We construct a cohomology theory for oriented links using singular cobordisms and a special type ... more We construct a cohomology theory for oriented links using singular cobordisms and a special type of 2-dimensional Topological Quantum Field Theory (TQFT), categorifying the quantum sl(2) invariant. In particular, we give a description of the universal dot-free sl(2) foam cohomology for links via a 2-dimensional TQFT.
Polyak proved that all oriented versions of Reidemeister moves for knot and link diagrams can be ... more Polyak proved that all oriented versions of Reidemeister moves for knot and link diagrams can be generated by a set of just four oriented Reidemeister moves, and that no fewer than four oriented Reidemeister moves generate them all. We refer to a set containing four oriented Reidemeister moves that collectively generate all of the other oriented Reidemeister moves as a minimal generating set. Polyak also proved that a certain set containing two Reidemeister moves of type 1, one move of type 2 and one move of type 3 form a minimal generating set for all oriented Reidemeister moves. We expand upon Polyak’s work by providing an additional eleven minimal, 4-element, generating sets of oriented Reidemeister moves, and we prove that these 12 sets represent all possible minimal generating sets of oriented Reidemeister moves. We also consider the Reidemeister-type moves that relate oriented spatial trivalent graph diagrams with trivalent vertices that are sources and sinks and prove that a ...
A quandle coloring quiver is a quiver structure, introduced by Cho and Nelson, which is defined o... more A quandle coloring quiver is a quiver structure, introduced by Cho and Nelson, which is defined on the set of quandle colorings of an oriented knot or link by a finite quandle. We study quandle coloring quivers of [Formula: see text]-torus knots and links with respect to dihedral quandles.
We show that the virtual singular braid monoid on n strands embeds in a group V SGn, which we cal... more We show that the virtual singular braid monoid on n strands embeds in a group V SGn, which we call the virtual singular braid group on n strands. The group V SGn contains a normal subgroup V SP Gn of virtual singular pure braids. We show that V SGn is a semi-direct product of V SP Gn and the symmetric group Sn. We provide a presentation for V SP Gn via generators and relations. We also represent V SP Gn as a semi-direct product of n -1 subgroups and study the structures of these subgroups. These results yield a normal form of words in the virtual singular braid group.
This paper begins with a survey of some applications of Khovanov homology to lowdimensional topol... more This paper begins with a survey of some applications of Khovanov homology to lowdimensional topology, with an eye toward extending these results to sl(n) homologies. We extend Levine and Zemke's ribbon concordance obstruction from Khovanov homology to sl(n) homology for n ≥ 2, including the universal sl(2) and sl(3) homology theories. Inspired by Alishahi and Dowlin's bounds for the unknotting number coming from Khovanov homology and relying on spectral sequence arguments, we produce bounds on the alternation number of a knot. Lee and Bar-Natan spectral sequences also provide lower bounds on Turaev genus.
This paper begins with a survey of some applications of Khovanov homology to lowdimensional topol... more This paper begins with a survey of some applications of Khovanov homology to lowdimensional topology, with an eye toward extending these results to sl(n) homologies. We extend Levine and Zemke's ribbon concordance obstruction from Khovanov homology to sl(n) homology for n ≥ 2, including the universal sl(2) and sl(3) homology theories. Inspired by Alishahi and Dowlin's bounds for the unknotting number coming from Khovanov homology and relying on spectral sequence arguments, we produce bounds on the alternation number of a knot. Lee and Bar-Natan spectral sequences also provide lower bounds on Turaev genus.
An enhanced trivalent tangle is a trivalent tangle with some of its edges labeled. We use enhance... more An enhanced trivalent tangle is a trivalent tangle with some of its edges labeled. We use enhanced trivalent tangles and classical knot theory to provide a recipe for constructing invariants for trivalent tangles, and in particular, for knotted trivalent graphs. Our method also yields invariants of, what we refer to as, enhanced handlebody-tangles and enhanced handlebody-links.
The L-move for classical braids extends naturally to trivalent braids. We follow the L-move appro... more The L-move for classical braids extends naturally to trivalent braids. We follow the L-move approach to the Markov Theorem, to prove a one-move Markov-type theorem for trivalent braids. We also reformulate this L-Move Markov theorem and prove a more algebraic Markov-type theorem for trivalent braids. Along the way, we provide a proof of the Alexander's theorem analogue for spatial trivalent graphs and trivalent braids.
We extend the Kamada-Miyazawa polynomial to virtual singular links, which is valued in Z[A 2 , A ... more We extend the Kamada-Miyazawa polynomial to virtual singular links, which is valued in Z[A 2 , A -2 , h]. The decomposition of the resulting polynomial into two components, one in Z[A 2 , A -2 ] and the other in Z[A 2 , A -2 ]h yields the decomposition of the Kauffman-Jones polynomial of virtual singular links into two components, one in Z[A 2 , A -2 ] and the other in Z[A 2 , A -2 ]A 2 , where both components are invariants for virtual singular links.
Journal of Knot Theory and Its Ramifications, 2018
We prove Alexander- and Markov-type theorems for virtual spatial trivalent graphs and virtual tri... more We prove Alexander- and Markov-type theorems for virtual spatial trivalent graphs and virtual trivalent braids. We provide two versions for the Markov-type theorem: one uses an algebraic approach similar to the case of classical braids and the other one is based on [Formula: see text]-moves.
Journal of Knot Theory and Its Ramifications, 2016
Two singular links are cobordant if one can be obtained from the other by singular link isotopy t... more Two singular links are cobordant if one can be obtained from the other by singular link isotopy together with a combination of births or deaths of simple unknotted curves, and saddle point transformations. A movie description of a singular link cobordism in 4-space is a sequence of singular link diagrams obtained from a projection of the cobordism into 3-space by taking 2-dimensional cross-sections perpendicular to a fixed direction. We present a set of movie moves that are sufficient to connect any two movies of isotopic singular link cobordisms.
We employ a solution of the Yang-Baxter equation to construct invariants for knot-like objects. S... more We employ a solution of the Yang-Baxter equation to construct invariants for knot-like objects. Specifically, we consider a Yang-Baxter state model for the sl(n) polynomial of classical links and extend it to oriented singular links and balanced oriented 4-valent knotted graphs with rigid vertices. We also define a representation of the singular braid monoid into a matrix algebra, and seek conditions for extending further the invariant to contain topological knotted graphs. In addition, we show that the resulting Yang-Baxter-type invariant for singular links yields a version of the Murakami-Ohtsuki-Yamada state model for the sl(n) polynomial for classical links.
We provide a formula for the Dubrovnik polynomial of a rational knot in terms of the entries of t... more We provide a formula for the Dubrovnik polynomial of a rational knot in terms of the entries of the tuple associated with a braid-form diagram of the knot. Our calculations can be easily carried out using a computer algebra system. 2010 Mathematics Subject Classification. 57M27; 57M25. Key words and phrases. Kauffman two-variable polynomial of knots and links, rational knots and links, knot invariants.
In this paper we extend the Kauffman bracket to singular links. Specifically, we define a polynom... more In this paper we extend the Kauffman bracket to singular links. Specifically, we define a polynomial invariant for singular links, and in doing this, we consider three approaches to our extended Kauffman bracket polynomial: (1) using skein relations involving singular link diagrams, (2) using representations of the singular braid monoid, (3) via a Yang-Baxter state model. We also study some properties of the extended Kauffman bracket. MSC2010: 57M25, 57M27.
We construct a bigraded cohomology theory which depends on one parameter a and whose graded Euler... more We construct a bigraded cohomology theory which depends on one parameter a and whose graded Euler characteristic is the quantum sl(2) link invariant. We follow Bar-Natan's approach to tangles on one side, and Khovanov's sl(3) theory for foams on the other side. Our theory is properly functorial under tangle cobordisms, and a version of the Khovanov sl(2) invariant (or Lee's variant of it) corresponds to a = 0 (or a = 1). In particular, our construction naturally resolves the sign ambiguity in functoriality of Khovanov's sl(2) homology theory.
Virtual singular braids are generalizations of singular braids and virtual braids. We define the ... more Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander-and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer generators.
We employ the sl(2) foam cohomology to define a cohomology theory for oriented framed tangles who... more We employ the sl(2) foam cohomology to define a cohomology theory for oriented framed tangles whose components are labeled by irreducible representations of Uq(sl(2)). We show that the corresponding colored invariants of tangles can be assembled into invariants of bigger tangles. For the case of knots and links, the corresponding theory is a categorification of the colored Jones polynomial, and provides a tool for efficient computations of the resulting colored invariant of knots and links. Our theory is defined over the Gaussian integers Z[i] (and more generally over Z [i][a, h], where a, h are formal parameters), and enhances the existing categorifications of the colored Jones polynomial.
We introduce the category of singular 2-dimensional cobordisms and show that it admits a complete... more We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on a twin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra (𝐶, 𝑊, 𝑧, 𝑧 * ) consists of a commutative Frobenius algebra C, a symmetric Frobenius algebra W, and an algebra homomorphism 𝑧 : 𝐶 → 𝑊 with dual 𝑧 * : 𝑊 → 𝐶, satisfying some extra conditions. We also introduce a generalized 2-dimensional Topological Quantum Field Theory defined on singular 2-dimensional cobordisms and show that it is equivalent to a twin Frobenius algebra in a symmetric monoidal category.
We introduce and study the singular Temperley-Lieb category overℤ[q,q-1], which is a free pivotal... more We introduce and study the singular Temperley-Lieb category overℤ[q,q-1], which is a free pivotal category over two self-dual generators and is an extension of the (classical) Temperley-Lieb category. Our construction is motivated by a state model for thesl(2) polynomial of an oriented link and provides a categorical perspective to this link invariant. We also construct a couple of polynomial invariants for oriented tangles from category theory point of view.
François Jaeger presented the two-variable Kauffman polynomial of an unoriented link L as a weigh... more François Jaeger presented the two-variable Kauffman polynomial of an unoriented link L as a weighted sum of HOMFLY-PT polynomials of oriented links associated with L. Murakami, Ohtsuki and Yamada (MOY) used planar graphs and a recursive evaluation of these graphs to construct a state model for the sl(n)link invariant (a one-variable specialization of the HOMFLY-PT polynomial). We apply the MOY framework to Jaeger's work, and construct a state summation model for the SO(2n) Kauffman polynomial.
We construct a cohomology theory for oriented links using singular cobordisms and a special type ... more We construct a cohomology theory for oriented links using singular cobordisms and a special type of 2-dimensional Topological Quantum Field Theory (TQFT), categorifying the quantum sl(2) invariant. In particular, we give a description of the universal dot-free sl(2) foam cohomology for links via a 2-dimensional TQFT.
Polyak proved that all oriented versions of Reidemeister moves for knot and link diagrams can be ... more Polyak proved that all oriented versions of Reidemeister moves for knot and link diagrams can be generated by a set of just four oriented Reidemeister moves, and that no fewer than four oriented Reidemeister moves generate them all. We refer to a set containing four oriented Reidemeister moves that collectively generate all of the other oriented Reidemeister moves as a minimal generating set. Polyak also proved that a certain set containing two Reidemeister moves of type 1, one move of type 2 and one move of type 3 form a minimal generating set for all oriented Reidemeister moves. We expand upon Polyak’s work by providing an additional eleven minimal, 4-element, generating sets of oriented Reidemeister moves, and we prove that these 12 sets represent all possible minimal generating sets of oriented Reidemeister moves. We also consider the Reidemeister-type moves that relate oriented spatial trivalent graph diagrams with trivalent vertices that are sources and sinks and prove that a ...
A quandle coloring quiver is a quiver structure, introduced by Cho and Nelson, which is defined o... more A quandle coloring quiver is a quiver structure, introduced by Cho and Nelson, which is defined on the set of quandle colorings of an oriented knot or link by a finite quandle. We study quandle coloring quivers of [Formula: see text]-torus knots and links with respect to dihedral quandles.
We show that the virtual singular braid monoid on n strands embeds in a group V SGn, which we cal... more We show that the virtual singular braid monoid on n strands embeds in a group V SGn, which we call the virtual singular braid group on n strands. The group V SGn contains a normal subgroup V SP Gn of virtual singular pure braids. We show that V SGn is a semi-direct product of V SP Gn and the symmetric group Sn. We provide a presentation for V SP Gn via generators and relations. We also represent V SP Gn as a semi-direct product of n -1 subgroups and study the structures of these subgroups. These results yield a normal form of words in the virtual singular braid group.
This paper begins with a survey of some applications of Khovanov homology to lowdimensional topol... more This paper begins with a survey of some applications of Khovanov homology to lowdimensional topology, with an eye toward extending these results to sl(n) homologies. We extend Levine and Zemke's ribbon concordance obstruction from Khovanov homology to sl(n) homology for n ≥ 2, including the universal sl(2) and sl(3) homology theories. Inspired by Alishahi and Dowlin's bounds for the unknotting number coming from Khovanov homology and relying on spectral sequence arguments, we produce bounds on the alternation number of a knot. Lee and Bar-Natan spectral sequences also provide lower bounds on Turaev genus.
This paper begins with a survey of some applications of Khovanov homology to lowdimensional topol... more This paper begins with a survey of some applications of Khovanov homology to lowdimensional topology, with an eye toward extending these results to sl(n) homologies. We extend Levine and Zemke's ribbon concordance obstruction from Khovanov homology to sl(n) homology for n ≥ 2, including the universal sl(2) and sl(3) homology theories. Inspired by Alishahi and Dowlin's bounds for the unknotting number coming from Khovanov homology and relying on spectral sequence arguments, we produce bounds on the alternation number of a knot. Lee and Bar-Natan spectral sequences also provide lower bounds on Turaev genus.
An enhanced trivalent tangle is a trivalent tangle with some of its edges labeled. We use enhance... more An enhanced trivalent tangle is a trivalent tangle with some of its edges labeled. We use enhanced trivalent tangles and classical knot theory to provide a recipe for constructing invariants for trivalent tangles, and in particular, for knotted trivalent graphs. Our method also yields invariants of, what we refer to as, enhanced handlebody-tangles and enhanced handlebody-links.
The L-move for classical braids extends naturally to trivalent braids. We follow the L-move appro... more The L-move for classical braids extends naturally to trivalent braids. We follow the L-move approach to the Markov Theorem, to prove a one-move Markov-type theorem for trivalent braids. We also reformulate this L-Move Markov theorem and prove a more algebraic Markov-type theorem for trivalent braids. Along the way, we provide a proof of the Alexander's theorem analogue for spatial trivalent graphs and trivalent braids.
We extend the Kamada-Miyazawa polynomial to virtual singular links, which is valued in Z[A 2 , A ... more We extend the Kamada-Miyazawa polynomial to virtual singular links, which is valued in Z[A 2 , A -2 , h]. The decomposition of the resulting polynomial into two components, one in Z[A 2 , A -2 ] and the other in Z[A 2 , A -2 ]h yields the decomposition of the Kauffman-Jones polynomial of virtual singular links into two components, one in Z[A 2 , A -2 ] and the other in Z[A 2 , A -2 ]A 2 , where both components are invariants for virtual singular links.
Journal of Knot Theory and Its Ramifications, 2018
We prove Alexander- and Markov-type theorems for virtual spatial trivalent graphs and virtual tri... more We prove Alexander- and Markov-type theorems for virtual spatial trivalent graphs and virtual trivalent braids. We provide two versions for the Markov-type theorem: one uses an algebraic approach similar to the case of classical braids and the other one is based on [Formula: see text]-moves.
Journal of Knot Theory and Its Ramifications, 2016
Two singular links are cobordant if one can be obtained from the other by singular link isotopy t... more Two singular links are cobordant if one can be obtained from the other by singular link isotopy together with a combination of births or deaths of simple unknotted curves, and saddle point transformations. A movie description of a singular link cobordism in 4-space is a sequence of singular link diagrams obtained from a projection of the cobordism into 3-space by taking 2-dimensional cross-sections perpendicular to a fixed direction. We present a set of movie moves that are sufficient to connect any two movies of isotopic singular link cobordisms.
We employ a solution of the Yang-Baxter equation to construct invariants for knot-like objects. S... more We employ a solution of the Yang-Baxter equation to construct invariants for knot-like objects. Specifically, we consider a Yang-Baxter state model for the sl(n) polynomial of classical links and extend it to oriented singular links and balanced oriented 4-valent knotted graphs with rigid vertices. We also define a representation of the singular braid monoid into a matrix algebra, and seek conditions for extending further the invariant to contain topological knotted graphs. In addition, we show that the resulting Yang-Baxter-type invariant for singular links yields a version of the Murakami-Ohtsuki-Yamada state model for the sl(n) polynomial for classical links.
We provide a formula for the Dubrovnik polynomial of a rational knot in terms of the entries of t... more We provide a formula for the Dubrovnik polynomial of a rational knot in terms of the entries of the tuple associated with a braid-form diagram of the knot. Our calculations can be easily carried out using a computer algebra system. 2010 Mathematics Subject Classification. 57M27; 57M25. Key words and phrases. Kauffman two-variable polynomial of knots and links, rational knots and links, knot invariants.
In this paper we extend the Kauffman bracket to singular links. Specifically, we define a polynom... more In this paper we extend the Kauffman bracket to singular links. Specifically, we define a polynomial invariant for singular links, and in doing this, we consider three approaches to our extended Kauffman bracket polynomial: (1) using skein relations involving singular link diagrams, (2) using representations of the singular braid monoid, (3) via a Yang-Baxter state model. We also study some properties of the extended Kauffman bracket. MSC2010: 57M25, 57M27.
We construct a bigraded cohomology theory which depends on one parameter a and whose graded Euler... more We construct a bigraded cohomology theory which depends on one parameter a and whose graded Euler characteristic is the quantum sl(2) link invariant. We follow Bar-Natan's approach to tangles on one side, and Khovanov's sl(3) theory for foams on the other side. Our theory is properly functorial under tangle cobordisms, and a version of the Khovanov sl(2) invariant (or Lee's variant of it) corresponds to a = 0 (or a = 1). In particular, our construction naturally resolves the sign ambiguity in functoriality of Khovanov's sl(2) homology theory.
Virtual singular braids are generalizations of singular braids and virtual braids. We define the ... more Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander-and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer generators.
We employ the sl(2) foam cohomology to define a cohomology theory for oriented framed tangles who... more We employ the sl(2) foam cohomology to define a cohomology theory for oriented framed tangles whose components are labeled by irreducible representations of Uq(sl(2)). We show that the corresponding colored invariants of tangles can be assembled into invariants of bigger tangles. For the case of knots and links, the corresponding theory is a categorification of the colored Jones polynomial, and provides a tool for efficient computations of the resulting colored invariant of knots and links. Our theory is defined over the Gaussian integers Z[i] (and more generally over Z [i][a, h], where a, h are formal parameters), and enhances the existing categorifications of the colored Jones polynomial.
We introduce the category of singular 2-dimensional cobordisms and show that it admits a complete... more We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on a twin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra (𝐶, 𝑊, 𝑧, 𝑧 * ) consists of a commutative Frobenius algebra C, a symmetric Frobenius algebra W, and an algebra homomorphism 𝑧 : 𝐶 → 𝑊 with dual 𝑧 * : 𝑊 → 𝐶, satisfying some extra conditions. We also introduce a generalized 2-dimensional Topological Quantum Field Theory defined on singular 2-dimensional cobordisms and show that it is equivalent to a twin Frobenius algebra in a symmetric monoidal category.
We introduce and study the singular Temperley-Lieb category overℤ[q,q-1], which is a free pivotal... more We introduce and study the singular Temperley-Lieb category overℤ[q,q-1], which is a free pivotal category over two self-dual generators and is an extension of the (classical) Temperley-Lieb category. Our construction is motivated by a state model for thesl(2) polynomial of an oriented link and provides a categorical perspective to this link invariant. We also construct a couple of polynomial invariants for oriented tangles from category theory point of view.
François Jaeger presented the two-variable Kauffman polynomial of an unoriented link L as a weigh... more François Jaeger presented the two-variable Kauffman polynomial of an unoriented link L as a weighted sum of HOMFLY-PT polynomials of oriented links associated with L. Murakami, Ohtsuki and Yamada (MOY) used planar graphs and a recursive evaluation of these graphs to construct a state model for the sl(n)link invariant (a one-variable specialization of the HOMFLY-PT polynomial). We apply the MOY framework to Jaeger's work, and construct a state summation model for the SO(2n) Kauffman polynomial.
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