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During the talk, we consider singular links in the 3-space up to a
suitable equivalence relation defined by isotopy of 3-space. It is
well known that every such link \(L\) can be represented as the closer
\(\hat{b}\) of a singular braid \(b\). The minimal possible number \(k\)
of strands in a braid for such representation is called the braid
index of \(L\).
There are known different methods of estimating the braid index of
classical links: the Morton-Frank-Williams inequalities
(MFW-inequalities) or their cable versions (in terms of HOMFLYPT
polynomial); the KR-MFW-inequalities (in terms of Khowanov-Rozansky
homology), combinatorial methods etc.}
We address a question of whether the methods mentioned above can be
used for estimating the braid index of singular links. In this
relation, we consider the extended versions of HOMFLYPT polynomial
for singular links, described by L. Paris and L. Rabenda, and
L. $\,$ H. Kauffman and P. Vogel. We also analyze some combinatorial
tools, such as graphs associated with diagrams of singular
links. The question of whether computing the braid index of a
singular link \(L\) can be reduced (may be partially) to estimating
the stabilized braid index for a family of classical links
associated with \(L\) is a
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References- J.S. Birman, New point of view in knot theory ,
Bull. Amer. Math. Soc.(N.S.) 28, 1993, 253-287.
- K. Kawamuro Khovanov-Rozansky homology and the
braid index , Proc. Amer. Math. Soc., 137, 2009, 2459-2469.
- L. Paris, L. Rabenda B., Singular Hecke algebras,
Markov traces, and HOMFLY-type invariants , arXiv [math/GT]:
0707.0400v1.
- L.H. Kauffman, P. Vogel, Link polynomials and a
graphical calculus , J. Knot Theory Ramification 1, 1992,
59-104.
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