The ν+\nu^+-equivalence classes of genus one knots

The $\nu^+$-equivalence is an equivalence relation on the knot concordance group. This relation can be seen as a certain stable equivalence on knot Floer complexes $CFK^{\infty}$, and many concordance invariants derived from Heegaard Floer theory are invariant under the equivalence. In this paper, we show that any genus one knot is $\nu^+$-equivalent to one of the trefoil, its mirror and the unknot.Comment: 46 pages, 8 figures;(v2)typos correcte

A family of slice-torus invariants from the divisibility of reduced Lee classes

We give a family of slice-torus invariants, one defined for each prime element $c$ in a principal ideal domain $R$, from the $c$-divisibility of the reduced Lee class in a variant of reduced Khovanov homology. It is proved that this family contains the Rasmussen invariant $s^F$ over any field $F$. Moreover, computational results show that the invariants corresponding to $(R, c) = (\mathbb{Z}, 2)$, $(\mathbb{Z}, 3)$ and $(\mathbb{Z}[i], 1 + i)$ are distinct from $s^\mathbb{Q}$.Comment: 38 page

Non-orientable genus of a knot in punctured CP2\mathbb{C}P ^2

For any knot $K$ which bounds non-orientable and null-homologous surfaces $F$ in punctured $n\mathbb{C}P^2$, we construct a lower bound of the first Betti number of $F$ which consists of the signature of $K$ and the Heegaard Floer $d$-invariant of the integer homology sphere obtained by $1$-surgery along $K$. By using this lower bound, we prove that for any integer $k$, a certain knot cannot bound any surface which satisfies the above conditions and whose first Betti number is less than $k$.Comment: 11 pages, 6 figure