INTRODUCTION TO BIPOLAR FILTRATION OF TOPOLOGICALLYSLICE KNOTS
MIN HOON KIM

Abstract. This is a note of a talk in mini workshop on concordance, Tokyo.

1. Definition of bipolar filtration
For a 4-manifold V , we define slice knots in V .
Definition. We say a knot K ⊂ S 3 is slice in V if (S 3 , K) = ∂(V, ∆) where ∆ is a smooth,properly embedded disk in V and [∆] = 0 ∈ H2 (V, S 3 ; Z).
Using this definition, T. Cochran and S. Harvey and P. Horn defined n-bipolar knots[CHH13].
Definition (Cochran-Harvey-Horn). K is n-positive (resp. n-negative) if K is slice in V
(V is called an n-positon (resp. n-negaton)) such that(1) π1 (V ) = 0(2) V is positive definite (resp. V is negative definite).
m(3) There exist disjointly embedded surfaces {Si }mi=1 in V − ∆ such that {[Si ]}i=1 ⊂H2 (V ; Z) form a basis for H2 (V ; Z) and π1 (Si ) ⊂ π1 (V − ∆)(n) where G(n) is then-th lower-central series of a group G with the convention G(0) = G.
K is n-bipolar if K is n-positive and n-negative.
Here are some remarks about n-bipolar knots.
Remark.
(1) If K is smoothly slice in D4 , then K is n-bipolar for all n.
(2) If V is n-positon or n-negaton, then the interesection form of V , QV : H2 (V ; Z) ×
H2 (V ; Z) → Z is diagonal because Si are disjointly embedded and {[Si ]}mi=1 ⊂H2 (V ; Z) form a basis for H2 (V ; Z). So, we can say that the condition (3) in theabove definition is modelled on Donaldson’s the diagonalizability theorem [Don83].
(3) By Freedman’s homeomorphism classifcation of simply connected topological 4manifolds [Fre82], n-position (resp. n-negaton) V is homeomorphic to punctured
2m2#m
i=1 CP (resp. punctured #i=1 CP ). Further, if the surfaces Si were actuallyspheres, then they could be excised by blow-down and K is smoothly slice in somehomotopy 4-ball.
1

INTRODUCTION TO BIPOLAR FILTRATION OF TOPOLOGICALLY SLICE KNOTS

2Let T be the smooth concordance classes of topologically slice knots. That is,
T = {topologically slice knots in S 3 }/smooth concordance
So, T measures the difference between smooth and topological concordance. Using thenotion of n-bipolar knots, Cochran-Harvey-Horn defined the bipolar filtration on T .
Tn = {topologically slice, n-bipolar knots in S 3 }/smooth concordance
Remark. One may ask what happens if we take T ∩ Fnsmooth where Fnsmooth is the set ofsmooth n-solvable knots in S 3 . But, nothing happens (that is, T ∩ Fnsmooth = T for all n)
because of the smoothing theory in [FQ90, Section 8.6].
2. Properties and Construction of n-bipolar knots
Even though we didn’t used smooth n-solvable knots, we can apply obsturction forn-solvable knots because of the following Proposition.
Proposition. If K is n-positive or n-negative, then K is integral n-solvable.
Using satellite construction, we can construct more n-bipolar knots.
Proposition. Suppose a knot K is n-positive (resp. K is n-negative) and η is an unknotted
simple closed curve in S 3 disjoint from K. If [η] ∈ π1 (S 3 − K)(k) and J is (n − k)-positive(resp. J is (n − k) negative), then K(η, J) is n-positive (resp. n-negative).
Remark. This propostion can be compared with the following: If K is n-solvable and η
is an unknotted simple closed curve in S 3 disjoint from K. If [η] ∈ π1 (S 3 − K)(k) and J
is (n − k)-solvable, then K(η, J) is n-solvable.
The following theorem gives a construcion of 0-bipolar knots.
Theorem (Cochran-Lickorish [CL86]). Suppose K is an oriented knot in S 3 . If K can bechanged into the unknot via positive crossing changes (resp. negative crossing changes),then K is 0-positive (resp. K is 0-negative). Here, positive crossing change (resp. negativecrossing change) is a crossing change from postive crossing to negative crossing (resp. fromnegative crossing to positive crossing).
Example.
(1) The right handed trefoil knot is 0-positive because the right handed trefoil can bechanged into the unknot using one postive crossing change. The figure eight knot
is 0-bipolar because the figure eight knot can be changed into the unknot either
using one positive crossing or one negative crossing.
(2) Take K to be one component in Whitehead link (so, K is the unknot) and η tobe the other component. Then, η is unknotted and [η] ∈ π1 (S 3 − K)(n) for alln. Then, K(η, J) is exactly the untwisted Whitehead double of J. So, if J is 0positive (resp. 0-negative), then the untwisted Whitehead double of J is n-positivefor all n (resp. n-negative for all n).

INTRODUCTION TO BIPOLAR FILTRATION OF TOPOLOGICALLY SLICE KNOTS

3

Many concordance invariatns can be used to prove a some knot K is not n-bipolar forsome n.
Proposition (Cochran-Harvey-Horn).
(1) If K is 0-bipolar, then the Levine-Tristram signature function of K vanishes and3 (K)) = (K) = 0.
τ (K) = s(K) = d(S±1(2) If K is 0-bipolar and 2r -signatures vanish, then Hedden-Kirk slicing obstruction(extending the Fintushel-Stern invariants) vanishes(3) If K is 1-bipolar and Y is the pr -fold cyclic cover of S 3 branched over K, then there
exists a metabolizer G < H1 (Y ; Z) and spinc -structure s0 such that d(Y, s0 + zb) = 0for all z ∈ G and zb is the Poincare dual of z.
(4) If K is 2-bipolar, then the Casson-Gordon invariant vanishes.
(5) If K is n-bipolar and Γ is a poly-torsion free abelian group, then ρ(MK , φ) = 0 for
φ : π1 (MK ) → Γ which extends to π1 (V − ∆) and φ(π1 (V − ∆)(n) ) = 0.
Now, we discuss some facts on T . First, Casson observed (unpublished) that T =6 0using [Fre82] and [Don83]. Endo [End95] observed that T is an infinite rank using thework of Furuta and Fintushel-Stern. M. Hedden, Se Goo Kim and C. Livingston [HL]observed that T contains a Z/2∞ -subgroup using d-invariant of 2-fold branched cover.
Actually, examples in [End95] show T /T0 ⊃ Z∞ and [HL] show T /T1 ⊃ Z/2∞ .
3. Survey of the recent results
So, we have nice filtration Tn on smooth concordance classes of topologically slice knots
T . Now, it is natural to ask about the nontriviality of this filtration.
Main question. Tn /Tn+1 6= 0 for all n?
We state theroems about this question without proof and refer the reader to the original
beautiful proofs.
Theorem. [CH12, Theorem 1.2] T0 /T1 has infinite rank. That is, T0 /T1 ⊃ Z∞ .
Theorem. [CHH13, Theorem 8.1] T1 /T2 has positive rank. That is, T1 /T2 ⊃ Z.
For links, J. C. Cha and M. Powell proved that Tn /Tn+1 is nontrivial for all n.
Theorem. [CP, Theorem 1.2] For m ≥ 2 and n ≥ 0, there exists an m-component link
which are n-bipolar but not (n + 1)-bipolar which has unknotted components for n ≥ 1.
References[CH12]

T. D. Cochran and P. D. Horn, Structure in the bipolar filtration of topologically slice knots,
arXiv:1208.5788, 2012.
[CHH13] T. D. Cochran, S. Harvey, and P. D. Horn, Filtering smooth concordance classes of topologicallyslice knots, Geometry and Topology 17 (2013), 2103–2162.
[CL86] T. D. Cochran and W. B. R. Lickorish, Unknotting information from 4-manifolds, Trans. Amer.
Math. Soc. 297 (1986), no. 1, 125–142.

INTRODUCTION TO BIPOLAR FILTRATION OF TOPOLOGICALLY SLICE KNOTS

4[CP]J. C. Cha and M. Powell, Covering link calculus and the bipolar filtration of topologically slice
links, arXiv:1212.5011.
[Don83] S. K. Donaldson, An application of gauge theory to four-dimensional topology, J. DifferentialGeom. 18 (1983), no. 2, 279–315.
[End95] H . Endo, Linear independence of topologically slice knots in the smooth cobordism group, Topology Appl. 63 (1995), no. 3, 257–262.
[FQ90] M. H. Freedman and F. Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39,Princeton University Press, Princeton, NJ, 1990.
[Fre82] M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982),no. 3, 357–453.
[HL]
Kim S. G. Hedden, M. and C. Livingston, Topologically slice knots of smooth concordance order
two, arxiv:1212.6628.
Department of Mathematics, Pohang University of Science and Technology, Pohang,
Gyungbuk 790-784, Republic of Korea
E-mail address: kminhoon@gmail.com