L11a143

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+t(2)^{4}-6t(1)^{2}t(2)^{3}+11t(1)t(2)^{3}-4t(2)^{3}+7t(1)^{2}t(2)^{2}-15t(1)t(2)^{2}+7t(2)^{2}-4t(1)^{2}t(2)+11t(1)t(2)-6t(2)+t(1)^{2}-3t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {27}{q^{9/2}}}-{\frac {27}{q^{7/2}}}+{\frac {22}{q^{5/2}}}+q^{3/2}-{\frac {17}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {5}{q^{17/2}}}-{\frac {11}{q^{15/2}}}+{\frac {18}{q^{13/2}}}-{\frac {24}{q^{11/2}}}-4{\sqrt {q}}+{\frac {9}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{5}+a^{7}z^{3}-a^{7}z-a^{7}z^{-1}-a^{5}z^{7}-2a^{5}z^{5}+a^{5}z^{3}+5a^{5}z+3a^{5}z^{-1}-a^{3}z^{7}-3a^{3}z^{5}-5a^{3}z^{3}-6a^{3}z-2a^{3}z^{-1}+az^{5}+2az^{3}+az$ (db)
Kauffman polynomial	$-z^{5}a^{11}-5z^{6}a^{10}+4z^{4}a^{10}-11z^{7}a^{9}+14z^{5}a^{9}-4z^{3}a^{9}-14z^{8}a^{8}+19z^{6}a^{8}-5z^{4}a^{8}-2z^{2}a^{8}+a^{8}-10z^{9}a^{7}+3z^{7}a^{7}+17z^{5}a^{7}-12z^{3}a^{7}+3za^{7}-a^{7}z^{-1}-3z^{10}a^{6}-20z^{8}a^{6}+47z^{6}a^{6}-24z^{4}a^{6}-3z^{2}a^{6}+3a^{6}-17z^{9}a^{5}+24z^{7}a^{5}+4z^{5}a^{5}-17z^{3}a^{5}+10za^{5}-3a^{5}z^{-1}-3z^{10}a^{4}-13z^{8}a^{4}+37z^{6}a^{4}-23z^{4}a^{4}+3a^{4}-7z^{9}a^{3}+6z^{7}a^{3}+11z^{5}a^{3}-16z^{3}a^{3}+9za^{3}-2a^{3}z^{-1}-7z^{8}a^{2}+13z^{6}a^{2}-6z^{4}a^{2}-4z^{7}a+9z^{5}a-7z^{3}a+2za-z^{6}+2z^{4}-z^{2}$ (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

6

1

-5

-2

11

3

8

-4

12

7

-5

-6

15

10

5

-8

13

0

-10

11

14

-3

-12

7

13

6

-14

4

11

-7

-16

1

7

6

-18

4

-4

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-4$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-3$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{15}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$