L11a80

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_16,8,17,7 X_22,18,5,17 X_18,14,19,13 X_20,9,21,10 X_14,20,15,19 X_8,21,9,22 X_10,16,11,15 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -8, 6, -9, 11, -2, 5, -7, 9, -3, 4, -5, 7, -6, 8, -4}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{5}-2t(2)^{5}-4t(1)t(2)^{4}+8t(2)^{4}+9t(1)t(2)^{3}-11t(2)^{3}-11t(1)t(2)^{2}+9t(2)^{2}+8t(1)t(2)-4t(2)-2t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+10q^{7/2}-16q^{5/2}+20q^{3/2}-23{\sqrt {q}}+{\frac {22}{\sqrt {q}}}-{\frac {19}{q^{3/2}}}+{\frac {13}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+8az^{3}-9z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-5a^{3}z+8az-9za^{-1}+3za^{-3}+a^{5}z^{-1}-2a^{3}z^{-1}+3az^{-1}-3a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-4a^{5}z^{5}+4z^{5}a^{-5}+6a^{5}z^{3}-4a^{5}z+a^{5}z^{-1}+3a^{4}z^{8}-10a^{4}z^{6}+10z^{6}a^{-4}+11a^{4}z^{4}-8z^{4}a^{-4}-4a^{4}z^{2}+4z^{2}a^{-4}-a^{-4}+4a^{3}z^{9}-8a^{3}z^{7}+16z^{7}a^{-3}-4a^{3}z^{5}-23z^{5}a^{-3}+16a^{3}z^{3}+13z^{3}a^{-3}-9a^{3}z-4za^{-3}+2a^{3}z^{-1}+a^{-3}z^{-1}+2a^{2}z^{10}+8a^{2}z^{8}+16z^{8}a^{-2}-39a^{2}z^{6}-24z^{6}a^{-2}+42a^{2}z^{4}+6z^{4}a^{-2}-15a^{2}z^{2}+4z^{2}a^{-2}+2a^{2}-2a^{-2}+13az^{9}+9z^{9}a^{-1}-25az^{7}-6az^{5}-33z^{5}a^{-1}+27az^{3}+30z^{3}a^{-1}-15az-14za^{-1}+3az^{-1}+3a^{-1}z^{-1}+2z^{10}+21z^{8}-63z^{6}+46z^{4}-11z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

7

1

-6

6

9

3

6

4

11

7

-4

2

12

9

3

0

11

12

1

-2

8

11

-3

-4

5

11

6

-6

3

8

-5

-8

1

6

5

-10

2

-2

-12

1

Integral Khovanov Homology

(db, data source)

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