L9a3

L9a2

L9a4

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a3 at Knotilus!
	L9a3 is $9_{33}^{2}$ in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a3's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X8493 X14,10,15,9 X10,14,11,13 X18,12,5,11 X2,16,3,15
Gauss code	{1, -9, 5, -3}, {4, -1, 2, -5, 6, -7, 8, -4, 7, -6, 9, -2, 3, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-6q^{9/2}+8q^{7/2}-10q^{5/2}+{\frac {1}{q^{5/2}}}+9q^{3/2}-{\frac {3}{q^{3/2}}}-q^{13/2}+4q^{11/2}-9{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}+z^{5}a^{-3}-az^{3}+2z^{3}a^{-1}+z^{3}a^{-3}-z^{3}a^{-5}-az+3za^{-1}-2za^{-3}+2a^{-1}z^{-1}-3a^{-3}z^{-1}+a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-2z^{8}a^{-2}-2z^{8}a^{-4}-4z^{7}a^{-1}-9z^{7}a^{-3}-5z^{7}a^{-5}-4z^{6}a^{-2}-4z^{6}a^{-4}-4z^{6}a^{-6}-4z^{6}-3az^{5}+2z^{5}a^{-1}+16z^{5}a^{-3}+10z^{5}a^{-5}-z^{5}a^{-7}-a^{2}z^{4}+9z^{4}a^{-2}+13z^{4}a^{-4}+8z^{4}a^{-6}+3z^{4}+4az^{3}+3z^{3}a^{-1}-6z^{3}a^{-3}-4z^{3}a^{-5}+z^{3}a^{-7}+a^{2}z^{2}-2z^{2}a^{-2}-3z^{2}a^{-4}-2z^{2}a^{-6}-2az-5za^{-1}-3za^{-3}-3a^{-2}-3a^{-4}-a^{-6}+2a^{-1}z^{-1}+3a^{-3}z^{-1}+a^{-5}z^{-1}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 14                   1 1 12                 3   -3 10               3 1   2 8             5 3     -2 6           5 3       2 4         4 5         1 2       5 5           0 0     2 6             4 -2   1 3               -2 -4   2                 2 -6 1                   -1

Integral Khovanov Homology (db, data source)

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

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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