Postive knots and braids


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 postive if it has a diagram in which all crossings are positive.

Strongly Quasipositve A braid is called strongly quasipostive if it is the product of conjugates of positive generators of the braid group (sigma_i), where each conjugating element is of the form (sigma_j sigma_{j+1} . . . sigma_{i-1}). A knot is called strongly quasipositive if it is the closure of a strongly quasipositive braid.

Quasipositive A braid is called quasipostive if it is the product of conjugates of positive generators of the braid group (sigma_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 [Yamada] and [Vogel].

The main obstructions used to rule out possible positive representations of a knot are the following:

(1) If K positive then its Conway polynomial has all coefficients nonnegative. See [Cromwell].

(2) If K positive, let h(P) be the highest degree coefficient of Homfly; this coefficient is a polynomial in v. Then all coefficients of h(P) are nonnegative or all nonpositive. This is a result of Trackyz, quoted in [Cromwell].

(3) If K positive then twice the 3-genus is equal to the maximum z exponent in the Homfly polynomial, which also equals the minimun v exponent in the Homfly polynomial. [Cromwell].

(4) Positive braids are fibered. [Stallings]

(5) For a positive braid, t^(-g) * Jones = 1 + t^2 + kt^3, where -1 <= k <= 3*(2g-1)/2 = 3g -3/2, where g is the three-genus. Note, this implies that -1 <= k <=3g-2. [Stoimenow2]

(6) For strongly quasipositive, 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. [Baader]


Further references concerning positivity and Heegaard Floer homology include [Hedden] and [Livingston]. Also, the positive notations for 12n638 was found by T. Abe, K. Tagami, and K. Moroi. The positive notaton for 11n183 was found by [Stoimenow1].


References

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

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

[Hedden] Hedden, M., "Notions of positivity and the Ozsvath0Szabo concordance invariant," J. Knot Theory Ramification 19 (2010), 617-629.

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

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

[Stoimenow1] 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.

[Stoimenow2] Stoimenow, A., "On polynomials and surfaces of variously positive links," J. Eur. Math. Soc. (JEMS) 7 (2005), 477-509.

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

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