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 match 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 ( σ_{i}), where each conjugating element is of the form
(σ_{j} 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 ( σ_{i}). 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 + t^{2} + kt^{3} + (higher order terms), where -1 ≤ k ≤ 3*(2g-1)/2 =
3g -3/2, where g is the three-genus. Note, this implies that -1 ≤ k ≤ 3g-2; see [10]. Results concerning the coefficients of the Homfly polynomial of positive braids can be found in [14].

(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 4-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 12n_{638} was found by T. Abe, K. Tagami, and K. Moroi.
The positive notaton for 11n_{183} was found in [8?].

In the tables, if a knot has any particular positivity property, if an explicit representative is available, we include it. In [4], Hamer, Ito, and Kawamuro provide tools to find such representatives for strong quasi-positivity. Examples a listed below.

10_{132}

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

11n_{183}

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

12n_{638}

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

This was also proved independently by Katsumi Moroi.

{8_{20}, 8_{21}, 9_{45}, 9_{46}, 10_{126}, 10_{127}, 10_{131},
10_{133}, 10_{140}, 10_{143}, 10_{145}, 10_{148}, 10_{149}, 10_{155},
10_{157}, 10_{159}, 10_{165}}

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

10_{145}, 12n_{276}, 12n_{329}, and 12n_{402}

Ken Baker has assisted us in identifying these and other knots as strongly quasipositive.

12n_{0113}, 12n_{0114}, 12n_{0190}, 12n_{0191}, 12n_{0233}, 12n_{0234},
12n_{0344}, 12n_{0345}, 12n_{0466}, 12n_{0467}, 12n_{0570}, 12n_{0604},
12n_{0666}, 12n_{0674}, 12n_{0683}, 12n_{0684}, 12n_{0707}, 12n_{0708},
12n_{0721}, 12n_{0722}, 12n_{0747}, 12n_{0748}, 12n_{0767}, 12n_{0820},
12n_{0822}, 12n_{0829}, 12n_{0831}, 12n_{0882}, 12n_{0887}

Paolo Lisca has identified all of these knots as quasipositive. These follow from Theorem 2.1 from [6].

12n_{750}

Peter Feller identified this knot as strongly quasipositive.

12n_{340}

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].

11n_{17}, 11n_{91}, 11n_{99}, 11n_{113},
11n_{162}, 12n_{171}, 12n_{176}, 12n_{247}, 12n_{270}, 12n_{383},
12n_{441}, 12n_{496}, 12n_{520}, 12n_{564}, 12n_{626}, 12n_{698},
12n_{699}, 12n_{700}, 12n_{701}, 12n726, 12n734, 12n735,
12n_{796}, 12n_{797}, 12n_{814}, 12n_{863}, and 12n_{867}.

Steve Boyer, Cameron Gordon, and Michel Boileau obstruct strong quasipositivity for these knots in [2].

Mikami Hirasawa provided the strongly quasipositive braid notation for 10a_{145}, 12n_{276},
12n_{329}, and 12n_{402}.

Jesse Hamer, Tetsuya Ito and Keiko Kawamuro [4] have found strongly quasipositive braid notation for 12n_{148,149,293,321,332,366,404,432,528,642,660,801,830}.

12n_{148}, 12n_{276}, 12n_{329}, 12n_{366},
12n_{402}, 12n_{528}, 12n_{660}.

Lizzie Buchanan proves these are not positive in [15]. These knots were also resolved by Stoimenow.

[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] Hammer, J., Ito, T., and Kawamuro, K., "Positivities of Knots and Links and the Defect of Bennequin Inequality," Experimental Mathematics, published online 29 April, 2019.

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

[6] Lewark, L., http://lewark.de/lukas/foamho.html and "sl3-foam homology calculations," Arxiv preprint.

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

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

[9] 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.

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

[11] 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.

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

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

[14] Ito, Tetsuya, "A note on HOMFLY polynomial of positive braid links," Arxiv preprint.

[15] Buchanan, Lizzie, "A new condition on the Jones polynomial of a fibered positive link" arxiv preprint.