L10a97

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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_14,5,15,6 X_6,9,7,10 X_18,7,19,8 X_20,16,9,15 X_16,20,17,19 X_8,17,1,18 X_4,13,5,14
Gauss code	{1, -2, 3, -10, 4, -5, 6, -9}, {5, -1, 2, -3, 10, -4, 7, -8, 9, -6, 8, -7}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}v^{3}-2u^{3}v^{2}-2u^{2}v^{3}+3u^{2}v^{2}-2u^{2}v-2uv^{2}+3uv-2u-2v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {4}{q^{7/2}}}-{\frac {3}{q^{5/2}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{21/2}}}+{\frac {2}{q^{19/2}}}-{\frac {4}{q^{17/2}}}+{\frac {5}{q^{15/2}}}-{\frac {6}{q^{13/2}}}+{\frac {6}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{9}+3za^{9}+a^{9}z^{-1}-2z^{5}a^{7}-8z^{3}a^{7}-7za^{7}-a^{7}z^{-1}+z^{7}a^{5}+5z^{5}a^{5}+7z^{3}a^{5}+3za^{5}-z^{5}a^{3}-4z^{3}a^{3}-3za^{3}$ (db)
Kauffman polynomial	$-z^{3}a^{13}+za^{13}-2z^{4}a^{12}+z^{2}a^{12}-3z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-3z^{6}a^{10}+3z^{4}a^{10}-z^{2}a^{10}-3z^{7}a^{9}+6z^{5}a^{9}-6z^{3}a^{9}+5za^{9}-a^{9}z^{-1}-2z^{8}a^{8}+3z^{6}a^{8}+2z^{4}a^{8}-2z^{2}a^{8}+a^{8}-z^{9}a^{7}+9z^{5}a^{7}-12z^{3}a^{7}+7za^{7}-a^{7}z^{-1}-4z^{8}a^{6}+16z^{6}a^{6}-17z^{4}a^{6}+5z^{2}a^{6}-z^{9}a^{5}+2z^{7}a^{5}+5z^{5}a^{5}-9z^{3}a^{5}+2za^{5}-2z^{8}a^{4}+10z^{6}a^{4}-14z^{4}a^{4}+5z^{2}a^{4}-z^{7}a^{3}+5z^{5}a^{3}-7z^{3}a^{3}+3za^{3}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

1

-1

-4

2

1

-6

3

2

-1

-8

3

1

2

-10

3

0

-12

3

0

-14

2

3

1

-16

2

3

-1

-18

2

-20

1

2

-1

-22

1

Integral Khovanov Homology

(db, data source)

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