L11a511

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Planar diagram presentation	X₈₁₉₂ X_14,3,15,4 X_20,12,21,11 X_18,10,19,9 X_22,16,13,15 X_12,20,7,19 X_10,22,11,21 X_16,6,17,5 X₂₇₃₈ X_4,13,5,14 X_6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 4, -7, 3, -6}, {10, -2, 5, -8, 11, -4, 6, -3, 7, -5}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)^{2}t(3)^{3}+t(1)t(3)^{3}+t(1)^{2}t(2)t(3)^{3}-2t(1)t(2)t(3)^{3}+t(2)t(3)^{3}+2t(1)^{2}t(3)^{2}+t(1)^{2}t(2)^{2}t(3)^{2}-2t(1)t(2)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}-3t(1)^{2}t(2)t(3)^{2}+5t(1)t(2)t(3)^{2}-3t(2)t(3)^{2}+t(3)^{2}-t(1)^{2}t(3)-t(1)^{2}t(2)^{2}t(3)+2t(1)t(2)^{2}t(3)-2t(2)^{2}t(3)+2t(1)t(3)+3t(1)^{2}t(2)t(3)-5t(1)t(2)t(3)+3t(2)t(3)-t(3)-t(1)t(2)^{2}+t(2)^{2}-t(1)^{2}t(2)+2t(1)t(2)-t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+11q^{5}-14q^{4}+17q^{3}-16q^{2}+15q-10+7q^{-1}-3q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-2z^{2}a^{-6}-a^{-6}+z^{6}a^{-4}+3z^{4}a^{-4}+5z^{2}a^{-4}+a^{-4}z^{-2}+4a^{-4}+z^{6}a^{-2}+z^{4}a^{-2}+a^{2}z^{2}-3z^{2}a^{-2}-2a^{-2}z^{-2}+a^{2}-5a^{-2}-2z^{4}-3z^{2}+z^{-2}+1$ (db)
Kauffman polynomial	$z^{5}a^{-9}-2z^{3}a^{-9}+3z^{6}a^{-8}-6z^{4}a^{-8}+2z^{2}a^{-8}+5z^{7}a^{-7}-10z^{5}a^{-7}+6z^{3}a^{-7}-za^{-7}+6z^{8}a^{-6}-14z^{6}a^{-6}+17z^{4}a^{-6}-9z^{2}a^{-6}+2a^{-6}+4z^{9}a^{-5}-4z^{7}a^{-5}-2z^{5}a^{-5}+9z^{3}a^{-5}-3za^{-5}+z^{10}a^{-4}+10z^{8}a^{-4}-35z^{6}a^{-4}+53z^{4}a^{-4}-35z^{2}a^{-4}-a^{-4}z^{-2}+10a^{-4}+7z^{9}a^{-3}-13z^{7}a^{-3}+7z^{5}a^{-3}+5z^{3}a^{-3}-8za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+8z^{8}a^{-2}+a^{2}z^{6}-27z^{6}a^{-2}-3a^{2}z^{4}+35z^{4}a^{-2}+3a^{2}z^{2}-29z^{2}a^{-2}-2a^{-2}z^{-2}-a^{2}+12a^{-2}+3z^{9}a^{-1}+3az^{7}-z^{7}a^{-1}-8az^{5}-10z^{5}a^{-1}+5az^{3}+9z^{3}a^{-1}-6za^{-1}+2a^{-1}z^{-1}+4z^{8}-8z^{6}+2z^{4}-2z^{2}-z^{-2}+4$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

2

13

4

1

-3

11

7

2

5

9

8

5

-3

7

9

6

3

5

8

9

1

3

7

8

-1

1

4

9

5

-1

3

6

-3

1

5

4

-5

2

-2

-7

1

Integral Khovanov Homology

(db, data source)

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