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CYCLIC BRANCHED COVERS OF ALTERNATING KNOTS AND L-SPACES
TERAGAITO, MASAKAZU Korean Mathematical Society 2015 대한수학회보 Vol.52 No.4
For any alternating knot, it is known that the double branched cover of the 3-sphere branched over the knot is an L-space. We show that the three-fold cyclic branched cover is also an L-space for any genus one alternating knot.
CYCLIC BRANCHED COVERS OF ALTERNATING KNOTS AND L-SPACES
Masakazu Teragaito 대한수학회 2015 대한수학회보 Vol.52 No.4
For any alternating knot, it is known that the double branched cover of the 3-sphere branched over the knot is an L-space. We show that the three-fold cyclic branched cover is also an L-space for any genus one alternating knot.
The secondary Upsilon function of L--space knots is a concave conjugate
Masakazu Teragaito 대한수학회 2024 대한수학회보 Vol.61 No.2
For a knot in the $3$--sphere, the Upsilon invariant is a piecewise linear function defined on the interval $[0,2]$. It is known that this invariant of an L--space knot is the Legendre--Fenchel transform (or, convex conjugate) of a certain gap function derived from the Alexander polynomial. To recover an information lost in the Upsilon invariant, Kim and Livingston introduced the secondary Upsilon invariant. In this note, we prove that the secondary Upsilon invariant of an L--space knot is a concave conjugate of a restricted gap function. Also, a similar argument gives an alternative proof of the above fact that the Upsilon invariant of an L--space knot is a convex conjugate of a gap function.
NEW FAMILIES OF HYPERBOLIC TWISTED TORUS KNOTS WITH GENERALIZED TORSION
Keisuke, Himeno,Masakazu, Teragaito Korean Mathematical Society 2023 대한수학회보 Vol.60 No.1
A generalized torsion element is an obstruction for a group to admit a bi-ordering. Only a few classes of hyperbolic knots are known to admit such an element in their knot groups. Among twisted torus knots, the known ones have their extra twists on two adjacent strands of torus knots. In this paper, we give several new families of hyperbolic twisted torus knots whose knot groups have generalized torsion. They have extra twists on arbitrarily large numbers of adjacent strands of torus knots.