L11a201

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_16,5,17,6 X_22,18,7,17 X_18,12,19,11 X_12,22,13,21 X_20,14,21,13 X_14,20,15,19 X₆₇₁₈ X_4,15,5,16
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 6, -7, 8, -9, 11, -4, 5, -6, 9, -8, 7, -5}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)^{2}t(2)^{4}-5t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}+4t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+4t(2)^{2}+4t(1)t(2)-5t(2)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{7/2}-2q^{5/2}+4q^{3/2}-7{\sqrt {q}}+{\frac {9}{\sqrt {q}}}-{\frac {11}{q^{3/2}}}+{\frac {11}{q^{5/2}}}-{\frac {11}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {6}{q^{11/2}}}+{\frac {3}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{3}z^{7}-az^{7}+a^{5}z^{5}-4a^{3}z^{5}-5az^{5}+z^{5}a^{-1}+3a^{5}z^{3}-4a^{3}z^{3}-9az^{3}+4z^{3}a^{-1}+2a^{5}z-a^{3}z-8az+4za^{-1}+a^{5}z^{-1}-2az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{3}+3a^{8}z^{4}+6a^{7}z^{5}-5a^{7}z^{3}+2a^{7}z+8a^{6}z^{6}-10a^{6}z^{4}+2a^{6}z^{2}+9a^{5}z^{7}-18a^{5}z^{5}+10a^{5}z^{3}-5a^{5}z+a^{5}z^{-1}+7a^{4}z^{8}-14a^{4}z^{6}+a^{4}z^{4}+3a^{4}z^{2}-a^{4}+4a^{3}z^{9}-6a^{3}z^{7}-9a^{3}z^{5}+9a^{3}z^{3}-a^{3}z+a^{2}z^{10}+6a^{2}z^{8}+z^{8}a^{-2}-34a^{2}z^{6}-6z^{6}a^{-2}+41a^{2}z^{4}+12z^{4}a^{-2}-16a^{2}z^{2}-9z^{2}a^{-2}+3a^{2}+2a^{-2}+6az^{9}+2z^{9}a^{-1}-26az^{7}-11z^{7}a^{-1}+35az^{5}+20z^{5}a^{-1}-21az^{3}-14z^{3}a^{-1}+10az+4za^{-1}-2az^{-1}-a^{-1}z^{-1}+z^{10}-18z^{6}+39z^{4}-26z^{2}+5$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

8

1

-1

6

1

4

3

1

-2

2

4

1

3

0

5

3

-2

6

4

2

-4

6

0

-6

5

0

-8

3

6

3

-10

3

5

-2

-12

3

-14

1

3

-2

-16

1

Integral Khovanov Homology

(db, data source)

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