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We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion, we derive a skein relation for a normalized form of this polynomial.

We introduce the virtual magnetic Kauffman bracket skein module, and show how to find a skein relation for the f-polynomial through the module. Furthermore, we give a unified approach to constructions of skein relations which hold under certain conditions.

In this paper, we introduce the tied links, i.e. ordinary links provided with some “ties” between strands. The motivation for introducing such objects originates from a diagrammatical interpretation of the defining generators of the so-called algebra of braids and ties; indeed, one half of such generators can be interpreted as the usual generators of the braid algebra, and the other half can be interpreted as ties between consecutive strands; this interpretation leads to the definition of tied braids. We define an invariant polynomial for the tied links via a skein relation. Furthermore, we introduce the monoid of tied braids and we prove the corresponding theorems of Alexander and Markov for tied links. Finally, we prove that the invariant of tied links we defined can be obtained also by using the Jones recipe.

The $(m,n)$-Turk’s head link is presented by the alternating diagram which is obtained from the standard diagram of the $(m,n)$-torus link by crossing changes. In this paper, we show that for any integers $m\ge 1$ and $n=2,3$, the coefficients of $z^i$ in the Conway polynomials of the $(m,n)$- and $(n,m)$-Turk’s head links coincide for $i\equiv 0,1(mod4)$ and differ by sign for $i\equiv 2,3(mod4)$. We conjecture that this property holds for any $n$.

Given any oriented link diagram, one can construct knot invariants using skein relations. Usually such a skein relation contains three or four terms. In this paper, the author introduces several new ways to smooth a crossings, and uses a system of skein equations to construct link invariants. This invariant can also be modified by writhe to get a more powerful invariant. The modified invariant is a generalization of both the HOMFLYPT polynomial and the two-variable Kauffman polynomial. Using the diamond lemma, a simplified version of the modified invariant is given. It is easy to compute and is a generalization of the two-variable Kauffman polynomial.

We show how the Alexander polynomial of links in lens spaces is related to the classical Alexander polynomial of a link in the 3-sphere, obtained by cutting out the exceptional lens space fiber. It follows from this relationship that a certain normalization of the Alexander polynomial satisfies a skein relation in lens spaces.

In the present paper, we develop a picture formalism which gives rise to an invariant that dominates several known invariants of classical and virtual knots: the Jones polynomial ${𝒳}\left(\cdot \right)$, the Kuperberg bracket ${\left|\cdot \right.〉}_{A_2}$, and the normalized arrow polynomial ${𝒲}\left[\cdot \right]$.