Studies on the Gordian complex and wild knots

Ichihara和我利用纽结的各种不变量和局部运动引入了新的单纯复形，从而推广了Gordian复形。重点研究了利用Alexander-Conway多项式和Delta-Move定义的单纯复形，证明了该单纯复形是Gromov双曲的拟等距实线，研究了交错纽结的Alexander多项式的刻画问题。特别地，我构造了无限多个Alexander多项式，它们满足成为交错纽结的Alexander多项式的必要条件，但不能通过交错纽结实现。

Ichihara and myself introduced new simplicial complexes by using various invariants and local moves for knots, which give generalizations of the Gordian complex. We focused on the simplicial complex defined by using the Alexander-Conway polynomial and the Delta-move, and showed that the simplicial complex is Gromov hyperbolic and quasi-isometric to the real line.I studied the characterization problem on the Alexander polynomial of an alternating knot. In particular, I constructed infinitely many Alexander polynomials which satisfy a necessary condition to become the Alexander polynomials of alternating knots but which cannot be realized by an alternating knot.