A knot K is a smooth embedding of the circle into the three-dimensional sphere; two knots are said to be concordant if they form the boundary of an annulus properly embedded into the product of the three-sphere with an interval. Heegaard Floer knot homology is an invariant of knots introduced by P. Ozsváth and Z. Szabó in the early 2000's which associates to a knot a filtered chain complex CFK(K), which improves on classical invariants of the knot. Involutive Heegaard Floer homology is a variant theory introduced in 2015 by K. Hendricks and C. Manolescu which additionally considers a chain map iota on CFK(K) induced by a conjugation operation, and extracts from this data two new numerical invariants of knot concordance. These new invariants are especially interesting because, unlike many other concordance invariants from Heegaard Floer homology, they do not necessarily vanish on knots of finite order in the group of concordance classes of knots. The chain map iota is in general difficult to compute, and computations have been carried out for relatively few knots. We give a complete computation of iota for 10 and 11-crossing knots satisfying a certain simplicity condition, called the (1,1)-knots.

Author Bio

Anna Antal is a graduating senior at Rutgers University, majoring in mathematics and minoring in physics. Her interests include algebraic topology and geometry. She participated in this research project at the DIMACS REU as a rising senior. Antal is a member of the Rutgers Undergraduate Mathematics Association, and attends lectures there in her free time. She is continuing her studies as a PhD student at Yale University.

Sarah Pritchard is a rising senior at the Georgia Institute of Technology majoring in mathematics. Her interests include geometry, knot theory, and low-dimensional topology. She participated in this research project at the DIMACS REU as a rising junior. In her spare time, she works as a teaching assistant in the School of Mathematics at Georgia Tech and organizes mathematical outreach events in the Atlanta area. She will be applying to graduate programs in mathematics during fall of 2022.