TY - JOUR

T1 - Picture-valued parity-biquandle bracket

AU - Ilyutko, Denis P.

AU - Manturov, Vassily O.

PY - 2020/2/1

Y1 - 2020/2/1

N2 - In V. O. Manturov, On free knots, preprint (2009), arXiv:math.GT/0901.2214], the second named author constructed the bracket invariant [[•]] of virtual knots valued in pictures (linear combinations of virtual knot diagrams with some crossing information omitted), such that for many diagrams K, the following formula holds: [K] = K, where K is the underlying graph of the diagram, i.e. the value of the invariant on a diagram equals the diagram itself with some crossing information omitted. This phenomenon allows one to reduce many questions about virtual knots to questions about their diagrams. In [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, preprint (2015), arXiv:math.GT/1508.06573], the authors discovered the following phenomenon: having a biquandle coloring of a certain knot, one can enhance various state-sum invariants (say, Kauffman bracket) by using various coefficients depending on colors. Taking into account that the parity can be treated in terms of biquandles, we bring together the two ideas from these papers and construct the picture-valued parity-biquandle bracket for classical and virtual knots. This is an invariant of virtual knots valued in pictures. Both the parity bracket and Nelson-Orrison-Rivera invariants are partial cases of this invariant, hence this invariant enjoys many properties of various kinds. Recently, the authors together with E. Horvat and S. Kim have found that the picture-valued phenomenon works in the classical case.

AB - In V. O. Manturov, On free knots, preprint (2009), arXiv:math.GT/0901.2214], the second named author constructed the bracket invariant [[•]] of virtual knots valued in pictures (linear combinations of virtual knot diagrams with some crossing information omitted), such that for many diagrams K, the following formula holds: [K] = K, where K is the underlying graph of the diagram, i.e. the value of the invariant on a diagram equals the diagram itself with some crossing information omitted. This phenomenon allows one to reduce many questions about virtual knots to questions about their diagrams. In [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, preprint (2015), arXiv:math.GT/1508.06573], the authors discovered the following phenomenon: having a biquandle coloring of a certain knot, one can enhance various state-sum invariants (say, Kauffman bracket) by using various coefficients depending on colors. Taking into account that the parity can be treated in terms of biquandles, we bring together the two ideas from these papers and construct the picture-valued parity-biquandle bracket for classical and virtual knots. This is an invariant of virtual knots valued in pictures. Both the parity bracket and Nelson-Orrison-Rivera invariants are partial cases of this invariant, hence this invariant enjoys many properties of various kinds. Recently, the authors together with E. Horvat and S. Kim have found that the picture-valued phenomenon works in the classical case.

KW - bracket

KW - coloring

KW - diagram

KW - invariant

KW - Knot

KW - parity

KW - picture biquandle

KW - Reidemeister moves

UR - http://www.scopus.com/inward/record.url?scp=85082422968&partnerID=8YFLogxK

U2 - 10.1142/S0218216520400040

DO - 10.1142/S0218216520400040

M3 - Article

AN - SCOPUS:85082422968

VL - 29

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 2

M1 - 2040004

ER -