## Algebraic & Geometric Topology

### Framed holonomic knots

#### Abstract

A holonomic knot is a knot in 3–space which arises as the 2–jet extension of a smooth function on the circle. A holonomic knot associated to a generic function is naturally framed by the blackboard framing of the knot diagram associated to the 1–jet extension of the function. There are two classical invariants of framed knot diagrams: the Whitney index (rotation number) $W$ and the self linking number $S$. For a framed holonomic knot we show that $W$ is bounded above by the negative of the braid index of the knot, and that the sum of $W$ and $|S|$ is bounded by the negative of the Euler characteristic of any Seifert surface of the knot. The invariant $S$ restricted to framed holonomic knots with $W=m$, is proved to split into $n$, where $n$ is the largest natural number with $n≤|m|2$, integer invariants. Using this, the framed holonomic isotopy classification of framed holonomic knots is shown to be more refined than the regular isotopy classification of their diagrams.

#### Article information

Source
Algebr. Geom. Topol., Volume 2, Number 1 (2002), 449-463.

Dates
Revised: 17 May 2002
Accepted: 28 May 2002
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882702

Digital Object Identifier
doi:10.2140/agt.2002.2.449

Mathematical Reviews number (MathSciNet)
MR1917062

Zentralblatt MATH identifier
1046.57011

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 58C25: Differentiable maps

#### Citation

Ekholm, Tobias; Wolff, Maxime. Framed holonomic knots. Algebr. Geom. Topol. 2 (2002), no. 1, 449--463. doi:10.2140/agt.2002.2.449. https://projecteuclid.org/euclid.agt/1513882702

#### References

• D. Bennequin, Entrelacements et équations de Pfaff, Soc. Math. de France, Astérisque 107–108 (1983) 87–161.
• J. S. Birman and N. C. Wrinkle, Holonomic and Legendrian parametrizations of knots, J. Knot Theory Ramifications 9 (2000), 293–309.
• L. Kauffman, Knots and Physics, World Scientific Publishing Co., Inc., River Edge, NJ (1991).
• B. Trace, On the Reidemeister moves of a classical knot, Proc. Amer. math. Soc. 89 (1983) 722–724.
• V. A. Vassiliev, Holonomic links and Smale principles for multisingularities, J. Knot Theory Ramifications 6 (1997) 115–123.
• H. Whitney, On regular closed curves in the plane, Composito Math. 4 (1936) 276–284.