L11n210

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_5,14,6,15 X_22,18,9,17 X_4,19,5,20 X_21,6,22,7 X_16,7,17,8 X_8,9,1,10 X_18,14,19,13 X_15,21,16,20
Gauss code	{1, -2, 3, -6, -4, 7, 8, -9}, {9, -1, 2, -3, 10, 4, -11, -8, 5, -10, 6, 11, -7, -5}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{3}v^{2}+u^{2}v^{5}-u^{2}v^{4}+u^{2}v^{3}-u^{2}v^{2}+u^{2}v+uv^{4}-uv^{3}+uv^{2}-uv+u+v^{3}}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	${\frac {1}{\sqrt {q}}}-{\frac {2}{q^{3/2}}}+{\frac {2}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {4}{q^{11/2}}}+{\frac {3}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+5z^{3}a^{7}+7za^{7}+2a^{7}z^{-1}-z^{7}a^{5}-7z^{5}a^{5}-17z^{3}a^{5}-15za^{5}-3a^{5}z^{-1}+z^{5}a^{3}+4z^{3}a^{3}+4za^{3}+a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{11}z^{5}-4a^{11}z^{3}+3a^{11}z+a^{10}z^{6}-3a^{10}z^{4}+a^{10}z^{2}+a^{9}z^{7}-3a^{9}z^{5}+2a^{9}z^{3}-a^{9}z+a^{8}z^{8}-4a^{8}z^{6}+6a^{8}z^{4}-3a^{8}z^{2}+a^{7}z^{9}-6a^{7}z^{7}+15a^{7}z^{5}-16a^{7}z^{3}+10a^{7}z-2a^{7}z^{-1}+2a^{6}z^{8}-11a^{6}z^{6}+23a^{6}z^{4}-15a^{6}z^{2}+3a^{6}+a^{5}z^{9}-7a^{5}z^{7}+21a^{5}z^{5}-28a^{5}z^{3}+18a^{5}z-3a^{5}z^{-1}+a^{4}z^{8}-6a^{4}z^{6}+15a^{4}z^{4}-14a^{4}z^{2}+3a^{4}+2a^{3}z^{5}-6a^{3}z^{3}+4a^{3}z-a^{3}z^{-1}+a^{2}z^{4}-3a^{2}z^{2}+a^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

χ

0

1

-1

-2

1

-4

2

0

-6

2

-8

2

0

-10

2

0

-12

1

2

1

-14

1

2

-1

-16

1

-18

1

0

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }$
$r=1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$