L9a32

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	[edit L9a32 Quick Notes] L9a32 is $9_{40}^{2}$ in the Rolfsen table of links.

[edit L9a32 Further Notes and Views]

A traditional symbol of the Christian Trinity (a triquetra interlaced with a circle, or "Trinity knot")		Symmetric version, with three lines touching at center		Symmetric version, with three lines touching at cicrumference
Logo of the Canadian Undergraduate Mathematics Conference		Circled hypotrochoid

Link Presentations

[edit Notes on L9a32's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(2)^{4}-3t(1)t(2)^{3}+3t(2)^{3}-3t(1)^{2}t(2)^{2}+5t(1)t(2)^{2}-3t(2)^{2}+3t(1)^{2}t(2)-3t(1)t(2)-t(1)^{2}}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {7}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {5}{q^{17/2}}}-{\frac {7}{q^{15/2}}}+{\frac {8}{q^{13/2}}}-{\frac {9}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4a^{9}z+3a^{9}z^{-1}-3a^{7}z^{3}-4a^{7}z-2a^{7}z^{-1}-3a^{5}z^{3}-3a^{5}z-a^{3}z^{3}$ (db)
Kauffman polynomial	$a^{12}z^{6}-3a^{12}z^{4}+3a^{12}z^{2}-a^{12}+3a^{11}z^{7}-10a^{11}z^{5}+9a^{11}z^{3}-2a^{11}z+a^{11}z^{-1}+2a^{10}z^{8}-a^{10}z^{6}-12a^{10}z^{4}+14a^{10}z^{2}-3a^{10}+9a^{9}z^{7}-26a^{9}z^{5}+22a^{9}z^{3}-11a^{9}z+3a^{9}z^{-1}+2a^{8}z^{8}+5a^{8}z^{6}-21a^{8}z^{4}+14a^{8}z^{2}-3a^{8}+6a^{7}z^{7}-10a^{7}z^{5}+6a^{7}z^{3}-6a^{7}z+2a^{7}z^{-1}+7a^{6}z^{6}-9a^{6}z^{4}+3a^{6}z^{2}+6a^{5}z^{5}-6a^{5}z^{3}+3a^{5}z+3a^{4}z^{4}+a^{3}z^{3}$ (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

χ

-2

1

-4

3

1

-2

-6

3

-8

4

3

-1

-10

5

3

2

-12

3

4

1

-14

4

5

-1

-16

2

4

2

-18

1

3

-2

-20

2

-22

1

-1

Integral Khovanov Homology

(db, data source)

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