L11a372

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Planar diagram presentation	X_12,1,13,2 X_2,13,3,14 X_14,3,15,4 X_16,5,17,6 X_10,11,1,12 X_4,15,5,16 X_22,17,11,18 X_18,10,19,9 X_20,8,21,7 X_8,20,9,19 X_6,22,7,21
Gauss code	{1, -2, 3, -6, 4, -11, 9, -10, 8, -5}, {5, -1, 2, -3, 6, -4, 7, -8, 10, -9, 11, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{3}t(1)^{4}-t(2)^{2}t(1)^{4}+t(2)^{4}t(1)^{3}-t(2)^{3}t(1)^{3}+t(2)^{2}t(1)^{3}-t(2)t(1)^{3}-t(2)^{4}t(1)^{2}+t(2)^{3}t(1)^{2}-t(2)^{2}t(1)^{2}+t(2)t(1)^{2}-t(1)^{2}-t(2)^{3}t(1)+t(2)^{2}t(1)-t(2)t(1)+t(1)-t(2)^{2}+t(2)}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {5}{q^{9/2}}}+{\frac {4}{q^{7/2}}}-{\frac {4}{q^{5/2}}}-q^{3/2}+{\frac {3}{q^{3/2}}}+{\frac {1}{q^{19/2}}}-{\frac {2}{q^{17/2}}}+{\frac {3}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+{\sqrt {q}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{7}-4z^{3}a^{7}-3za^{7}+z^{7}a^{5}+5z^{5}a^{5}+6z^{3}a^{5}+za^{5}+z^{7}a^{3}+6z^{5}a^{3}+11z^{3}a^{3}+7za^{3}+a^{3}z^{-1}-z^{5}a-5z^{3}a-6za-az^{-1}$ (db)
Kauffman polynomial	$-z^{2}a^{12}-2z^{3}a^{11}-3z^{4}a^{10}+2z^{2}a^{10}-4z^{5}a^{9}+7z^{3}a^{9}-2za^{9}-4z^{6}a^{8}+9z^{4}a^{8}-2z^{2}a^{8}-4z^{7}a^{7}+13z^{5}a^{7}-9z^{3}a^{7}+2za^{7}-3z^{8}a^{6}+11z^{6}a^{6}-8z^{4}a^{6}-2z^{9}a^{5}+8z^{7}a^{5}-5z^{5}a^{5}-5z^{3}a^{5}+2za^{5}-z^{10}a^{4}+4z^{8}a^{4}-z^{6}a^{4}-7z^{4}a^{4}+3z^{2}a^{4}-3z^{9}a^{3}+20z^{7}a^{3}-44z^{5}a^{3}+37z^{3}a^{3}-11za^{3}+a^{3}z^{-1}-z^{10}a^{2}+7z^{8}a^{2}-16z^{6}a^{2}+13z^{4}a^{2}-2z^{2}a^{2}-a^{2}-z^{9}a+8z^{7}a-22z^{5}a+24z^{3}a-9za+az^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

4

1

2

0

2

1

-2

1

-1

-4

3

2

1

-6

2

0

-8

3

2

1

-10

2

3

1

-12

2

0

-14

1

2

1

-16

1

2

-1

-18

1

-20

1

-1

Integral Khovanov Homology

(db, data source)

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