Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory. Vassily Olegovich Manturov

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4.12-knots and links

       4.2Surface knots

       4.3Other types of 2-dimensional knotted surfaces

       4.4Smoothing on 2-dimensional knots

       4.4.1The notion of smoothing

       4.4.2The smoothing process in terms of the framing change

       4.4.3Generalised F-lemma

       Parity Theory

       5.Parity in Knot Theories. The Parity Bracket

       5.1The Gaußian parity and the parity bracket

       5.1.1The Gaußian parity

       5.1.2Smoothings of knot diagrams

       5.1.3The parity bracket invariant

       5.1.4The bracket invariant with integer coefficients

       5.2The parity axioms

       5.3Parity in terms of category theory

       5.4The L-invariant

       5.5Parities on 2-knots and links

       5.5.1The Gaußian parity

       5.5.2General parity principle

       5.6Parity Projection. Weak Parity

       5.6.1Gaußian parity and parity projection

       5.6.2The notion of weak parity

       5.6.3Functorial mapping for Gaußian parity

       5.6.4The parity hierarchy on virtual knots

       6.Cobordisms

       6.1Cobordism in knot theories

       6.1.1Basic definitions

       6.1.2Cobordism types

       6.2Sliceness criteria

       6.2.1Odd framed graphs

       6.2.2Iteratively odd framed graphs

       6.2.3Multicomponent links

       6.2.4Other results on free knot cobordisms

       6.3L-invariant as an obstruction to sliceness

       The Groups

       7.General Theory of Invariants of Dynamical Systems

       7.1Dynamical systems and their properties

       7.2Free k-braids

       7.3The main theorem

       7.4Pictures

       8.Groups and Their Homomorphisms

       8.1Homomorphism of pure braids into

       8.2Homomorphism of pure braids into

       8.3Homomorphism into a free group

       8.4Free groups and crossing numbers

       8.5Proof of Proposition 8.3

       9.Generalisations of the Groups

       9.1Indices from and Brunnian braids

       9.2Groups with parity and points

       9.2.1Connection between and

      

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