L11n186

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_18,10,19,9 X_19,22,20,7 X_13,20,14,21 X_21,14,22,15 X_10,16,11,15 X_16,6,17,5 X₂₇₃₈ X_4,11,5,12 X_6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 3, -7, 10, -2, -5, 6, 7, -8, 11, -3, -4, 5, -6, 4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

2

-2

4

3

1

2

4

2

-2

0

5

3

2

-2

4

5

1

-4

4

0

-6

2

5

3

-8

1

3

-2

-10

2

-12

1

-1

Integral Khovanov Homology

(db, data source)

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