EXTENSIONS OF QUANDLES AND COCYCLE KNOT INVARIANTS by
نویسندگان
چکیده
Approved: Major Professor: Masahiko Saito, Ph.D. Associate Professor, Department of Mathematics Date Approved:
منابع مشابه
Quandle coloring and cocycle invariants of composite knots and abelian extensions.
Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle coloring of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation t...
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Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and is extended to links component-wise.
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The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to group extension theories for group cocycles. Explicit formulas for cocycles cor...
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Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite family of quandles, the non-triviality of quandle homology groups is proved for all odd dimensions.
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A tangle is a pair (B,A) where A is a set of properly embedded arcs in a 3-ball B. A tangle will have four end points throughout the article, unless otherwise specified. A tangle T is embedded in a link (or a knot) L if there is an embedded ball B in 3-space such that T is equivalent to the pair (B,B ∩ L). All maps are assumed to be smooth. Tangles are represented by diagrams in a manner simila...
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