L11a278

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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_8,9,1,10 X_4,13,5,14 X_20,17,21,18 X_18,8,19,7 X_6,20,7,19 X_22,15,9,16 X_16,21,17,22
Gauss code	{1, -2, 3, -6, 4, -9, 8, -5}, {5, -1, 2, -3, 6, -4, 10, -11, 7, -8, 9, -7, 11, -10}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(2)^{3}t(1)^{3}-2t(2)^{2}t(1)^{3}-2t(2)^{3}t(1)^{2}+3t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}-2t(2)^{2}t(1)+3t(2)t(1)-2t(1)-2t(2)+2}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {3}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {6}{q^{11/2}}}-{\frac {7}{q^{13/2}}}+{\frac {6}{q^{15/2}}}-{\frac {5}{q^{17/2}}}+{\frac {3}{q^{19/2}}}-{\frac {2}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{5}\right)-4a^{9}z^{3}-3a^{9}z+a^{7}z^{7}+5a^{7}z^{5}+7a^{7}z^{3}+3a^{7}z+a^{5}z^{7}+5a^{5}z^{5}+7a^{5}z^{3}+4a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-5a^{3}z^{3}-6a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{14}z^{4}-2a^{14}z^{2}+2a^{13}z^{5}-4a^{13}z^{3}+a^{13}z+2a^{12}z^{6}-3a^{12}z^{4}+a^{12}z^{2}+2a^{11}z^{7}-4a^{11}z^{5}+4a^{11}z^{3}+2a^{10}z^{8}-6a^{10}z^{6}+8a^{10}z^{4}-a^{10}z^{2}+2a^{9}z^{9}-9a^{9}z^{7}+17a^{9}z^{5}-11a^{9}z^{3}+2a^{9}z+a^{8}z^{10}-4a^{8}z^{8}+7a^{8}z^{6}-6a^{8}z^{4}+a^{8}z^{2}+3a^{7}z^{9}-15a^{7}z^{7}+27a^{7}z^{5}-22a^{7}z^{3}+5a^{7}z+a^{6}z^{10}-5a^{6}z^{8}+11a^{6}z^{6}-16a^{6}z^{4}+8a^{6}z^{2}+a^{5}z^{9}-3a^{5}z^{7}-2a^{5}z^{5}+8a^{5}z^{3}-5a^{5}z+a^{5}z^{-1}+a^{4}z^{8}-4a^{4}z^{6}+2a^{4}z^{4}+3a^{4}z^{2}-a^{4}+a^{3}z^{7}-6a^{3}z^{5}+11a^{3}z^{3}-7a^{3}z+a^{3}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

0

-4

3

1

2

-6

2

1

-1

-8

3

2

1

-10

3

2

-1

-12

4

3

1

-14

3

4

1

-16

2

3

-1

-18

1

3

2

-20

1

2

-1

-22

1

-24

1

-1

Integral Khovanov Homology

(db, data source)

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