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Journal articles on the topic 'Knot concordance invariants'

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Published: 10 March 2023

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1

KEARNEY, M. KATE. "THE CONCORDANCE GENUS OF 11-CROSSING KNOTS." Journal of Knot Theory and Its Ramifications 22, no. 13 (November 2013): 1350077. http://dx.doi.org/10.1142/s0218216513500776.

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The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the concordance genus of 11-crossing prime knots. This analysis resolves the concordance genus of 533 of the 552 prime 11-crossing knots. The appendix to the paper gives concordance diagrams for 59 knots found to be concordant to knots of lower genus, including null-concordances for the 30 11-crossing knots known to be slice.

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2

Kim, Taehee. "Knots having the same Seifert form and primary decomposition of knot concordance." Journal of Knot Theory and Its Ramifications 26, no. 14 (December 2017): 1750103. http://dx.doi.org/10.1142/s0218216517501036.

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We show that for each Seifert form of an algebraically slice knot with nontrivial Alexander polynomial, there exists an infinite family of knots having the Seifert form such that the knots are linearly independent in the knot concordance group and not concordant to any knot with coprime Alexander polynomial. Key ingredients for the proof are Cheeger–Gromov–von Neumann [Formula: see text]-invariants for amenable groups developed by Cha–Orr and polynomial splittings of metabelian [Formula: see text]-invariants.

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3

Hom, Jennifer. "A survey on Heegaard Floer homology and concordance." Journal of Knot Theory and Its Ramifications 26, no. 02 (February 2017): 1740015. http://dx.doi.org/10.1142/s0218216517400156.

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In this survey paper, we discuss several different knot concordance invariants coming from the Heegaard Floer homology package of Ozsváth and Szabó. Along the way, we prove that if two knots are concordant, then their knot Floer complexes satisfy a certain type of stable equivalence.

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4

Park, Kyungbae. "On independence of iterated Whitehead doubles in the knot concordance group." Journal of Knot Theory and Its Ramifications 27, no. 01 (January 2018): 1850003. http://dx.doi.org/10.1142/s0218216518500037.

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Let [Formula: see text] be the positively clasped untwisted Whitehead double of a knot [Formula: see text], and [Formula: see text] be the [Formula: see text] torus knot. We show that [Formula: see text] and [Formula: see text] are linearly independent in the smooth knot concordance group [Formula: see text] for each [Formula: see text]. Further, [Formula: see text] and [Formula: see text] generate a [Formula: see text] summand in the subgroup of [Formula: see text] generated by topologically slice knots. We use the concordance invariant [Formula: see text] of Manolescu and Owens, using Heegaard Floer correction term. Interestingly, these results are not easily shown using other concordance invariants such as the [Formula: see text]-invariant of knot Floer theory and the [Formula: see text]-invariant of Khovanov homology. We also determine the infinity version of the knot Floer complex of [Formula: see text] for any [Formula: see text] generalizing a result for [Formula: see text] of Hedden, Kim and Livingston.

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5

Kim, Se-Goo, and Mi Jeong Yeon. "Rasmussen and Ozsváth–Szabó invariants of a family of general pretzel knots." Journal of Knot Theory and Its Ramifications 24, no. 03 (March 2015): 1550017. http://dx.doi.org/10.1142/s0218216515500170.

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We compute integer valued knot concordance invariants of a family of general pretzel knots if the invariants are equal to the negative values of signatures for alternating knots. Examples of such invariants are Rasmussen s-invariants and twice Ozsváth–Szabó knot Floer homology τ-invariants. We use the crossing change inequalities of Livingston and the fact that pretzel knots are almost alternating. As a consequence, for the family of pretzel knots given in this paper, s-invariants are twice τ-invariants.

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6

Cochran, Tim D., and Peter Teichner. "Knot concordance and von Neumann $\rho$ -invariants." Duke Mathematical Journal 137, no. 2 (April 2007): 337–79. http://dx.doi.org/10.1215/s0012-7094-07-13723-2.

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7

Gilmer, Pat, and Charles Livingston. "An algebraic link concordance group for (p, 2p−1)-links in S2p+1." Proceedings of the Edinburgh Mathematical Society 34, no. 3 (October 1991): 455–62. http://dx.doi.org/10.1017/s0013091500005228.

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A concordance classification of links of , p < 1, is given in terms of an algebraically defined group, Φ±, which is closely related to Levine's algebraic knot concordance group. For p=1,Φ_ captures certain obstructions to two component links in S3 being concordant to boundary links, the generalized Sato-Levine invariants defined by Cochran. As a result, purely algebraic proofs of properties of these invariants are derived.

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8

Dasbach, Oliver T., and Adam M. Lowrance. "Turaev genus, knot signature, and the knot homology concordance invariants." Proceedings of the American Mathematical Society 139, no. 7 (December 22, 2010): 2631–45. http://dx.doi.org/10.1090/s0002-9939-2010-10698-6.

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9

Truong, Linh. "Truncated Heegaard Floer homology and knot concordance invariants." Algebraic & Geometric Topology 19, no. 4 (August 16, 2019): 1881–901. http://dx.doi.org/10.2140/agt.2019.19.1881.

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10

Lewark, Lukas, and Andrew Lobb. "Upsilon-like concordance invariants from 𝔰𝔩n knot cohomology." Geometry & Topology 23, no. 2 (April 8, 2019): 745–80. http://dx.doi.org/10.2140/gt.2019.23.745.

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11

Carter, J. Scott, Masahico Saito, and Shin Satoh. "Ribbon concordance of surface-knots via quandle cocycle invariants." Journal of the Australian Mathematical Society 80, no. 1 (February 2006): 131–47. http://dx.doi.org/10.1017/s1446788700011423.

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AbstractWe give necessary conditions of a surface-knot to be ribbon concordant to another, by introducing a new variant of the cocycle invariant of surface-knots in addition to using the invariant already known. We demonstrate that twist-spins of some torus knots are not ribbon concordant to their orientation reversed images.

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12

Vance, Katherine. "Tau invariants for balanced spatial graphs." Journal of Knot Theory and Its Ramifications 29, no. 09 (August 2020): 2050066. http://dx.doi.org/10.1142/s0218216520500662.

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In 2003, Ozsváth and Szabó defined the concordance invariant [Formula: see text] for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of [Formula: see text] for knots in [Formula: see text] and a combinatorial proof that [Formula: see text] gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory for transverse spatial graphs in [Formula: see text], extending HFK for knots. We define a [Formula: see text]-filtered chain complex for balanced spatial graphs whose associated graded chain complex has homology determined by Harvey and O’Donnol’s graph Floer homology. We use this to show that there is a well-defined [Formula: see text] invariant for balanced spatial graphs generalizing the [Formula: see text] knot concordance invariant. In particular, this defines a [Formula: see text] invariant for links in [Formula: see text]. Using techniques similar to those of Sarkar, we show that our [Formula: see text] invariant is an obstruction to a link being slice.

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13

LIVINGSTON, CHARLES. "THE ALGEBRAIC CONCORDANCE ORDER OF A KNOT." Journal of Knot Theory and Its Ramifications 19, no. 12 (December 2010): 1693–711. http://dx.doi.org/10.1142/s0218216510008571.

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The algebraic concordance group contains elements of order two, four, and of infinite order. Elements of infinite order are detected by the signature function. This paper develops computable invariants to simplify the computation of the order of torsion classes. The results are applied to determine the algebraic orders of all prime knots of 12 or fewer crossings.

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14

BORODZIK, MACIEJ, and MATTHEW HEDDEN. "The ϒ function ofL–space knots is a Legendre transform." Mathematical Proceedings of the Cambridge Philosophical Society 164, no. 3 (March 20, 2017): 401–11. http://dx.doi.org/10.1017/s030500411700024x.

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AbstractGiven anL–space knot we show that its ϒ function is the Legendre transform of a counting function equivalent to thed–invariants of its large surgeries. The unknotting obstruction obtained for the ϒ function is, in the case ofL–space knots, contained in thed–invariants of large surgeries. Generalisations apply for connected sums ofL–space knots, which imply that the slice obstruction provided by ϒ on the subgroup of concordance generated byL–space knots is no finer than that provided by thed–invariants.

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15

Mio, Washington. "On the geometry of homotopy invariants of links." Mathematical Proceedings of the Cambridge Philosophical Society 111, no. 2 (March 1992): 291–98. http://dx.doi.org/10.1017/s0305004100075381.

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One of the central problems in higher-dimensional knot theory is the classification of links up to concordance. In 14, Le Dimet constructed a universal model for (disk) link complements, which allowed him to formulate this problem in the framework of surgery theory by applying the Cappell-Shaneson program for studying codimension two embeddings of manifolds 1. The concordance classification was reduced to questions in L-theory (-groups 1) and homotopy theory (of Vogel local spaces 14). While recent results of Cochran and Orr2 (see also 18) provide rich information on the -theoretic part of the problem (in particular, they settle the question of the existence of links not concordant to boundary links), little is known about Le Dimet's homotopy invariant of links; for example, it is not known whether it may ever be non-trivial, or phrasing it more geometrically (according to 19), whether there are links that are not concordant to sublinks of homology boundary links. This motivated us to look at simpler classes of links, for which a more direct geometric approach to the problem is also possible, in an attempt to get some insight on the geometry carried by the homotopy invariants.

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16

Grigsby, J. Elisenda, Daniel Ruberman, and Sašo Strle. "Knot concordance and Heegaard Floer homology invariants in branched covers." Geometry & Topology 12, no. 4 (September 2, 2008): 2249–75. http://dx.doi.org/10.2140/gt.2008.12.2249.

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17

Kim, Min Hoon, Se-Goo Kim, and Taehee Kim. "Primary decomposition of knot concordance and von Neumann rho-invariants." Proceedings of the American Mathematical Society 149, no. 1 (October 20, 2020): 439–47. http://dx.doi.org/10.1090/proc/15282.

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18

KIM, SE-GOO. "ALEXANDER POLYNOMIALS AND ORDERS OF HOMOLOGY GROUPS OF BRANCHED COVERS OF KNOTS." Journal of Knot Theory and Its Ramifications 18, no. 07 (July 2009): 973–84. http://dx.doi.org/10.1142/s0218216509007300.

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Fox showed that the order of homology of a cyclic branched cover of a knot is determined by its Alexander polynomial. We find examples of knots with relatively prime Alexander polynomials such that the first homology groups of their q-fold cyclic branched covers are of the same order for every prime power q. Furthermore, we show that these knots are linearly independent in the knot concordance group using the polynomial splitting property of the Casson–Gordon–Gilmer invariants.

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19

MEILHAN, JEAN-BAPTISTE, and AKIRA YASUHARA. "Characterization of finite type string link invariants of degree <5." Mathematical Proceedings of the Cambridge Philosophical Society 148, no. 3 (March 16, 2010): 439–72. http://dx.doi.org/10.1017/s0305004110000046.

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AbstractWe give a complete set of finite type string link invariants of degree <5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closures of (cabled) string links. We show that finite type invariants classify string links up toCk-moves fork≤ 5, which proves, at low degree, a conjecture due to Goussarov and Habiro. We also give a similar classification of string links up toCk-moves and concordance fork≤ 6.

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20

Cochran, Tim D., and Taehee Kim. "Higher-order Alexander invariants and filtrations of the knot concordance group." Transactions of the American Mathematical Society 360, no. 03 (March 1, 2008): 1407–42. http://dx.doi.org/10.1090/s0002-9947-07-04177-3.

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21

KIM, TAEHEE. "Filtration of the classical knot concordance group and Casson–Gordon invariants." Mathematical Proceedings of the Cambridge Philosophical Society 137, no. 2 (September 2004): 293–306. http://dx.doi.org/10.1017/s0305004104007686.

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22

Davis, Christopher William. "Von Neumann rho invariants and torsion in the topological knot concordance group." Algebraic & Geometric Topology 12, no. 2 (April 12, 2012): 753–89. http://dx.doi.org/10.2140/agt.2012.12.753.

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23

Chrisman, Micah. "Virtual Seifert surfaces." Journal of Knot Theory and Its Ramifications 28, no. 06 (May 2019): 1950039. http://dx.doi.org/10.1142/s0218216519500391.

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A virtual knot that has a homologically trivial representative [Formula: see text] in a thickened surface [Formula: see text] is said to be an almost classical (AC) knot. [Formula: see text] then bounds a Seifert surface [Formula: see text]. Seifert surfaces of AC knots are useful for computing concordance invariants and slice obstructions. However, Seifert surfaces in [Formula: see text] are difficult to construct. Here, we introduce virtual Seifert surfaces of AC knots. These are planar figures representing [Formula: see text]. An algorithm for constructing a virtual Seifert surface from a Gauss diagram is given. This is applied to computing signatures and Alexander polynomials of AC knots. A canonical genus of AC knots is also studied. It is shown to be distinct from the virtual canonical genus of Stoimenow–Tchernov–Vdovina.

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24

Tobin, Joshua P. "Knot Floer filtration classes of topologically slice knots." Journal of Knot Theory and Its Ramifications 23, no. 09 (August 2014): 1450047. http://dx.doi.org/10.1142/s0218216514500473.

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The knot Floer complex and the concordance invariant ε can be used to define a filtration on the smooth concordance group. We exhibit an ordered subset of this filtration that is isomorphic to ℕ × ℕ and consists of topologically slice knots.

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25

Chrisman, Micah. "Band-passes and long virtual knot concordance." Journal of Knot Theory and Its Ramifications 26, no. 10 (September 2017): 1750057. http://dx.doi.org/10.1142/s0218216517500572.

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Every classical knot is band-pass equivalent to the unknot or the trefoil. The band-pass class of a knot is a concordance invariant. Every ribbon knot, for example, is band-pass equivalent to the unknot. Here we introduce the long virtual knot concordance group [Formula: see text]. It is shown that for every concordance class [Formula: see text], there is a [Formula: see text] that is not band-pass equivalent to [Formula: see text] and an [Formula: see text] that is not band-pass equivalent to either the long unknot or any long trefoil. This is accomplished by proving that [Formula: see text] is a band-pass invariant but not a concordance invariant of long virtual knots, where [Formula: see text] and [Formula: see text] generate the degree two Polyak group for long virtual knots.

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26

LIVINGSTON, CHARLES, and CORNELIA A. VAN COTT. "Concordance of Bing Doubles and Boundary Genus." Mathematical Proceedings of the Cambridge Philosophical Society 151, no. 3 (July 18, 2011): 459–70. http://dx.doi.org/10.1017/s0305004111000442.

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AbstractCha and Kim proved that if a knot K is not algebraically slice, then no iterated Bing double of K is concordant to the unlink. We prove that if K has nontrivial signature σ, then the n–iterated Bing double of K is not concordant to any boundary link with boundary surfaces of genus less than 2n−1σ. The same result holds with σ replaced by 2τ, twice the Ozsváth–Szabó knot concordance invariant.

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27

HEDDEN, MATTHEW. "NOTIONS OF POSITIVITY AND THE OZSVÁTH–SZABÓ CONCORDANCE INVARIANT." Journal of Knot Theory and Its Ramifications 19, no. 05 (May 2010): 617–29. http://dx.doi.org/10.1142/s0218216510008017.

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In this paper we examine the relationship between various types of positivity for knots and the concordance invariant τ discovered by Ozsváth and Szabó and independently by Rasmussen. The main result shows that, for fibered knots, τ characterizes strong quasipositivity. This is quantified by the statement that for K fibered, τ(K) = g(K) if and only if K is strongly quasipositive. A corollary is that any knot admitting a lens space surgery or, more generally, an L-space surgery, is strongly quasipositive. In addition, we survey existing results regarding τ and forms of positivity and highlight several consequences concerning the types of knots which are (strongly) (quasi) positive.

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28

Livingston, Charles. "Notes on the knot concordance invariant Upsilon." Algebraic & Geometric Topology 17, no. 1 (January 26, 2017): 111–30. http://dx.doi.org/10.2140/agt.2017.17.111.

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29

Miller, Allison N. "Distinguishing mutant pretzel knots in concordance." Journal of Knot Theory and Its Ramifications 26, no. 07 (March 31, 2017): 1750041. http://dx.doi.org/10.1142/s0218216517500419.

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We prove that many four-strand pretzel knots of the form [Formula: see text] are not topologically slice, even though their positive mutants [Formula: see text] are ribbon. We use the sliceness obstruction of Kirk and Livingston [Twisted Alexander invariants, Reidemeister torsion, and Casson–Gordon invariants, Topology 38 (1999) 635–661], related to the twisted Alexander polynomials associated to prime power cyclic covers of knots.

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30

Livingston, Charles. "Computations of the Ozsváth–Szabó knot concordance invariant." Geometry & Topology 8, no. 2 (May 17, 2004): 735–42. http://dx.doi.org/10.2140/gt.2004.8.735.

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31

Kronheimer, P. B., and T. S. Mrowka. "Instantons and some concordance invariants of knots." Journal of the London Mathematical Society 104, no. 2 (January 22, 2021): 541–71. http://dx.doi.org/10.1112/jlms.12439.

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32

Chrisman, Micah. "Milnor’s concordance invariants for knots on surfaces." Algebraic & Geometric Topology 22, no. 5 (October 25, 2022): 2293–353. http://dx.doi.org/10.2140/agt.2022.22.2293.

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33

FARBER, MICHAEL S., and JONATHAN A. HILLMAN. "DOUBLY NULL CONCORDANT KNOTS HAVE HYPERBOLIC STABLE ISOMETRY STRUCTURES." Journal of Knot Theory and Its Ramifications 02, no. 02 (June 1993): 125–40. http://dx.doi.org/10.1142/s0218216593000088.

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We shall show that the stable isometry structure determined by a suitable Seifert hypersurface of a doubly null concordant knot is hyperbolic and we prove a converse for stable knots. This suggests a “universal” source for the known homological invariants of DNC-equivalence. As an application of our main result we shall show that if the homology of the universal cover of the complement of a stable n-knot is torsion, involving only primes >(n+10)/6, and is 0 in the middle dimensions then the knot is doubly null concordant.

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34

Bosman, Anthony M. "Shake slice and shake concordant links." Journal of Knot Theory and Its Ramifications 29, no. 12 (October 2020): 2050087. http://dx.doi.org/10.1142/s021821652050087x.

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We can construct a [Formula: see text]-manifold by attaching [Formula: see text]-handles to a [Formula: see text]-ball with framing [Formula: see text] along the components of a link in the boundary of the [Formula: see text]-ball. We define a link as [Formula: see text]-shake slice if there exists embedded spheres that represent the generators of the second homology of the [Formula: see text]-manifold. This naturally extends [Formula: see text]-shake slice, a generalization of slice that has previously only been studied for knots, to links of more than one component. We also define a relative notion of shake[Formula: see text]-concordance for links and versions with stricter conditions on the embedded spheres that we call strongly[Formula: see text]-shake slice and strongly[Formula: see text]-shake concordance. We provide infinite families of links that distinguish concordance, shake concordance, and strong shake concordance. Moreover, for [Formula: see text] we completely characterize shake slice and shake concordant links in terms of concordance and string link infection. This characterization allows us to prove that the first non-vanishing Milnor [Formula: see text] invariants are invariants of shake concordance. We also argue that shake concordance does not imply link homotopy.

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35

Dey, Subhankar, and Hakan Doğa. "A combinatorial description of the knot concordance invariant epsilon." Journal of Knot Theory and Its Ramifications 30, no. 06 (May 2021): 2150036. http://dx.doi.org/10.1142/s021821652150036x.

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In this paper, we give a combinatorial description of the concordance invariant [Formula: see text] defined by Hom, prove some properties of this invariant using grid homology techniques. We compute the value of [Formula: see text] for [Formula: see text] torus knots and prove that [Formula: see text] if [Formula: see text] is a grid diagram for a positive braid. Furthermore, we show how [Formula: see text] behaves under [Formula: see text]-cabling of negative torus knots.

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36

Kauffman, Louis H. "Virtual knot cobordism and the affine index polynomial." Journal of Knot Theory and Its Ramifications 27, no. 11 (October 2018): 1843017. http://dx.doi.org/10.1142/s0218216518430174.

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This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relation to these phenomena. Information on determinations of the four-ball genus of some virtual knots is obtained by via the affine index polynomial in conjunction with results on the genus of positive virtual knots using joint work with Dye and Kaestner.

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37

HANCOCK, STEPHEN, JENNIFER HOM, and MICHAEL NEWMAN. "ON THE KNOT FLOER FILTRATION OF THE CONCORDANCE GROUP." Journal of Knot Theory and Its Ramifications 22, no. 14 (December 2013): 1350084. http://dx.doi.org/10.1142/s0218216513500843.

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The knot Floer complex together with the associated concordance invariant ε can be used to define a filtration on the smooth concordance group. We show that the indexing set of this filtration contains ℕ × ℤ as an ordered subset.

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38

Lobb, Andrew. "Computable bounds for Rasmussen’s concordance invariant." Compositio Mathematica 147, no. 2 (December 13, 2010): 661–68. http://dx.doi.org/10.1112/s0010437x10005117.

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AbstractGiven a diagram D of a knot K, we give easily computable bounds for Rasmussen’s concordance invariant s(K). The bounds are not independent of the diagram D chosen, but we show that for diagrams satisfying a given condition the bounds are tight. As a corollary we improve on previously known Bennequin-type bounds on the slice genus.

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39

Kim, Se-Goo, and Kwan Yong Lee. "Concordance invariants of doubled knots and blowing up." Proceedings of the American Mathematical Society 147, no. 4 (January 9, 2019): 1781–88. http://dx.doi.org/10.1090/proc/14320.

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40

Kholodenko, Arkady. "Black magic session of concordance: Regge mass spectrum from Casson’s invariant." International Journal of Modern Physics A 30, no. 33 (November 26, 2015): 1550189. http://dx.doi.org/10.1142/s0217751x15501894.

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Recently, there had been a great deal of interest in obtaining and describing all kinds of knots in links in hydrodynamics, electrodynamics, non-Abelian gauge field theories and gravity. Although knots and links are observables of the Chern–Simons (CS) functional, the dynamical conditions for their generation lie outside the scope of the CS theory. The nontriviality of dynamical generation of knotted structures is caused by the fact that the complements of all knots/links, say, in S3are 3-manifolds which have positive, negative or zero curvature. The ability to curve the ambient space is thus far attributed to masses. The mass theorem of general relativity requires the ambient 3-manifolds to be of nonnegative curvature. Recently, we established that, in the absence of boundaries, complements of dynamically generated knots/links are represented by 3-manifolds of nonnegative curvature. This fact opens the possibility to discuss masses in terms of dynamically generated knotted/linked structures. The key tool is the notion of knot/link concordance. The concept of concordance is a specialization of the concept of cobordism to knots and links. The logic of implementation of the concordance concept to physical masses results in new interpretation of Casson’s surgery formula in terms of the Regge trajectories. The latest thoroughly examined Chew–Frautschi (CF) plots associated with these trajectories demonstrate that the hadron mass spectrum for both mesons and baryons is nicely described by the data on the corresponding CF plots. The physics behind Casson’s surgery formula is similar but not identical to that described purely phenomenologically by Keith Moffatt in 1990. The developed topological treatment is fully consistent with available rigorous mathematical and experimentally observed results related to physics of hadrons.

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41

TAMULIS, ANDRIUS. "KNOTS OF TEN OR FEWER CROSSINGS OF ALGEBRAIC ORDER 2." Journal of Knot Theory and Its Ramifications 11, no. 02 (March 2002): 211–22. http://dx.doi.org/10.1142/s0218216502001585.

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The concordance orders of many algebraic order two knots of ten or fewer crossings have been heretofore unknown. We use Casson-Gordon invariants and twisted Alexander polynomials to find that, in all but one case, these knots do not have concordance order two. We also find that a certain family of algebraic order two twisted doubles of the unknot have infinite concordance order.

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42

CHA, JAE CHOON, CHARLES LIVINGSTON, and DANIEL RUBERMAN. "Algebraic and Heegaard–Floer invariants of knots with slice Bing doubles." Mathematical Proceedings of the Cambridge Philosophical Society 144, no. 2 (March 2008): 403–10. http://dx.doi.org/10.1017/s0305004107000795.

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AbstractIf the Bing double of a knotKis slice, thenKis algebraically slice. In addition the Heegaard–Floer concordance invariants τ, developed by Ozsváth–Szabó, and δ, developed by Manolescu and Owens, vanish onK.

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43

NAIK, SWATEE. "CASSON-GORDON INVARIANTS OF GENUS ONE KNOTS AND CONCORDANCE TO REVERSES." Journal of Knot Theory and Its Ramifications 05, no. 05 (October 1996): 661–77. http://dx.doi.org/10.1142/s0218216596000382.

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For genus one knots the Casson-Gordon invariant τ is expressed in terms of the classical signatures, generalizing an earlier result of P. Gilmer. As an application it is shown that the pretzel knots K(3, –5, 7) and K(3, –5, 17) are not concordant to their reverses.

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44

Martin, Taylor, and Carolyn Otto. "Splittings of link concordance groups." Journal of Knot Theory and Its Ramifications 26, no. 02 (February 2017): 1740007. http://dx.doi.org/10.1142/s0218216517400077.

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We establish several results about two short exact sequences involving lower terms of the [Formula: see text]-solvable filtration, [Formula: see text] of the string link concordance group [Formula: see text]. We utilize the Thom–Pontryagin construction to show that the Sato–Levine invariants [Formula: see text] must vanish for 0.5-solvable links. Using this result, we show that the short exact sequence [Formula: see text] does not split for links of two or more components, in contrast to the fact that it splits for knots. Considering lower terms of the filtration [Formula: see text] in the short exact sequence [Formula: see text], we show that while the sequence does not split for [Formula: see text], it does indeed split for [Formula: see text]. This allows us to determine that the quotient [Formula: see text].

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45

Wang, Shida. "A note on the concordance invariants Upsilon and phi." Communications in Contemporary Mathematics, December 9, 2021. http://dx.doi.org/10.1142/s021919972150098x.

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Dai, Hom, Stoffregen and Truong defined a family of concordance invariants [Formula: see text]. The example of a knot with zero Upsilon invariant but nonzero epsilon invariant previously given by Hom also has nonzero phi invariant. We show there are infinitely many such knots that are linearly independent in the smooth concordance group. In the opposite direction, we build infinite families of linearly independent knots with zero phi invariant but nonzero Upsilon invariant. We also give a recursive formula for the phi invariant of torus knots.

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46

"Upsilon Invariants from Cyclic Branched Covers." Studia Scientiarum Mathematicarum Hungarica 58, no. 4 (December 3, 2021): 457–88. http://dx.doi.org/10.1556/012.2021.01515.

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We extend the construction of Y-type invariants to null-homologous knots in rational homology three-spheres. By considering m-fold cyclic branched covers with m a prime power, this extension provides new knot concordance invariants of knots in S3. We give computations of some of these invariants for alternating knots and reprove independence results in the smooth concordance group.

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47

Cha, Jae Choon. "Primary decomposition in the smooth concordance group of topologically slice knots." Forum of Mathematics, Sigma 9 (2021). http://dx.doi.org/10.1017/fms.2021.46.

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Abstract We address primary decomposition conjectures for knot concordance groups, which predict direct sum decompositions into primary parts. We show that the smooth concordance group of topologically slice knots has a large subgroup for which the conjectures are true and there are infinitely many primary parts, each of which has infinite rank. This supports the conjectures for topologically slice knots. We also prove analogues for the associated graded groups of the bipolar filtration of topologically slice knots. Among ingredients of the proof, we use amenable $L^2$ -signatures, Ozsváth-Szabó d-invariants and Némethi’s result on Heegaard Floer homology of Seifert 3-manifolds. In an appendix, we present a general formulation of the notion of primary decomposition.

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Lim, Geunho. "Enhanced Bounds for rho-invariants for both general and spherical 3-manifolds." Journal of Topology and Analysis, February 26, 2022, 1–51. http://dx.doi.org/10.1142/s1793525322500029.

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We establish enhanced bounds on Cheeger–Gromov [Formula: see text]-invariants for general 3-manifolds and yet stronger bounds for special classes of 3-manifold. As key ingredients, we construct chain null-homotopies whose complexity is linearly bounded by its boundary. This result can be regarded as an algebraic topological analogue of Gromov’s conjecture for quantitative topology. The author hopes for applications to various fields including the smooth knot concordance group, quantitative topology and complexity theory.

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Wang, Shida. "A Note on the Concordance Invariant Epsilon." Quarterly Journal of Mathematics, July 17, 2021. http://dx.doi.org/10.1093/qmath/haab033.

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Abstract We compare the smooth concordance invariants Upsilon, phi and epsilon. Previous work gave examples of knots with one of the Upsilon and phi invariants zero but the epsilon invariant nonzero. We build an infinite family of linearly independent knots with both the Upsilon and phi invariants zero but the epsilon invariant nonzero. This provides examples of knots with arbitrarily large concordance genus but vanishing bounds from the Upsilon and phi invariants.

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50

Jung, Hongtaek, Sungkyung Kang, and Seungwon Kim. "Concordance Invariants and the Turaev Genus." International Mathematics Research Notices, June 24, 2021. http://dx.doi.org/10.1093/imrn/rnab055.

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Abstract We show that the differences between various concordance invariants of knots, including Rasmussen’s $s$-invariant and its generalizations $s_n$-invariants, give lower bounds to the Turaev genus of knots. Using the fact that our bounds are nontrivial for some quasi-alternating knots, we show the additivity of Turaev genus for a certain class of knots. This leads us to the 1st example of an infinite family of quasi-alternating knots with Turaev genus exactly $g$ for any fixed positive integer $g$, solving a question of Champanerkar–Kofman.

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