Characterizes alternating links in thickened surfaces using Gordon-Litherland pairing.
problem Identifying alternating links in thickened surfaces.
method Extension of Gordon-Litherland pairing to thickened surfaces.
result A non-split link in a thickened surface is alternating if and only if it bounds two definite surfaces of opposite sign.
Gordon-Litherland pairing connects combinatorics and topology.
problem Unifying quadratic forms in link theory.
method Picture proof using Kirby diagrams.
result Their theorem has numerous applications in low-dimensional topology.
Extends Gordon-Litherland pairing to links in thickened surfaces, defining new invariants.
problem Defining invariants for links in thickened surfaces.
method Extending Gordon-Litherland pairing, defining new invariants based on spanning surfaces.
result Invariants depend only on S∗-equivalence class of spanning surfaces and give well-defined invariants of virtual links. The flyping theorem is extended to virtual links and surfaces.
problem Proving the flyping theorem for virtual links and surfaces.
method Adapting geometric proof for thickened surfaces and using diagrammatic correspondence.
result Established an isomorphism between Gordon-Litherland pairing and 4-manifold intersection form.
The Kashaev conjecture is proven for classical signatures and Alexander polynomials of links.
problem Proving the Kashaev conjecture for signatures and Alexander polynomials.
method Relating Kashaev's matrix to Gordon-Litherland's work and Kauffman's model.
result Proven Alexander polynomial and classical signature parts of the conjecture for arbitrary links, and full conjecture for definite knots.
Maximum Levine-Tristram signature of torus knots follows a reduction formula.
problem Determining the maximum Levine-Tristram signature for torus knots.
method Proved a reduction formula analogous to Gordon-Litherland-Murasugi's classical signature result.
result Maximum Levine-Tristram signature of torus knots satisfies a reduction formula.
In this paper, we consider generalizations of the Alexander polynomial and signature of 2-bridge knots by considering the Gordon-Litherland bilinear forms associated to essential state surfaces of the 2-bridge knots. We show that the resulting invariants are well-defined and explore properties of these invariants. Fina…
Algorithm calculates Jones polynomial from Goeritz matrix.
problem Calculating Jones polynomial from link diagrams.
method Explicit algorithm using Goeritz matrices.
result Jones polynomial can be recovered from orientable checkerboard surfaces.
The paper studies alternating links in thickened surfaces using flow lattices and disc mutations.
problem Understanding alternating links in thickened surfaces and their invariants.
method Using integer flows on Tait graphs and disc mutations, the paper proves invariants and compares link properties.
result Found alternating knots with isometric flow lattices but different linking forms.
Study invariants of null-homologous knots in thickened surfaces.
problem Understanding concordance properties of knot invariants.
method Using Gordon-Litherland pairing and cobordism results for closed unoriented surfaces.
result Brown invariants are invariant under concordance of spanning surfaces.
New unoriented algebraic concordance group defined using mock Seifert matrices.
problem Understanding unoriented algebraic concordance of knots in thickened surfaces.
method Introducing mock Seifert matrices and using them to define unoriented algebraic concordance.
result The unoriented algebraic concordance group is abelian and infinitely generated.
The paper calculates homology and intersection pairing of branched covers using disoriented homology.
problem Computing homology and intersection pairing of branched covers of 4-ball.
method Associate disoriented homology groups to projections of links and surfaces, show isomorphism to branched cover homology, define pairing on first disoriented homology of surfaces.
result Disoriented homology is isomorphic to the homology of branched cover and pairing is equal to the intersection pairing.
New bounds on virtual link genus using quantum supergroups.
problem Finding strong lower bounds on the minimal genus of virtual links.
method Defined a Uq(gl(m∣n)) invariant equivalent to the CSW polynomial, generalized to all Uq(gl(m∣n)). result Generalized CSW lower bounds to all quantum supergroups Uq(gl(m∣n)) with m,n>0.