A major question in Knot Theory concerns the process of trying to determine when two knots are different. A knot invariant is a quantity (number, polynomial, group, etc.) that does not change by continuous deformation of the knot. One of the simplest invariant of knots is colorability. In this thesis, we study Fox colorings of knots and knots that are colored by linear Alexander quandles. In recent years, there has been an interest in reducing Fox colorings to a minimum number of colors. We prove that any Fox coloring of a 13-colorable knot has a diagram that uses exactly five colors. The ideas behind the reduction of colors in a Fox coloring is extended to knots colored by linear Alexander quandles. Thus, we prove that any knot colored by either the linear Alexander quandle Z5[t]/(t − 2) or Z5[t]/(t − 3) has a diagram using only four colors.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21536号 / 理博第4443号 / 新制||理||1639(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 大槻 知忠, 教授 向井 茂, 教授 小野 薫 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM