L9a32

L9a31

L9a33

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a32 at Knotilus!
	L9a32 is $9_{40}^{2}$ in the Rolfsen table of links.

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 X12,3,13,4 X18,13,7,14 X14,9,15,10 X10,17,11,18 X16,5,17,6 X2738 X4,11,5,12 X6,15,1,16
Gauss code	{1, -7, 2, -8, 6, -9}, {7, -1, 4, -5, 8, -2, 3, -4, 9, -6, 5, -3}

A Braid Representative
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 1 -4                 3 1 -2 -6               3     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                 2 -22 1                   -1

Integral Khovanov Homology (db, data source)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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