L10a98

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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_18,7,19,8 X_20,15,9,16 X_16,19,17,20 X_8,9,1,10 X_4,13,5,14 X_6,17,7,18
Gauss code	{1, -2, 3, -9, 4, -10, 5, -8}, {8, -1, 2, -3, 9, -4, 6, -7, 10, -5, 7, -6}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(2)^{3}t(1)^{3}-t(2)^{2}t(1)^{3}-t(2)^{3}t(1)^{2}+3t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(1)+3t(2)t(1)-t(1)-t(2)+2}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}-{\frac {1}{q^{27/2}}}+{\frac {2}{q^{25/2}}}-{\frac {3}{q^{23/2}}}+{\frac {4}{q^{21/2}}}-{\frac {5}{q^{19/2}}}+{\frac {5}{q^{17/2}}}-{\frac {4}{q^{15/2}}}+{\frac {3}{q^{13/2}}}-{\frac {3}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{11}z^{5}+4a^{11}z^{3}+3a^{11}z-a^{9}z^{7}-5a^{9}z^{5}-6a^{9}z^{3}+a^{9}z^{-1}-a^{7}z^{7}-6a^{7}z^{5}-11a^{7}z^{3}-7a^{7}z-a^{7}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{17}+za^{17}-2z^{4}a^{16}+2z^{2}a^{16}-2z^{5}a^{15}+z^{3}a^{15}-2z^{6}a^{14}+2z^{4}a^{14}-z^{2}a^{14}-2z^{7}a^{13}+4z^{5}a^{13}-3z^{3}a^{13}+za^{13}-2z^{8}a^{12}+7z^{6}a^{12}-9z^{4}a^{12}+5z^{2}a^{12}-z^{9}a^{11}+3z^{7}a^{11}-3z^{5}a^{11}+5z^{3}a^{11}-3za^{11}-3z^{8}a^{10}+13z^{6}a^{10}-15z^{4}a^{10}+5z^{2}a^{10}-z^{9}a^{9}+4z^{7}a^{9}-3z^{5}a^{9}-z^{3}a^{9}+2za^{9}-a^{9}z^{-1}-z^{8}a^{8}+4z^{6}a^{8}-2z^{4}a^{8}-3z^{2}a^{8}+a^{8}-z^{7}a^{7}+6z^{5}a^{7}-11z^{3}a^{7}+7za^{7}-a^{7}z^{-1}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-6

1

-8

1

0

-10

2

-12

1

0

-14

3

2

1

-16

2

1

-1

-18

3

0

-20

1

2

1

-22

2

3

-1

-24

1

2

1

-26

1

-1

-28

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-8$	$i=-6$
$r=-10$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=-7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$