L11a529

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_14,3,15,4 X_12,13,7,14 X_16,7,17,8 X_22,17,13,18 X_18,5,19,6 X_20,9,21,10 X_10,19,11,20 X_4,11,5,12 X_2,15,3,16 X_6,21,1,22
Gauss code	{1, -10, 2, -9, 6, -11}, {4, -1, 7, -8, 9, -3}, {3, -2, 10, -4, 5, -6, 8, -7, 11, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(3)^{3}+t(2)^{2}t(3)^{3}-2t(1)t(3)^{3}+2t(1)t(2)t(3)^{3}-2t(2)t(3)^{3}+t(3)^{3}-3t(1)^{2}t(3)^{2}+3t(1)t(2)^{2}t(3)^{2}-3t(2)^{2}t(3)^{2}+5t(1)t(3)^{2}+3t(1)^{2}t(2)t(3)^{2}-7t(1)t(2)t(3)^{2}+5t(2)t(3)^{2}-2t(3)^{2}+3t(1)^{2}t(3)+2t(1)^{2}t(2)^{2}t(3)-5t(1)t(2)^{2}t(3)+3t(2)^{2}t(3)-3t(1)t(3)-5t(1)^{2}t(2)t(3)+7t(1)t(2)t(3)-3t(2)t(3)-t(1)^{2}-t(1)^{2}t(2)^{2}+2t(1)t(2)^{2}-t(2)^{2}+2t(1)^{2}t(2)-2t(1)t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-13}+5q^{-12}-10q^{-11}+17q^{-10}-22q^{-9}+26q^{-8}-25q^{-7}+23q^{-6}-16q^{-5}+10q^{-4}-4q^{-3}+q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{12}\left(-z^{2}\right)+a^{12}z^{-2}+2a^{12}+a^{10}z^{4}-6a^{10}z^{2}-2a^{10}z^{-2}-9a^{10}+6a^{8}z^{4}+11a^{8}z^{2}+a^{8}z^{-2}+7a^{8}+4a^{6}z^{4}+4a^{6}z^{2}+a^{4}z^{4}$ (db)
Kauffman polynomial	$z^{7}a^{15}-2z^{5}a^{15}+z^{3}a^{15}+5z^{8}a^{14}-15z^{6}a^{14}+13z^{4}a^{14}-z^{2}a^{14}-2a^{14}+8z^{9}a^{13}-23z^{7}a^{13}+17z^{5}a^{13}-z^{3}a^{13}+4z^{10}a^{12}+6z^{8}a^{12}-46z^{6}a^{12}+46z^{4}a^{12}-11z^{2}a^{12}-a^{12}z^{-2}+3a^{12}+21z^{9}a^{11}-53z^{7}a^{11}+26z^{5}a^{11}+9z^{3}a^{11}-9za^{11}+2a^{11}z^{-1}+4z^{10}a^{10}+20z^{8}a^{10}-76z^{6}a^{10}+70z^{4}a^{10}-33z^{2}a^{10}-2a^{10}z^{-2}+11a^{10}+13z^{9}a^{9}-13z^{7}a^{9}-17z^{5}a^{9}+19z^{3}a^{9}-9za^{9}+2a^{9}z^{-1}+19z^{8}a^{8}-35z^{6}a^{8}+28z^{4}a^{8}-19z^{2}a^{8}-a^{8}z^{-2}+7a^{8}+16z^{7}a^{7}-20z^{5}a^{7}+8z^{3}a^{7}+10z^{6}a^{6}-8z^{4}a^{6}+4z^{2}a^{6}+4z^{5}a^{5}+z^{4}a^{4}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over

r

).

\		r
	\
j		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

1

-5

4

1

-3

-7

6

-9

10

4

-6

-11

13

6

7

-13

12

10

-2

-15

14

13

1

-17

10

14

4

-19

7

12

-5

-21

4

11

7

-23

1

6

-5

-25

4

-27

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-5$	$i=-3$
$r=-11$	${\mathbb {Z} }$
$r=-10$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-8$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=-7$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-6$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{14}$
$r=-5$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$