L11a320

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(2)t(1)-t(1)+1)(t(1)t(2)-t(2)+1)(2t(2)t(1)-t(1)-t(2)+2)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {11}{q^{3/2}}}+{\frac {14}{q^{5/2}}}-{\frac {18}{q^{7/2}}}+{\frac {17}{q^{9/2}}}-{\frac {15}{q^{11/2}}}+{\frac {12}{q^{13/2}}}-{\frac {7}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{5}+3a^{7}z^{3}+3a^{7}z-a^{5}z^{7}-4a^{5}z^{5}-6a^{5}z^{3}-3a^{5}z+a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-6a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}+az^{5}+3az^{3}+2az$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+5z^{4}a^{10}-2z^{2}a^{10}-5z^{7}a^{9}+7z^{5}a^{9}-2z^{3}a^{9}-6z^{8}a^{8}+10z^{6}a^{8}-10z^{4}a^{8}+7z^{2}a^{8}-4z^{9}a^{7}+2z^{7}a^{7}+z^{5}a^{7}+z^{3}a^{7}-2za^{7}-z^{10}a^{6}-10z^{8}a^{6}+27z^{6}a^{6}-30z^{4}a^{6}+13z^{2}a^{6}-7z^{9}a^{5}+11z^{7}a^{5}-5z^{5}a^{5}+z^{3}a^{5}-za^{5}-a^{5}z^{-1}-z^{10}a^{4}-8z^{8}a^{4}+24z^{6}a^{4}-22z^{4}a^{4}+6z^{2}a^{4}+a^{4}-3z^{9}a^{3}+z^{7}a^{3}+11z^{5}a^{3}-12z^{3}a^{3}+5za^{3}-a^{3}z^{-1}-4z^{8}a^{2}+9z^{6}a^{2}-4z^{4}a^{2}-3z^{7}a+9z^{5}a-8z^{3}a+3za-z^{6}+3z^{4}-2z^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

0

4

1

-3

-2

7

2

5

-4

8

5

-3

-6

10

6

4

-8

7

8

1

-10

8

10

-2

-12

5

8

3

-14

2

7

-5

-16

1

5

4

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

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