L11a502

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)^{2}t(3)^{3}+2t(1)t(2)^{2}t(3)^{3}-t(2)^{2}t(3)^{3}+t(1)t(3)^{3}+2t(1)^{2}t(2)t(3)^{3}-3t(1)t(2)t(3)^{3}+t(2)t(3)^{3}+t(1)^{2}t(3)^{2}-2t(1)t(2)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}-2t(1)^{2}t(2)t(3)^{2}+4t(1)t(2)t(3)^{2}-2t(2)t(3)^{2}-t(1)^{2}t(3)+2t(1)t(2)^{2}t(3)-t(2)^{2}t(3)+2t(1)t(3)+2t(1)^{2}t(2)t(3)-4t(1)t(2)t(3)+2t(2)t(3)+t(1)^{2}-t(1)t(2)^{2}+t(2)^{2}-2t(1)-t(1)^{2}t(2)+3t(1)t(2)-2t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-3}-2q^{-4}+6q^{-5}-9q^{-6}+14q^{-7}-15q^{-8}+17q^{-9}-14q^{-10}+11q^{-11}-7q^{-12}+3q^{-13}-q^{-14}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}-a^{12}z^{4}-a^{12}z^{2}+4a^{12}z^{-2}+4a^{12}+a^{10}z^{6}+a^{10}z^{4}-6a^{10}z^{2}-5a^{10}z^{-2}-10a^{10}+2a^{8}z^{6}+7a^{8}z^{4}+7a^{8}z^{2}+2a^{8}z^{-2}+4a^{8}+a^{6}z^{6}+4a^{6}z^{4}+5a^{6}z^{2}+2a^{6}$ (db)
Kauffman polynomial	$z^{5}a^{17}-2z^{3}a^{17}+3z^{6}a^{16}-5z^{4}a^{16}+6z^{7}a^{15}-14z^{5}a^{15}+12z^{3}a^{15}-5za^{15}+a^{15}z^{-1}+7z^{8}a^{14}-18z^{6}a^{14}+22z^{4}a^{14}-12z^{2}a^{14}-a^{14}z^{-2}+4a^{14}+4z^{9}a^{13}-2z^{7}a^{13}-13z^{5}a^{13}+30z^{3}a^{13}-21za^{13}+5a^{13}z^{-1}+z^{10}a^{12}+9z^{8}a^{12}-30z^{6}a^{12}+44z^{4}a^{12}-33z^{2}a^{12}-4a^{12}z^{-2}+17a^{12}+6z^{9}a^{11}-8z^{7}a^{11}-9z^{5}a^{11}+35z^{3}a^{11}-33za^{11}+9a^{11}z^{-1}+z^{10}a^{10}+5z^{8}a^{10}-17z^{6}a^{10}+27z^{4}a^{10}-33z^{2}a^{10}-5a^{10}z^{-2}+20a^{10}+2z^{9}a^{9}+2z^{7}a^{9}-16z^{5}a^{9}+21z^{3}a^{9}-16za^{9}+5a^{9}z^{-1}+3z^{8}a^{8}-7z^{6}a^{8}+6z^{4}a^{8}-7z^{2}a^{8}-2a^{8}z^{-2}+6a^{8}+2z^{7}a^{7}-5z^{5}a^{7}+2z^{3}a^{7}+za^{7}+z^{6}a^{6}-4z^{4}a^{6}+5z^{2}a^{6}-2a^{6}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-5

1

-7

2

1

-1

-9

4

-11

5

2

-3

-13

9

4

5

-15

7

6

-1

-17

10

8

2

-19

6

9

3

-21

5

8

-3

-23

2

6

4

-25

1

5

-4

-27

2

-29

1

-1

Integral Khovanov Homology

(db, data source)

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