L10n15

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_14,10,15,9 X₈₄₉₃ X_5,13,6,12 X_13,5,14,20 X_16,12,17,11 X_10,16,11,15 X_2,18,3,17
Gauss code	{1, -10, 5, -3}, {-6, -1, 2, -5, 4, -9, 8, 6, -7, -4, 9, -8, 10, -2, 3, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{4}-v^{3}+v^{2}-v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$6q^{9/2}-7q^{7/2}+6q^{5/2}-7q^{3/2}+{\frac {1}{q^{3/2}}}+2q^{13/2}-4q^{11/2}+4{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$za^{-7}+a^{-7}z^{-1}-z^{5}a^{-5}-4z^{3}a^{-5}-5za^{-5}-2a^{-5}z^{-1}+z^{7}a^{-3}+5z^{5}a^{-3}+8z^{3}a^{-3}+5za^{-3}-z^{5}a^{-1}-3z^{3}a^{-1}-za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$3z^{2}a^{-8}-2a^{-8}+z^{5}a^{-7}+3z^{3}a^{-7}-2za^{-7}+a^{-7}z^{-1}+4z^{6}a^{-6}-9z^{4}a^{-6}+12z^{2}a^{-6}-5a^{-6}+5z^{7}a^{-5}-15z^{5}a^{-5}+18z^{3}a^{-5}-9za^{-5}+2a^{-5}z^{-1}+2z^{8}a^{-4}-12z^{4}a^{-4}+12z^{2}a^{-4}-3a^{-4}+8z^{7}a^{-3}-27z^{5}a^{-3}+26z^{3}a^{-3}-10za^{-3}+2z^{8}a^{-2}-3z^{6}a^{-2}-6z^{4}a^{-2}+5z^{2}a^{-2}+a^{-2}+3z^{7}a^{-1}-11z^{5}a^{-1}+11z^{3}a^{-1}-3za^{-1}-a^{-1}z^{-1}+z^{6}-3z^{4}+2z^{2}$ (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		\

-3

-2

-1

0

1

2

3

4

5

χ

14

2

-2

12

2

10

4

2

-2

8

3

2

1

6

3

4

1

4

3

1

2

5

3

0

1

2

-1

-2

2

-4

1

-1

Integral Khovanov Homology

(db, data source)

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