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On the PROB of Singular Braids Jonathan Beardsley / Link Invariants from Braided Monoidal On the PROB of Singular Braids Categories Singular Braid Monoids From Operads to PROBs Jonathan Beardsley (UNR), Suhyeon Lee (Berkeley), The PROB of Brendan Murphy (UW), Luke Trujillo (Harvey Mudd) Singular Braids Future Work September 18, 2020

On the PROB of Singular Braids Let ( C , ⌦ , 1 C , β ) denote a braided monoidal category, where Jonathan β is the symmetry transformation inducing the structure Beardsley isomorphisms β x , y : x ⌦ y ∼ ! y ⌦ x for all x , y 2 C . � Link Invariants Given any object x 2 C , we get a system of braid group from Braided Monoidal representations: Categories Singular Braid Monoids From Operads to ρ 1 : B 1 ! Aut C ( x ) e 7! ( id x : x ! x ) PROBs ρ 2 : B 2 ! Aut C ( x ⊗ 2 ) The PROB of 1 7! ( β x , x : x ⌦ x ! x ⌦ x ) Singular Braids β 1 7! ( β x , x ⌦ id x : x ⊗ 3 ! x ⊗ 3 ) ρ 3 : B 3 ! Aut C ( x ⊗ 3 ) Future Work β 2 7! ( id x ⌦ β x , x : x ⊗ 3 ! x ⊗ 3 ) β 1 7! ( β x , x ⌦ id x ⊗ 2 : x ⊗ 4 ! x ⊗ 4 ) ρ 4 : B 4 ! Aut C ( x ⊗ 4 ) β 2 7! ( id x ⌦ β x , x ⌦ id x : x ⊗ 4 ! x ⊗ 4 ) HI -5 ¥ . ¥ . .

On the PROB of Singular Braids Jonathan This manifests in the following equivalence of categories due Beardsley to Joyal and Street: Link Invariants from Braided Monoidal BMC ( B , C ) ' Cat (1 , C ) ' C Categories Singular Braid Monoids where BMC is the 2-category of braided monoidal categories From Operads to and braided monoidal functors, and B is the category defined PROBs in the following way: The PROB of Singular Braids Future Work obj ( B ) = N ( B n n = m B ( n , m ) = n 6 = m ∅ In other words, B is the free braided monoidal category on 1.

On the PROB of Singular Braids Jonathan Beardsley If our category C satisfies some additional conditions (e.g. having a suitable trace operator Link Invariants from Braided tr : Aut ( x ⊗ n ) ! Aut (1 c )) then the braid representations Monoidal Categories MELEE associated to x induce invariants of links: " ← Singular Braid Monoids ρ • tr { Aut ( x ⊗ n ) } { Braids } Aut (1 C ) From Operads to PROBs closure The PROB of t Singular Braids ← I { Braids 31M ¥ 49 dance { Links } " Future Work A category with the necessary structure to produce the above diagram is called a ribbon category, or sometimes a tortile category. These frequently arise as categories of modules over quantum groups.

On the PROB of Singular Braids Jonathan Beardsley If our category C satisfies some additional conditions (e.g. having a suitable trace operator Link Invariants from Braided tr : Aut ( x ⊗ n ) ! Aut (1 c )) then the braid representations Monoidal Categories associated to x induce invariants of links: Singular Braid Monoids ρ • tr { Aut ( x ⊗ n ) } { Braids } Aut (1 C ) From Operads to PROBs closure The PROB of Singular Braids { Links } Future Work A category with the necessary structure to produce the above diagram is called a ribbon category, or sometimes a tortile category. These frequently arise as categories of modules over quantum groups.

On the PROB of Singular Braids Jonathan Beardsley Goal Link Invariants from Braided We would like to mimic this procedure to produce invariants Monoidal Categories of singular knots and links, i.e. knots and links in which we Singular Braid allow a finite number of transverse self-intersections. Monoids From Operads to Remark PROBs i Singular knots and links can also be produced by “closing” The PROB of Singular Braids singular braids, and the redundancy is again classified by Future Work “singular Markov moves,” due to work of Gemein. Certain : ± invariants of singular braids, the so-called Vassiliev invariants, have a surprising relationship to the Grothendieck-Teichm¨ uller group.

On the PROB of Singular Braids Jonathan Beardsley Goal Link Invariants from Braided We would like to mimic this procedure to produce invariants Monoidal Categories of singular knots and links, i.e. knots and links in which we Singular Braid allow a finite number of transverse self-intersections. Monoids From Operads to Remark PROBs Singular knots and links can also be produced by “closing” The PROB of Singular Braids singular braids, and the redundancy is again classified by Future Work “singular Markov moves,” due to work of Gemein. Certain invariants of singular braids, the so-called Vassiliev invariants, have a surprising relationship to the Grothendieck-Teichm¨ uller group.

On the PROB of Singular Braids Jonathan Definition Beardsley Define the singular braid monoid on n-strands, denoted SB n , Link Invariants to be the monoid generated by the symbols b i , b − 1 from Braided , s i for Monoidal i Categories i = 1 , 2 , . . . , n � 1 subject to the following relations: Singular Braid Monoids b i b j = b j b i , if | i � j | > 1 , (1) From Operads to PROBs s i s j = s j s i , if | i � j | > 1 , (2) The PROB of s i b j = b j s i , for all 0 < i , j < n , (3) Singular Braids Future Work b i b i +1 b i = b i +1 b i b i +1 , (4) b i b i +1 s i = s i +1 b i b i +1 , (5) s i b i +1 b i = b i +1 b i s i +1 , (6) b i b − 1 = b − 1 b i = 1 . (7) i i

On the PROB of Singular Braids Jonathan Beardsley Some pictures of the singular braid relations: Sibi - bis , Sibzb , - bzbisz i ¥ ÷ ¥ f ¥¥ ÷ ¥¥ Link Invariants from Braided Monoidal Categories Singular Braid Monoids From Operads to PROBs The PROB of Singular Braids Future Work ' twist the " singularity down " " stretch

On the PROB of Singular Braids Jonathan Beardsley Goal Link Invariants from Braided Just as a choice of object in a braided monoidal category Monoidal Categories induces a system of representations of the braid groups, Singular Braid Monoids determine a structure on categories such that a choice of From Operads to object induces a system of representations of the singular PROBs braid monoids. Enddx Y The PROB of SBN Singular Braids Problem Future Work Unfortunately, for several reasons, this structure cannot be operadic in nature. In particular, it does not make sense to define a “singularly braided monoidal category.”

On the PROB of Singular Braids Jonathan Beardsley Goal Link Invariants from Braided Just as a choice of object in a braided monoidal category Monoidal Categories induces a system of representations of the braid groups, Singular Braid Monoids determine a structure on categories such that a choice of From Operads to object induces a system of representations of the singular PROBs braid monoids. The PROB of Singular Braids Problem Future Work Unfortunately, for several reasons, this structure cannot be operadic in nature. In particular, it does not make sense to define a “singularly braided monoidal category.” " " symmetry sisitfifsiqsis.it , ⇒ s cannot be a natural transform !

On the PROB of Singular Braids One solution to this problem is generalize from operads to Jonathan Beardsley PROPs, which can be thought of as operads with operations having multiple outputs and inputs (and are even more Link Invariants from Braided general than properads). In fact it will be useful to consider Monoidal Categories a more general (but less frequently studied) class of objects, Singular Braid Monoids PROBs. From Operads to Definition PROBs The PROB of A PROB (resp. PROP) is a braided (resp. symmetric) Singular Braids monoidal category whose objects are in bijection with the set Future Work N and whose monoidal structure corresponds to addition of natural numbers. Remark Joyal and Street’s category B is a PROB (in fact it is the initial PROB).

On the PROB of Singular Braids One solution to this problem is generalize from operads to Jonathan Beardsley PROPs, which can be thought of as operads with operations having multiple outputs and inputs (and are even more Link Invariants from Braided general than properads). In fact it will be useful to consider Monoidal Categories a more general (but less frequently studied) class of objects, Singular Braid Monoids PROBs. From Operads to Definition PROBs The PROB of A PROB (resp. PROP) is a braided (resp. symmetric) Singular Braids monoidal category whose objects are in bijection with the set Future Work N and whose monoidal structure corresponds to addition of natural numbers. Remark Joyal and Street’s category B is a PROB (in fact it is the initial PROB).

On the PROB of Singular Braids Jonathan Beardsley Remark Link Invariants I In a PROP or PROB P , the set P ( n , m ) is the set of from Braided Monoidal operations with n inputs and m ouputs. Categories I The category of braided (resp. symmetric) operads Singular Braid Monoids embeds fully faithfully into the category of PROBs From Operads to PROBs (resp. PROPs). The PROB of I An algebra over a PROB P is exactly a braided Singular Braids monoidal functor P ! C for some braided monoidal Future Work category C . I The term PROP is originally due to Mac Lane and stands for “PROduct and Permutation.” In PROB, the permutations are replaced by braid group actions.