L11a445

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{2}t(2)^{3}-t(3)^{2}t(2)^{3}-t(1)t(3)t(2)^{3}+2t(3)t(2)^{3}-t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-t(3)^{3}t(2)^{2}-5t(1)t(3)^{2}t(2)^{2}+5t(3)^{2}t(2)^{2}-t(1)t(2)^{2}+4t(1)t(3)t(2)^{2}-6t(3)t(2)^{2}+2t(2)^{2}-2t(1)t(3)^{3}t(2)+t(3)^{3}t(2)+6t(1)t(3)^{2}t(2)-4t(3)^{2}t(2)+t(1)t(2)-5t(1)t(3)t(2)+5t(3)t(2)-t(2)+t(1)t(3)^{3}-2t(1)t(3)^{2}+t(3)^{2}+t(1)t(3)-t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-9q^{6}+14q^{5}-17q^{4}+21q^{3}+q^{-3}-19q^{2}-3q^{-2}+17q+7q^{-1}-11$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-z^{2}a^{-6}-a^{-6}+z^{6}a^{-4}+2z^{4}a^{-4}+3z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}+z^{6}a^{-2}+z^{4}a^{-2}+a^{2}z^{2}-z^{2}a^{-2}-2a^{-2}z^{-2}+a^{2}-3a^{-2}-2z^{4}-3z^{2}+z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-9}-z^{3}a^{-9}+4z^{6}a^{-8}-5z^{4}a^{-8}+z^{2}a^{-8}+8z^{7}a^{-7}-13z^{5}a^{-7}+6z^{3}a^{-7}-2za^{-7}+9z^{8}a^{-6}-15z^{6}a^{-6}+11z^{4}a^{-6}-7z^{2}a^{-6}+3a^{-6}+5z^{9}a^{-5}+3z^{7}a^{-5}-20z^{5}a^{-5}+20z^{3}a^{-5}-7za^{-5}+z^{10}a^{-4}+16z^{8}a^{-4}-42z^{6}a^{-4}+47z^{4}a^{-4}-31z^{2}a^{-4}-a^{-4}z^{-2}+11a^{-4}+8z^{9}a^{-3}-6z^{7}a^{-3}-15z^{5}a^{-3}+24z^{3}a^{-3}-12za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+11z^{8}a^{-2}+a^{2}z^{6}-31z^{6}a^{-2}-3a^{2}z^{4}+35z^{4}a^{-2}+3a^{2}z^{2}-26z^{2}a^{-2}-2a^{-2}z^{-2}-a^{2}+11a^{-2}+3z^{9}a^{-1}+3az^{7}+2z^{7}a^{-1}-8az^{5}-17z^{5}a^{-1}+6az^{3}+17z^{3}a^{-1}-az-8za^{-1}+2a^{-1}z^{-1}+4z^{8}-7z^{6}+z^{4}-z^{-2}+3$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

3

13

6

1

-5

11

8

3

5

9

10

7

-3

7

11

7

4

5

8

10

2

3

9

11

-2

1

5

11

6

-1

2

6

-4

-3

1

5

4

-5

2

-2

-7

1

Integral Khovanov Homology

(db, data source)

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