TY - JOUR

T1 - New skein invariants of links

AU - Kauffman, Louis H.

AU - Lambropoulou, Sofia

PY - 2020/1/6

Y1 - 2020/1/6

N2 - We study new skein invariants of links based on a procedure where we first apply a given skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using the given invariant. A skein invariant can be computed on each link solely by the use of skein relations and a set of initial conditions. The new procedure, remarkably, leads to generalizations of the known skein invariants. We make skein invariants of classical links, H[R], K[Q] and D[T], based on the invariants of knots, R, Q and T, denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. We provide skein theoretic proofs of the well-definedness of these invariants. These invariants are also reformulated into summations of the generating invariants (R, Q, T) on sublinks of a given link L, obtained by partitioning L into collections of sublinks. These summations exhibit the tight and surprising relationship between our generalized skein-theoretic procedure and the structure of sublinks of a given link.

AB - We study new skein invariants of links based on a procedure where we first apply a given skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using the given invariant. A skein invariant can be computed on each link solely by the use of skein relations and a set of initial conditions. The new procedure, remarkably, leads to generalizations of the known skein invariants. We make skein invariants of classical links, H[R], K[Q] and D[T], based on the invariants of knots, R, Q and T, denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. We provide skein theoretic proofs of the well-definedness of these invariants. These invariants are also reformulated into summations of the generating invariants (R, Q, T) on sublinks of a given link L, obtained by partitioning L into collections of sublinks. These summations exhibit the tight and surprising relationship between our generalized skein-theoretic procedure and the structure of sublinks of a given link.

KW - 3-variable link invariant

KW - Classical links

KW - closed combinatorial formulae

KW - Dubrovnik polynomial

KW - Homflypt polynomial

KW - Kauffman polynomial

KW - mixed crossings

KW - Reidemeister moves

KW - skein invariants

KW - skein relations

KW - stacks of knots

KW - Yokonuma-Hecke algebras

KW - POLYNOMIAL INVARIANT

KW - BRAIDS

KW - HECKE ALGEBRAS

KW - FRAMIZATION

KW - ISOMORPHISM THEOREM

KW - KNOTS

KW - MODULE

KW - REPRESENTATION-THEORY

UR - http://www.scopus.com/inward/record.url?scp=85077895836&partnerID=8YFLogxK

U2 - 10.1142/S0218216519400182

DO - 10.1142/S0218216519400182

M3 - Article

AN - SCOPUS:85077895836

VL - 28

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 13

M1 - 1940018

ER -