scholarly journals Alternating links have at most polynomially many Seifert surfaces of fixed genus

2021 ◽  
Vol 70 (2) ◽  
pp. 525-534
Author(s):  
Joel Hass ◽  
Abigail Thompson ◽  
Anastasiia Tsvietkova
Keyword(s):  
Seifert Surfaces ◽  
Alternating Links ◽  
Fixed Genus

scholarly journals CONSTRUCTING SEIFERT SURFACES FROM n-BRIDGE LINK PROJECTIONS

2010 ◽  
Vol 19 (03) ◽  
pp. 313-334
Author(s):  
JOAN E. LICATA
Keyword(s):  
Large Classes ◽  
Link Complement ◽  
Seifert Surfaces ◽  
Alternating Links

This paper presents a new algorithm for constructing Seifert surfaces from n-bridge projections of links. The algorithm, 𝔄, produces minimal complexity surfaces for large classes of braids and alternating links. In addition, we consider a family of knots for which canonical genus is strictly greater than genus, (gc(K) > g(K)), and show that 𝔄 builds surfaces realizing the knot genus g(K). We also present a generalization of Seifert's algorithm which constructs surfaces representing arbitrary relative second homology classes in a link complement.

Finding Disjoint Seifert Surfaces

1988 ◽  
Vol 20 (1) ◽  
pp. 61-64 ◽  
Author(s):  
Martin Scharlemann ◽  
Abigail Thompson
Keyword(s):  
Seifert Surfaces

scholarly journals Almost alternating links

1992 ◽  
Vol 46 (2) ◽  
pp. 151-165 ◽  
Author(s):  
Colin C. Adams ◽  
Jeffrey F. Brock ◽  
John Bugbee ◽  
Timothy D. Comar ◽  
Keith A. Faigin ◽  
...  
Keyword(s):  
Alternating Links

scholarly journals SEIFERT SURFACES, COMMUTATORS AND VASSILIEV INVARIANTS

2007 ◽  
Vol 16 (10) ◽  
pp. 1295-1329
Author(s):  
E. KALFAGIANNI ◽  
XIAO-SONG LIN
Keyword(s):  
Fundamental Group ◽  
Lower Central Series ◽  
Central Series ◽  
Seifert Surface ◽  
Vassiliev Invariants ◽  
Seifert Surfaces

We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group. We also conjecture a characterization of knots whose invariants of all orders vanish in terms of their Seifert surfaces.

scholarly journals Quantum Computing, Seifert Surfaces, and Singular Fibers

2019 ◽  
Vol 1 (1) ◽  
pp. 12-22 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo M. Amaral ◽  
Klee Irwin
Keyword(s):  
Quantum Computing ◽  
Quantum Computer ◽  
Complete Information ◽  
Singular Fiber ◽  
Dynkin Diagrams ◽  
Singular Fibers ◽  
Seifert Surfaces ◽  
Trefoil Knot ◽  
Universal Quantum

The fundamental group π 1 ( L ) of a knot or link L may be used to generate magic states appropriate for performing universal quantum computation and simultaneously for retrieving complete information about the processed quantum states. In this paper, one defines braids whose closure is the L of such a quantum computer model and computes their braid-induced Seifert surfaces and the corresponding Alexander polynomial. In particular, some d-fold coverings of the trefoil knot, with d = 3 , 4, 6, or 12, define appropriate links L, and the latter two cases connect to the Dynkin diagrams of E 6 and D 4 , respectively. In this new context, one finds that this correspondence continues with Kodaira’s classification of elliptic singular fibers. The Seifert fibered toroidal manifold Σ ′ , at the boundary of the singular fiber E 8 ˜ , allows possible models of quantum computing.

scholarly journals Braid index bounds ropelength from below

2020 ◽  
Vol 29 (04) ◽  
pp. 2050019
Author(s):  
Yuanan Diao
Keyword(s):  
Lower Bound ◽  
General Formula ◽  
Crossing Number ◽  
Crossing Numbers ◽  
Braid Index ◽  
Alternating Links

For an unoriented link [Formula: see text], let [Formula: see text] be the ropelength of [Formula: see text]. It is known that in general [Formula: see text] is at least of the order [Formula: see text], and at most of the order [Formula: see text] where [Formula: see text] is the minimum crossing number of [Formula: see text]. Furthermore, it is known that there exist families of (infinitely many) links with the property [Formula: see text]. A long standing open conjecture states that if [Formula: see text] is alternating, then [Formula: see text] is at least of the order [Formula: see text]. In this paper, we show that the braid index of a link also gives a lower bound of its ropelength. More specifically, we show that there exists a constant [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] is the largest braid index among all braid indices corresponding to all possible orientation assignments of the components of [Formula: see text] (called the maximum braid index of [Formula: see text]). Consequently, [Formula: see text] for any link [Formula: see text] whose maximum braid index is proportional to its crossing number. In the case of alternating links, the maximum braid indices for many of them are proportional to their crossing numbers hence the above conjecture holds for these alternating links.

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