A zero polynomial of virtual knots

Jeong, Myeong-Ju

doi:10.1142/s0218216515500789

Cited by 23 publications

(14 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Theorem 7 of [BCG17] shows that the writhe polynomial W K (t) and Heinrich-Turaev polynomial P K (t) are concordance invariants. The following, which concerns the polynomials invariants Z n K (t) and Z K (t) introduced in [IK17] and [Jeo16], respectively, is an immediate consequence of combining the above observations with Theorem 5.9.…”

Section: Coverings and Paritymentioning

confidence: 88%

See 1 more Smart Citation

Concordance group of virtual knots

Bodén

Nagel²

2017

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

We introduce Tristram-Levine signatures of virtual knots and use them to investigate virtual knot concordance. The signatures are defined first for almost classical knots, which are virtual knots admitting homologically trivial representations. The signatures and ω-signatures are shown to give bounds on the topological slice genus of almost classical knots, and they are applied to address a recent question of Dye, Kaestner, and Kauffman on the virtual slice genus of classical knots. A conjecture on the topological slice genus is formulated and confirmed for all classical knots with up to 11 crossings and for 2150 out of 2175 of the 12 crossing knots.The Seifert pairing is used to define directed Alexander polynomials, which we show satisfy a Fox-Milnor criterion when the almost classical knot is slice. We introduce virtual disk-band surfaces and use them to establish realization theorems for Seifert matrices of almost classical knots. As a consequence, we deduce that any integral polynomial ∆(t) satisfying ∆(1) = 1 occurs as the Alexander polynomial of an almost classical knot.In the last section, we use parity projection and Turaev's coverings of knots to extend the Tristram-Levine signatures to all virtual knots. A key step is a theorem saying that parity projection preserves concordance of virtual knots. This theorem implies that the signatures, ω-signatures, and Fox-Milnor criterion can be lifted to give slice obstructions for all virtual knots. There are 76 almost classical knots with up to six crossings, and we use our invariants to determine the slice status for all of them and the slice genus for all but four.

show abstract

Section: Coverings and Paritymentioning

confidence: 88%

“…In [Jeo16], Jeong introduced the zero polynomial Z K (t), and in [IK17], Im and Kim give a sequence of polynomial invariants Z n K (t) for n a non-negative integer. They note that for n = 0, Z 0 K (t) = Z K (t), and they give a formula for computing Z n K (t) in terms of Gauss diagrams (see Definition 3.1 of [IK17]).…”

Section: Coverings and Paritymentioning

confidence: 99%

Concordance group of virtual knots

Bodén

Nagel²

2017

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…

In this work we describe a new invariant of virtual knots. We show that this transcendental function invariant generalizes several polynomial invariants of virtual knots, such as the writhe polynomial [2], the affine index polynomial [18] and the zero polynomial [13].1991 Mathematics Subject Classification. 57M27.

…”

mentioning

confidence: 89%

“…However there also exists one obvious drawback: if a real crossing has zero index then it has no contribution to the invariant. Recently this shortcoming was improved by Myeong-Ju Jeong in [13]. In [13], Jeong introduced the zero polynomial which focused on the real crossings with zero index.…”

Section: Introductionmentioning

confidence: 99%

A transcendental function invariant of virtual knots

Cheng¹

2017

J. Math. Soc. Japan

View full text Add to dashboard Cite

show abstract

“…The n-writhe is defined as a refinement of the odd writhe. Jeong defines an invariant called the zero polynomial of a virtual knot by using real crossings of index 0 [7]. In fact, Im and Kim prove that the zero polynomial is coincident with the writhe polynomial of the virtual knot obtained by replacing the real crossings whose indices are non-zero with virtual crossings [5].…”

Section: Introductionmentioning

confidence: 99%