Positivity

There are four notions of positivity presented in KnotInfo. In the table we enter Y, N, or Unknown if a knot satisfies the condition on positivity.

If a (strongly)(quasi)positive braid is marked with a letter M, it means that the braid represents the negative of the knot illustrated. The PD notation and braid notation matches the figure.

Positive Braids: A knot is called a positive braid knot if it can be represented as the closure of a positive braid.

Positive Knots: A knot is called positive if it has a diagram in which all crossings are positive.

Strongly Quasipositive: A braid is called strongly quasipositive if it is the product of conjugates of positive generators of the braid group ( σi), where each conjugating element is of the form (σj σ{j+1} . . . σ{i-1}). A knot is called strongly quasipositive if it is the closure of a strongly quasipositive braid.

Quasipositive: A braid is called quasipositive if it is the product of conjugates of positive generators of the braid group ( σi). A knot is called quasipositive if it is the closure of a quasipositive braid.

Each of these classes of knots is contained in the next. The only nontrivial inclusion follows from the result that a positive knot is strongly quasipositive, a fact that follows from results of [11] and [12].

The main obstructions used to rule out possible positive representations of a knot are the following:
(1) If K is positive then its Conway polynomial has all coefficients nonnegative. See [4?].
(2) If K is positive, let h(P) be the highest degree coefficient of Homfly; this coefficient is a polynomial in v. Then coefficients of h(P) are all nonnegative or all nonpositive. This is a result of Trackyz, quoted in [4?].
(3) If K is positive then twice the 3-genus is equal to the maximum z exponent in the Homfly polynomial, which also equals the minimum v exponent in the Homfly polynomial [4?].
(4) Positive braids are fibered [9?].
(5) For a positive braid, t-g * Jones = 1 + t2 + kt3, where -1 ≤ k ≤ 3*(2g-1)/2 = 3g -3/2, where g is the three-genus. Note, this implies that -1 ≤ k ≤ 3g-2. [8?]
(6) For a strongly quasipositive knot, the 4-genus equals the 3-genus. (This was proved by Rudolph.)
(7) For a quasipositive knot, twice the 3-genus is less than or equal to the minimum v degree in the Homfly polynomial [1].

Further references concerning positivity and Heegaard Floer homology include [4] and [6?]. Also, the positive notations for 12n638 was found by T. Abe, K. Tagami, and K. Moroi. The positive notaton for 11n183 was found in [8?].

Specific Knots

10132
   Stoimenow proved that this knot isn't quasipositive by considering its (2,0) cable. See [8?].

11n183
   Stoimenow showed that this knot is positive in [9?].

12n638
   Kenji Tagami and Tetsuyan Abe proved that this knot is positive. Here is a picture:


This was also proved independently by Katsumi Moroi.

{820, 821, 945, 946, 10126, 10127, 10131, 10133, 10140, 10143, 10145, 10148, 10149, 10155, 10157, 10159, 10165}

Sebastian Baader demonstrated the quasipositivity of these knots and found explicit descriptions of each as quasipositive braids [1].

10145, 12n276, 12n329, and 12n402
Ken Baker has assisted us in identifying these and other knots as strongly quasipositive.

12n0113, 12n0114, 12n0190, 12n0191, 12n0233, 12n0234, 12n0344, 12n0345, 12n0466, 12n0467, 12n0570, 12n0604, 12n0666, 12n0674, 12n0683, 12n0684, 12n0707, 12n0708, 12n0721, 12n0722, 12n0747, 12n0748, 12n0767, 12n0820, 12n0822, 12n0829, 12n0831, 12n0882, 12n0887
Paolo Lisca has identified all of these knots as quasipositive. These follow from Theorem 2.1 from [6].

12n750
Peter Feller identified this knot as strongly quasipositive.

12n340
Lukas Lewark showed that this knot is not quasipositive, because the original Rasmussen invariant and the generalised sl3-Rasmussen invariant are different for that knot, and these invariants agree on all quasipositive knots [5].

11n17, 11n91, 11n99, 11n113, 11n162, 12n171, 12n176, 12n247, 12n270, 12n383, 12n441, 12n496, 12n520, 12n564, 12n626, 12n698, 12n699, 12n700, 12n701, 12n726, 12n734, 12n735, 12n796, 12n797, 12n814, 12n863, and 12n867.
   Steve Boyer, Cameron Gordon, and Michel Boileau obstruct strong quasipositivity for these knots in [2].

Mikami Hirasawa provided the strongly quasipositive braid notation for 10a145, 12n276, 12n329, and 12n402.

Keiko Kawamuro and Jesse Hamer have found strongly quasipositive braid notation for 12n_{148,149,293,321,332,366,404,432,528,642,660,801,830}.

References

[1] Baader, S., "Slice and Gordian numbers of track knots," Osaka J. Math. 42, no. 1 (2005), 257-271.

[2] Boileau, M., Boyer, S., and Gordon, C., "Branched covers of quasipositive links and L-spaces," Arxiv preprint.

[3] Cromwell, P. R., "Homogeneous links," J. London Math. Soc. (2) 39 (1989), 535-552.

[4] Hedden, M., "Notions of positivity and the Ozsvath-Szabo concordance invariant," J. Knot Theory Ramification 19 (2010), 617-629.

[5] Lewark, L., http://lewark.de/lukas/foamho.html and Arxiv preprint.

[6] Lisca, P., "Stein fillable contact 3-manifolds and postive open books of genus one," Algebraic and Geometric Topology 14 (2014), 2411-2430.

[7] Livingston, C., "Computations of the Ozsvath-Szabo knot concordance invariant," Goem. Topol. 8 (2004), 735-742.

[8] Stallings, J., "Constructions of fibred knots and links," Algebraic and geometric topology, Proceedings of Symposia in Pure Mathematics 32 (American Mathematical Society, Providence, 1978) 55-60.

[9] Stoimenow, A., "On polynomials and surfaces of variously positive links," J. Eur. Math. Soc. (JEMS) 7 (2005), 477-509.

[10] Stoimenow, A., "On the crossing number of positive knots and braids and braid index criteria of Jones and Morton-Williams-Franks," Trans. Amer. Math. Soc. 354(10) (2002), 3927-3954.

[11] Vogel, P., "Representations of links by braids," Com. Math. Helv. 65 (1990), 104-113.

[12] Yamada, S., "The minimal number of Seifert circles equals the braid index," Inv. Math. 88 (1987), 347-356.