L11a277

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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_20,15,21,16 X_18,8,19,7 X_6,20,7,19 X_22,17,9,18 X_16,21,17,22
Gauss code	{1, -2, 3, -6, 4, -9, 8, -5}, {5, -1, 2, -3, 6, -4, 7, -11, 10, -8, 9, -7, 11, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{3}t(2)^{5}-t(1)^{2}t(2)^{5}-t(1)^{3}t(2)^{4}+t(1)^{2}t(2)^{4}-t(1)t(2)^{4}+t(1)^{3}t(2)^{3}-t(1)^{2}t(2)^{3}+t(1)t(2)^{3}-t(2)^{3}-t(1)^{3}t(2)^{2}+t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(2)^{2}-t(1)^{2}t(2)+t(1)t(2)-t(2)-t(1)+1}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {3}{q^{9/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{5/2}}}-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{25/2}}}-{\frac {2}{q^{23/2}}}+{\frac {3}{q^{21/2}}}-{\frac {4}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {5}{q^{15/2}}}+{\frac {5}{q^{13/2}}}-{\frac {4}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{7}\right)-6a^{9}z^{5}-11a^{9}z^{3}-7a^{9}z-a^{9}z^{-1}+a^{7}z^{9}+8a^{7}z^{7}+23a^{7}z^{5}+30a^{7}z^{3}+18a^{7}z+3a^{7}z^{-1}-a^{5}z^{7}-7a^{5}z^{5}-16a^{5}z^{3}-13a^{5}z-2a^{5}z^{-1}$ (db)
Kauffman polynomial	$-z^{2}a^{16}-2z^{3}a^{15}-3z^{4}a^{14}+2z^{2}a^{14}-4z^{5}a^{13}+7z^{3}a^{13}-2za^{13}-4z^{6}a^{12}+9z^{4}a^{12}-2z^{2}a^{12}-4z^{7}a^{11}+12z^{5}a^{11}-5z^{3}a^{11}-za^{11}-4z^{8}a^{10}+17z^{6}a^{10}-19z^{4}a^{10}+7z^{2}a^{10}-a^{10}-3z^{9}a^{9}+15z^{7}a^{9}-22z^{5}a^{9}+13z^{3}a^{9}-6za^{9}+a^{9}z^{-1}-z^{10}a^{8}+2z^{8}a^{8}+12z^{6}a^{8}-34z^{4}a^{8}+23z^{2}a^{8}-3a^{8}-4z^{9}a^{7}+27z^{7}a^{7}-61z^{5}a^{7}+56z^{3}a^{7}-22za^{7}+3a^{7}z^{-1}-z^{10}a^{6}+6z^{8}a^{6}-9z^{6}a^{6}-3z^{4}a^{6}+11z^{2}a^{6}-3a^{6}-z^{9}a^{5}+8z^{7}a^{5}-23z^{5}a^{5}+29z^{3}a^{5}-15za^{5}+2a^{5}z^{-1}$ (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

-2

1

-4

0

-6

3

1

2

-8

1

0

-10

3

2

1

-12

2

1

-1

-14

3

0

-16

2

3

1

-18

2

0

-20

1

2

1

-22

1

2

-1

-24

1

-26

1

-1

Integral Khovanov Homology

(db, data source)

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