L11a358

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{3}t(1)^{4}-t(2)^{2}t(1)^{4}+t(2)^{4}t(1)^{3}-3t(2)^{3}t(1)^{3}+3t(2)^{2}t(1)^{3}-t(2)t(1)^{3}-t(2)^{4}t(1)^{2}+3t(2)^{3}t(1)^{2}-3t(2)^{2}t(1)^{2}+3t(2)t(1)^{2}-t(1)^{2}-t(2)^{3}t(1)+3t(2)^{2}t(1)-3t(2)t(1)+t(1)-t(2)^{2}+t(2)}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$-q^{15/2}+3q^{13/2}-5q^{11/2}+7q^{9/2}-9q^{7/2}+9q^{5/2}-9q^{3/2}+7{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {2}{q^{5/2}}}+{\frac {1}{q^{7/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-1}+z^{7}a^{-3}-az^{5}+5z^{5}a^{-1}+4z^{5}a^{-3}-z^{5}a^{-5}-4az^{3}+8z^{3}a^{-1}+3z^{3}a^{-3}-3z^{3}a^{-5}-3az+6za^{-1}-za^{-3}-za^{-5}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}-2az^{9}-5z^{9}a^{-1}-3z^{9}a^{-3}-a^{2}z^{8}-2z^{8}a^{-2}-5z^{8}a^{-4}+2z^{8}+12az^{7}+24z^{7}a^{-1}+6z^{7}a^{-3}-6z^{7}a^{-5}+6a^{2}z^{6}+19z^{6}a^{-2}+11z^{6}a^{-4}-6z^{6}a^{-6}+8z^{6}-24az^{5}-37z^{5}a^{-1}+2z^{5}a^{-3}+10z^{5}a^{-5}-5z^{5}a^{-7}-11a^{2}z^{4}-23z^{4}a^{-2}-4z^{4}a^{-4}+7z^{4}a^{-6}-3z^{4}a^{-8}-20z^{4}+19az^{3}+25z^{3}a^{-1}-z^{3}a^{-3}-2z^{3}a^{-5}+4z^{3}a^{-7}-z^{3}a^{-9}+6a^{2}z^{2}+7z^{2}a^{-2}+z^{2}a^{-4}-z^{2}a^{-6}+z^{2}a^{-8}+10z^{2}-5az-9za^{-1}-3za^{-3}-za^{-7}-a^{-2}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

16

1

14

2

-2

12

3

1

2

10

4

2

-2

8

5

3

2

6

5

0

4

0

2

4

6

2

0

2

3

-1

-2

1

4

3

-4

1

2

-1

-6

1

-8

1

-1

Integral Khovanov Homology

(db, data source)

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