L11a441

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(3)^{3}t(2)^{3}-t(1)t(3)^{2}t(2)^{3}+2t(3)^{2}t(2)^{3}+t(1)t(3)t(2)^{3}-t(3)t(2)^{3}-t(1)t(3)^{3}t(2)^{2}+2t(3)^{3}t(2)^{2}+4t(1)t(3)^{2}t(2)^{2}-6t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-4t(1)t(3)t(2)^{2}+4t(3)t(2)^{2}-t(2)^{2}+t(1)t(3)^{3}t(2)-t(3)^{3}t(2)-4t(1)t(3)^{2}t(2)+4t(3)^{2}t(2)-2t(1)t(2)+6t(1)t(3)t(2)-4t(3)t(2)+t(2)+t(1)t(3)^{2}-t(3)^{2}+t(1)-2t(1)t(3)+t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-7}+3q^{-6}-6q^{-5}+q^{4}+11q^{-4}-4q^{3}-14q^{-3}+9q^{2}+19q^{-2}-13q-18q^{-1}+17$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-2a^{6}+3a^{4}z^{4}+9a^{4}z^{2}+a^{4}z^{-2}+7a^{4}-2a^{2}z^{6}-8a^{2}z^{4}+z^{4}a^{-2}-13a^{2}z^{2}-2a^{2}z^{-2}+z^{2}a^{-2}-9a^{2}-z^{6}-z^{4}+3z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+8a^{3}z^{9}+5az^{9}+3a^{6}z^{8}+9a^{4}z^{8}+16a^{2}z^{8}+10z^{8}+a^{7}z^{7}-6a^{5}z^{7}-13a^{3}z^{7}+6az^{7}+12z^{7}a^{-1}-12a^{6}z^{6}-41a^{4}z^{6}-48a^{2}z^{6}+9z^{6}a^{-2}-10z^{6}-4a^{7}z^{5}-6a^{5}z^{5}-11a^{3}z^{5}-29az^{5}-16z^{5}a^{-1}+4z^{5}a^{-3}+16a^{6}z^{4}+55a^{4}z^{4}+47a^{2}z^{4}-9z^{4}a^{-2}+z^{4}a^{-4}-2z^{4}+5a^{7}z^{3}+17a^{5}z^{3}+24a^{3}z^{3}+20az^{3}+7z^{3}a^{-1}-z^{3}a^{-3}-9a^{6}z^{2}-34a^{4}z^{2}-29a^{2}z^{2}+4z^{2}a^{-2}-2a^{7}z-7a^{5}z-12a^{3}z-8az-za^{-1}+3a^{6}+11a^{4}+11a^{2}-a^{-2}+3+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

9

1

7

3

-3

5

6

1

5

3

7

3

-4

1

10

6

4

-1

11

10

-1

-3

8

7

1

-5

6

11

5

-7

5

8

-3

-9

2

7

5

-11

1

4

-3

-13

2

-15

1

-1

Integral Khovanov Homology

(db, data source)

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