}-5a^{2}z^{-2}+11a^{2}+z^{5}a+z^{3}a-7za+5az^{-1}+z^{4}-4z^{2}-2z^{-2}+5$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

1

0

-1

6

1

5

-3

4

0

-5

5

2

3

-7

3

4

1

-9

4

5

-1

-11

2

3

1

-13

1

4

-3

-15

2

-17

1

-1

Integral Khovanov Homology

(db, data source)

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