L11a135

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Planar diagram presentation	X₆₁₇₂ X_2,9,3,10 X_14,3,15,4 X_10,5,11,6 X_20,11,21,12 X_22,13,5,14 X_12,21,13,22 X_4,19,1,20 X_18,16,19,15 X_16,8,17,7 X_8,18,9,17
Gauss code	{1, -2, 3, -8}, {4, -1, 10, -11, 2, -4, 5, -7, 6, -3, 9, -10, 11, -9, 8, -5, 7, -6}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)t(2)^{5}-2t(2)^{5}-6t(1)t(2)^{4}+6t(2)^{4}+10t(1)t(2)^{3}-9t(2)^{3}-9t(1)t(2)^{2}+10t(2)^{2}+6t(1)t(2)-6t(2)-2t(1)+2}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {8}{\sqrt {q}}}-{\frac {14}{q^{3/2}}}+{\frac {19}{q^{5/2}}}-{\frac {23}{q^{7/2}}}+{\frac {22}{q^{9/2}}}-{\frac {20}{q^{11/2}}}+{\frac {15}{q^{13/2}}}-{\frac {9}{q^{15/2}}}+{\frac {4}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{5}+2a^{7}z^{3}+a^{7}z-a^{5}z^{7}-3a^{5}z^{5}-3a^{5}z^{3}-a^{5}z+a^{5}z^{-1}-a^{3}z^{7}-3a^{3}z^{5}-3a^{3}z^{3}-2a^{3}z-a^{3}z^{-1}+az^{5}+2az^{3}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+z^{3}a^{11}-4z^{6}a^{10}+5z^{4}a^{10}-2z^{2}a^{10}-8z^{7}a^{9}+12z^{5}a^{9}-7z^{3}a^{9}+za^{9}-9z^{8}a^{8}+10z^{6}a^{8}-z^{4}a^{8}-z^{2}a^{8}-6z^{9}a^{7}-2z^{7}a^{7}+14z^{5}a^{7}-6z^{3}a^{7}-2z^{10}a^{6}-13z^{8}a^{6}+27z^{6}a^{6}-14z^{4}a^{6}+3z^{2}a^{6}-12z^{9}a^{5}+19z^{7}a^{5}-9z^{5}a^{5}+4z^{3}a^{5}+za^{5}-a^{5}z^{-1}-2z^{10}a^{4}-11z^{8}a^{4}+32z^{6}a^{4}-24z^{4}a^{4}+4z^{2}a^{4}+a^{4}-6z^{9}a^{3}+9z^{7}a^{3}-4z^{3}a^{3}+2za^{3}-a^{3}z^{-1}-7z^{8}a^{2}+18z^{6}a^{2}-14z^{4}a^{2}+2z^{2}a^{2}-4z^{7}a+10z^{5}a-6z^{3}a-z^{6}+2z^{4}$ (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

9

3

6

-4

11

6

-5

-6

12

8

4

-8

10

11

1

-10

10

12

-2

-12

6

11

5

-14

3

9

-6

-16

1

6

5

-18

3

-3

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=-3$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$