Hyperbolic Knots and the Volume Conjecture II: Khovanov Homology Mathematics REU at Rutgers University 2013 July 19 Advisor: Professor Feng Luo, Department of Mathematics, Rutgers University
Overview 1 Hyperbolic knots Review The volume conjecture The Jones polynomial 2 Idea Homology 3 Goal Construction of 4 Future work
Review: Definitions Review The volume conjecture The Jones polynomial Recall from my June presentation: Definition: (hyperbolic knot) A knot K is said to be hyperbolic if its complement S 3 \ K is equipped with a complete hyperbolic metric. Definition: (knot volume) The volume of a knot K is the hyperbolic volume of the knot complement S 3 \ K. Knot volume is a useful knot invariant. It distinguishes knots fairly well, and there are algorithms that can compute it, like SNAPPEA.
The volume conjecture Review The volume conjecture The Jones polynomial The volume conjecture was stated by R. Kashaev, H. Murakami, and J. Murakami. It relates the volume of a knot to the asymptotic growth of its colored Jones polynomial J N,K. Conjecture vol(s 3 \ K) = 2π lim N log J N,K (e2πi/n ). (1) N The conjecture should be understood as assigning a quantum invariant interpretation to hyperbolic volume.
The Jones polynomial Review The volume conjecture The Jones polynomial The Jones polynomial V K is an important knot invariant. It can be computed by recursive application of its defining skein relations: Figure 1 : The axioms (skein relations) for the Jones polynomial in the variable t. [K] The colored Jones polynomial is defined using the Jones polynomial.
Idea Hyperbolic knots Idea Homology Categorification is the process of lifting from set-theoretic theorems to category-theoretic analogues. (There is a good analogy with the idea of a combinatorial proof: a counting argument lifts a number-theoretic theorem to a set-theoretic analogue.) We seek to perform categorification on the Jones polynomial, and obtain a novel homology theory that may offer deeper insights than the Jones polynomial alone.
Homology Idea Homology Definition: (chain complex) n+2 n+1... C n+1 C n n 1 n C n 1... A chain complex (C, ) is a sequence of abelian groups (modules) (C n ) and group (module) homomorphisms ( n ) such that the above diagram commutes and n+1 n = 0 for all n.
Homology, continued Idea Homology Given a chain complex (C, ), we know that im n+1 is a subgroup of ker n. Thus it makes sense to speak of the quotient groups ker n / im n+1. Definition: (homology group) The n-th homology group of the complex C is the quotient group H n = ker n / im n+1. Homology comes in many flavors. Most commonly, a chain complex C(X) is associated to a topological space X, and the homology groups are obtained from this chain complex. Most importantly, homology groups are topological invariants up to isomorphism.
Homology, continued Idea Homology By the Nielsen-Schreier theorem, homology groups are free abelian. The rank of the n-th homology group H n is denoted β n and called the n-th Betti number. If there are finitely many nontrivial homology groups, then the alternating sum of the Betti numbers is called the Euler characteristic χ. Euler characteristic is a familiar topological invariant, and homology in fact generalizes Euler characteristic, but is much stronger.
Goal Hyperbolic knots Goal Construction of In the late 1990s, M. Khovanov introduced a novel theory called, which associates a homology to knots, links, and tangles from their planar diagrams. It has been shown that is a much stronger knot invariant than many existing knot invariants: it encompasses the Jones, Alexander, and HOMFLY-PT polynomials, but distinguishes knots that these polynomials fail to distinguish. It also detects the unknot. It is not yet clear how much information there is contained in. We studied in the hope that it would lead to interesting directions.
Construction of Goal Construction of Figure 2 : knot. [BN] Constructing the Khovanov chain complex for the trefoil
Goal Construction of Construction of, continued Each crossing in a planar diagram of a knot can be smoothed in one of two ways: Figure 3 : The two smoothings of a knot crossing. [K] These smoothings can be consistently assigned orientations 1 and 0 across the entire knot diagram. If χ is the set of crossings in a knot diagram, then there are 2 χ smoothings of the knot diagram. We can label each smoothing by a vertex of the hypercube {0, 1} χ.
Goal Construction of Construction of, continued Let V be a graded vector space with basis {q 1, q}. We can assign a vector space to each vertex of the hypercube in the following way: 1 For each connected component of the associated smoothing diagram, take a tensor product of V with itself; e.g. for the 110 smoothing of the trefoil, the associated vector space is V 2. 2 Let r be the number of 1-s in the coordinates of the vertex, called the height; e.g. r = 2 for the 110 smoothing. Perform a height shift operation by this height r on the associated vector space. Group the smoothings by height. The r-th chain vector space is the direct sum of the vector spaces attached to a vertex of height r.
Goal Construction of Construction of, continued If there are maps that make Figure 2 commute, then we can take formal sums of these maps and sprinkle signs to give us boundary homomorphisms. This is easily checked. In conclusion, we have constructed a chain complex of graded vector spaces, and therefore we have defined a homology. It can be shown that the homology is invariant under the Reidemeister moves. The graded dimensions of the homology spaces are the analogues of the Betti numbers. The Hilbert-Poincaré series of the chain complex, a polynomial in q, q 1, t, is the graded Euler characteristic.
Why study? Goal Construction of 1 encompasses many known invariants. For example, the t = 1 case of the graded Euler characteristic is the Jones polynomial. Alexander and HOMFLY-PT are also cases of the graded Euler characteristic for a suitably constructed Khovanov homology. 2 is strictly stronger than these invariants. 3 can be constructed in a category-theoretic setting, and gives us a way to consider cobordisms of links and tangles.
Future work Future work We will continue looking into and try to see what it can tell us about quantum invariants.
Future work I would like to thank Professor Feng Luo, Professor Christopher Woodward s research group, and the DIMACS/Math REU for this opportunity. [B] Bar-Natan, Dror: On Khovanov s categorification of the Jones polynomial. Algebr. Geom. Topol., 2, 1433-1499. [K] Kauffman, Louis H: New Invariants in the Theory of Knots. Amer. Math. Monthly, 95, 195-242.