Mathematics

The Mathematics of Knots: Theory and Application by Markus Banagl, Denis Vogel

By Markus Banagl, Denis Vogel

The current quantity grew out of the Heidelberg Knot thought Semester, prepared by way of the editors in wintry weather 2008/09 at Heidelberg college. The contributed papers convey the reader brand new at the at present so much actively pursued components of mathematical knot conception and its functions in mathematical physics and phone biology. either unique learn and survey articles are awarded; a number of illustrations aid the textual content. The e-book could be of significant curiosity to researchers in topology, geometry, and mathematical physics, graduate scholars focusing on knot conception, and mobile biologists drawn to the topology of DNA strands.

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The gluing is effected by a PL-homeomorphism f ( 1 , 0 ) : F → F , which is the identity on ∂F = S k . Set F = F ∪∂F D k+1 . Since every f ( 1 , 0 ) is the identity on ∂F , we can extend it to a PL-homeomorphism 1 Knots, Singular Embeddings, and Monodromy 29 f ( 1 , 0 ) : F → F by taking f to be the identity on D k+1 . Using the system {f ( 1 , 0 )} to glue the blocks × F , ∈ K, we obtain the total space M k+2 of an F -block bundle over S 1 . This manifold M is the result of surgery on the knot.

Atti. Sem. Mat. Fis. Univ. Modena Suppl. : A 2-variable polynomial invariant for a virtual link derived from magnetic graphs. Hiroshima Math. J. : What is a virtual link? Algebraic. Geom. Topol. : Kauffman–like polynomial and curves in 2-surfaces. J. Knot Theory Ramif. : A multi-variable polynomial invariant for virtual knots and links. J. Knot Theory Ramif. 17(11), 1311–1326 (2008) Chapter 3 A Survey of Twisted Alexander Polynomials Stefan Friedl and Stefano Vidussi Abstract We give a short introduction to the theory of twisted Alexander polynomials of a 3-manifold associated to a representation of its fundamental group.

In the following, given w ∈ Q(R[F ]) we write . τ (N, α) = w if there exists a representative of τ (N, α) which equals w. Note that if γ : π1 (N ) → GL(k, R) is a representation and ψ : π1 (N ) → F a homomorphism to a free abelian group, then we get a tensor representation γ ⊗ ψ : π1 (N ) → GL(k, R[F ]) g → γ (g) · ψ(g) and the corresponding Reidemeister torsion τ (N, γ ⊗ ψ). Except for parts of Sect. 2 we will always consider the twisted Reidemeister torsion corresponding to such a tensor representation.

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