L11a257

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{5}-t(1)t(2)^{5}+t(1)^{3}t(2)^{4}-4t(1)^{2}t(2)^{4}+4t(1)t(2)^{4}-t(2)^{4}-2t(1)^{3}t(2)^{3}+7t(1)^{2}t(2)^{3}-6t(1)t(2)^{3}+2t(2)^{3}+2t(1)^{3}t(2)^{2}-6t(1)^{2}t(2)^{2}+7t(1)t(2)^{2}-2t(2)^{2}-t(1)^{3}t(2)+4t(1)^{2}t(2)-4t(1)t(2)+t(2)-t(1)^{2}+t(1)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {18}{q^{9/2}}}-{\frac {19}{q^{7/2}}}+{\frac {16}{q^{5/2}}}+q^{3/2}-{\frac {13}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {6}{q^{15/2}}}+{\frac {11}{q^{13/2}}}-{\frac {16}{q^{11/2}}}-4{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{5}+3a^{7}z^{3}+2a^{7}z-a^{5}z^{7}-4a^{5}z^{5}-6a^{5}z^{3}-3a^{5}z+a^{5}z^{-1}-a^{3}z^{7}-3a^{3}z^{5}-2a^{3}z^{3}-a^{3}z-a^{3}z^{-1}+az^{5}+2az^{3}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-3z^{6}a^{10}+6z^{4}a^{10}-2z^{2}a^{10}-5z^{7}a^{9}+10z^{5}a^{9}-7z^{3}a^{9}+2za^{9}-6z^{8}a^{8}+12z^{6}a^{8}-11z^{4}a^{8}+2z^{2}a^{8}-5z^{9}a^{7}+8z^{7}a^{7}-6z^{5}a^{7}-3z^{3}a^{7}+za^{7}-2z^{10}a^{6}-5z^{8}a^{6}+18z^{6}a^{6}-21z^{4}a^{6}+6z^{2}a^{6}-10z^{9}a^{5}+24z^{7}a^{5}-25z^{5}a^{5}+12z^{3}a^{5}-a^{5}z^{-1}-2z^{10}a^{4}-5z^{8}a^{4}+17z^{6}a^{4}-10z^{4}a^{4}+2z^{2}a^{4}+a^{4}-5z^{9}a^{3}+7z^{7}a^{3}+2z^{5}a^{3}+za^{3}-a^{3}z^{-1}-6z^{8}a^{2}+13z^{6}a^{2}-4z^{4}a^{2}-z^{2}a^{2}-4z^{7}a+10z^{5}a-6z^{3}a-z^{6}+2z^{4}-z^{2}$ (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

3

0

5

1

-4

-2

8

3

5

-4

9

6

-3

-6

10

7

3

-8

8

9

1

-10

8

10

-2

-12

4

9

5

-14

2

7

-5

-16

1

4

3

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$