L11a405

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Link Presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(2)^{2}t(3)^{4}+t(1)t(3)^{4}-t(1)t(2)t(3)^{4}+2t(2)t(3)^{4}-t(3)^{4}-t(1)t(2)^{2}t(3)^{3}+2t(2)^{2}t(3)^{3}-2t(1)t(3)^{3}+4t(1)t(2)t(3)^{3}-4t(2)t(3)^{3}+t(3)^{3}+t(1)t(2)^{2}t(3)^{2}-2t(2)^{2}t(3)^{2}+2t(1)t(3)^{2}-4t(1)t(2)t(3)^{2}+4t(2)t(3)^{2}-t(3)^{2}-t(1)t(2)^{2}t(3)+2t(2)^{2}t(3)-2t(1)t(3)+4t(1)t(2)t(3)-4t(2)t(3)+t(3)+t(1)t(2)^{2}-t(2)^{2}+t(1)-2t(1)t(2)+t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$1-3q^{-1}+7q^{-2}-10q^{-3}+16q^{-4}-16q^{-5}+18q^{-6}-15q^{-7}+11q^{-8}-7q^{-9}+3q^{-10}-q^{-11}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-z^{2}a^{10}-a^{10}z^{-2}-2a^{10}+3z^{4}a^{8}+9z^{2}a^{8}+4a^{8}z^{-2}+9a^{8}-2z^{6}a^{6}-8z^{4}a^{6}-14z^{2}a^{6}-5a^{6}z^{-2}-14a^{6}-z^{6}a^{4}-z^{4}a^{4}+5z^{2}a^{4}+2a^{4}z^{-2}+7a^{4}+z^{4}a^{2}+2z^{2}a^{2}$ (db)
Kauffman polynomial	$a^{13}z^{5}-2a^{13}z^{3}+a^{13}z+3a^{12}z^{6}-5a^{12}z^{4}+a^{12}z^{2}+5a^{11}z^{7}-8a^{11}z^{5}+4a^{11}z^{3}-4a^{11}z+a^{11}z^{-1}+6a^{10}z^{8}-11a^{10}z^{6}+11a^{10}z^{4}-10a^{10}z^{2}-a^{10}z^{-2}+5a^{10}+4a^{9}z^{9}-18a^{9}z^{5}+33a^{9}z^{3}-21a^{9}z+5a^{9}z^{-1}+a^{8}z^{10}+13a^{8}z^{8}-48a^{8}z^{6}+72a^{8}z^{4}-48a^{8}z^{2}-4a^{8}z^{-2}+18a^{8}+8a^{7}z^{9}-14a^{7}z^{7}-6a^{7}z^{5}+36a^{7}z^{3}-29a^{7}z+9a^{7}z^{-1}+a^{6}z^{10}+12a^{6}z^{8}-50a^{6}z^{6}+77a^{6}z^{4}-57a^{6}z^{2}-5a^{6}z^{-2}+21a^{6}+4a^{5}z^{9}-6a^{5}z^{7}-5a^{5}z^{5}+13a^{5}z^{3}-13a^{5}z+5a^{5}z^{-1}+5a^{4}z^{8}-15a^{4}z^{6}+18a^{4}z^{4}-18a^{4}z^{2}-2a^{4}z^{-2}+9a^{4}+3a^{3}z^{7}-8a^{3}z^{5}+4a^{3}z^{3}+a^{2}z^{6}-3a^{2}z^{4}+2a^{2}z^{2}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

2

-2

-3

5

1

4

-5

6

3

-3

-7

10

4

6

-9

8

0

-11

10

8

2

-13

6

9

3

-15

5

9

-4

-17

2

6

4

-19

1

5

-4

-21

2

-23

1

-1

Integral Khovanov Homology

(db, data source)

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