Narrow Results

Homomorphisms on quandle cohomology groups that raise the dimensions by one are studied in relation to the cocycle state-sum invariants of knots and knotted surfaces. Skein relations are also studied.

We give characterizations of the skein polynomial for links (as well as Jones and Alexander–Conway polynomials derivable from it), avoiding the usual “smoothing of a crossing” move. As by-products, we have characterizations of these polynomials for knots and for links with any given number of components.

Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk. We derive some formulas for $A_1$ and $A_2$ clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formulas and bubble skein expansion formulas. We calculate the ${𝔰𝔩}_2$ and ${𝔰𝔩}_3$ colored Jones polynomials of $2$-bridge knots and links explicitly using twist formulas.

An alternative framework underlying connection between tensor ${\mathrm{\text{sl}}}_2$-calculus and spin networks is suggested. New sign convention for the inner product in the dual spinor space leads to a simpler and direct set of initial rules for the diagrammatic recoupling methods. Yet, it preserves the standard chromatic graph evaluations. In contrast with the standard formulation, the background space is that of symmetric tensor spaces, which seems to be in accordance with the representation theory of $\mathrm{\text{SL}}(2)$. An example of Apollonian disk packing is shown to be a source of spin networks. The graph labeling is extended to non-integer values, resulting in the complex values of chromatic evaluations.

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.

The colored ${𝔰𝔩}_3$ Jones polynomials $J_{(n_1,n_2)}^{{𝔰𝔩}_3}(L;q)$ are given by a link and an $(n_1,n_2)$-irreducible representation of ${𝔰𝔩}_3$. In general, it is hard to calculate $J_{(n_1,n_2)}^{{𝔰𝔩}_3}(L;q)$ for an oriented link $L$. However, we calculate the one-row-colored ${𝔰𝔩}_3$ Jones polynomials $J_{(n,0)}^{{𝔰𝔩}_3}(P(\alpha ,\beta ,\gamma );q)$ for three-parameter families of oriented pretzel links $P(\alpha ,\beta ,\gamma )$ by using Kuperberg’s linear skein theory by setting $n_2=0$. Furthermore, we show the existence of the tails of $J_{(n,0)}^{{𝔰𝔩}_3}(P(2\alpha +1,2\beta +1,2\gamma );q)$ for the alternating pretzel knots $P(2\alpha +1,2\beta +1,2\gamma )$.

We solve the planar generalizations of the Yang-Baxter equation using Kauffman's bracket methods. These solutions are applied to get representations of certain Coxeter groups that are quotients of generalized braid groups.

Pulling back the weight systems associated with the exceptional Lie algebras and their standard representations by a modification of the universal Vassiliev-Kontsevich invariant yields link invariants; extending them to coloured 3-nets, we derive for each of them a skein relation.