L11n146

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_9,19,10,18 X₆₇₁₈ X_19,7,20,22 X_5,13,6,12 X_3,10,4,11 X_15,5,16,4 X_11,16,12,17 X_13,21,14,20 X_21,15,22,14 X_17,2,18,3
Gauss code	{1, 11, -6, 7, -5, -3}, {3, -1, -2, 6, -8, 5, -9, 10, -7, 8, -11, 2, -4, 9, -10, 4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

10

1

8

3

-3

6

3

1

2

4

5

3

-2

2

5

3

2

0

4

6

2

-2

4

0

-4

2

5

3

-6

1

3

-2

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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