Bridge numbers for virtual and welded knots

Abstract

Abstract. Using Gauss diagrams, one can define the virtual bridge number vb(K) and the welded bridge number wb(K), invariants of virtual and welded knots satisfying wb(K) ≤ vb(K). If K is a classical knot, Chernov and Manturov showed that vb(K) = br(K), the bridge number as a classical knot, and we ask whether the same thing is true for welded knots. The welded bridge number is bounded below by the meridional rank of the knot group G K , and we use this to relate this question to a conjecture of Cappell and Shaneson. We show how to use other virtual and welded invariants to further investigate bridge numbers. Among them are Manturov's parity and the reduced virtual knot group G K , and we apply these methods to address Questions 6.1, 6.2, 6.3 & 6.5 raised by Hirasawa, Kamada and Kamada in their paper Bridge presentation of virtual knots, J. Knot Theory Ramifications 20 (2011), no. 6, 881-893. Virtual and welded knotsThe classical theory of knots involves studying embeddings of S 1 → S 3 up to ambient isotopy. Under projection to a generic two-plane, any given knot is mapped to an immersion S 1 R 2 with finitely many immersion points, each of which is a transverse double point. The 3-dimensional knot can be reconstructed from its projection by keeping track of crossing information at each double point. The knot projection, together with the crossing information, is called the knot diagram. Two knot diagrams correspond to isotopic knots if and only if they are related by a sequence of Reidemeister moves.To every oriented knot diagram corresponds a Gauss word, which records the order in which the crossings are encountered starting from a fixed basepoint on the knot, along with the sign of each crossing. The Gauss word is conveniently represented by a decorated trivalent graph called the Gauss diagram, consisting of a circle oriented counterclockwise together with n signed, directed arcs connecting n distinct pairs of points on the circle. The directed arcs, or chords, point from an overcrossing (arrowtail) to an undercrossing (arrowhead). The sign of the chord indicates whether the crossing is right-handed (positive) or left-handed (negative). The Reidemeister moves on knot diagrams can be translated into corresponding moves on the Gauss 2010 Mathematics Subject Classification. Primary: 57M25, Secondary: 57M27.

Minimal crossing number implies minimal supporting genus

“…with m n. Our assumption on the length of the sequence of equation (2) implies that D n−1 → D n is an RII move of the form…”

“…In fact, the meridional rank conjecture implies the stronger statement that the bridge number of a classical link does not decrease when it is considered as a welded link. (For further details see [2]. )…”

“…To verify that p(D i ) → p(D i+1 ) preserves the Carter surface, we employ the topological definition of the C -parity. The move D i → D i+1 is associated to a move D j → D j+1 of equation (2). By assumption this latter move occurs on a disc neighbourhood of Σ, and D j , D j+1 are cellularly embedded.…”

“…) with m n. Our assumption on the length of the sequence of equation(2) implies that D n−1 → D n is an RII move of the form We may further assume that γ (depicted by the red arc) is disjoint to D n−1 outside of the disc neighbourhood depicted above. Observe that D n−1 has intersection number 0 with γ, and that there exists a γ-colouring of D n−1 such that all components possess the same colour, except in the region supporting the RII move, where the colouring is:Denote this distinguished γ-colouring by C .…”

A journey through loop braid groups

“…This is done using the isomorphisms between loop braid groups and welded braid groups. Though Gauss diagrams has already been used as an equivalent formulation of welded objects (see for example [5,7,12]), to formally prove the isomorphism between the groups of welded Gauss diagrams and welded braid diagrams we need to use results from [17] on virtual braids.…”