Narrow Results

Computing unlinking number is usually very difficult and complex problem, therefore we define BJ-unlinking number and recall Bernhard–Jablan conjecture stating that the classical unknotting/unlinking number is equal to the BJ-unlinking number. We compute BJ-unlinking number for various families of knots and links for which the unlinking number is unknown. Furthermore, we define BJ-unlinking gap and construct examples of links with arbitrarily large BJ-unlinking gap. Experimental results for BJ-unlinking gap of rational links up to 16 crossings, and all alternating links up to 12 crossings are obtained using programs LinKnot and K2K. Moreover, we propose families of rational links with arbitrarily large BJ-unlinking gap and polyhedral links with constant non-trivial BJ-unlinking gap. Computational results suggest existence of families of non-alternating links with arbitrarily large BJ-unlinking gap.

Let K be a knot or link and let p = det(K). Using integral colorings of rational tangles, Kauffman and Lambropoulou showed that every rational K has a mod p coloring with distinct colors. If p is prime this holds for all mod p colorings. Harary and Kauffman conjectured that this should hold for prime, alternating knot diagrams without nugatory crossings for p prime. Asaeda, Przyticki and Sikora proved the conjecture for Montesinos knots. In this paper, we use an elementary combinatorial argument to prove the conjecture for prime alternating algebraic knots with prime determinant. We also give a procedure for coloring any prime alternating knot or link diagram and demonstrate the conjecture for non-algebraic examples.

We give explicit palindrome presentations of the groups of rational knots, i.e. presentations with relators which read the same forwards and backwards. This answers a question posed by Hilden, Tejada and Toro in 2002. Using such presentations we obtain simple alternative proofs of some classical results concerning the Alexander polynomial of all rational knots and the character variety of certain rational knots. Finally, we derive a new recursive description of the SL(2, ℂ) character variety of twist knots.

The volume density of a hyperbolic link is defined as the ratio of hyperbolic volume to crossing number. We study its properties and a closely-related invariant called the determinant density. It is known that the sets of volume densities and determinant densities of links are dense in the interval $[0,v_{\mathrm{\text{oct}}}]$. We construct sequences of alternating knots whose volume and determinant densities both converge to any $x\in [0,v_{\mathrm{\text{oct}}}]$. We also investigate the distributions of volume and determinant densities for hyperbolic rational links, and establish upper bounds and density results for these invariants.

In the paper, we introduce the concept of loop numbers of knot diagrams and we show that loop numbers are knot invariants for a large family of alternating knots. We also discuss relationships of loop numbers of knots with some classic knot invariants.

We propose a method to determine the character variety of a class $J(m,n)$ of rational knots, which includes the twist knots. The defining polynomials depend only on the variables $m$ and $n$. This answers for these classes of knots a question posed in [H. M. Hilden, M. T. Lozano and J. M. Montesinos–Amilibia, On the character variety of group representations of a 2-bridge link $p/3$ into $PSL(2,ℂ)$, Bol. Soc. Mat. Mexicana 37(2) (1992) 241–262], and allows us to give an easy geometrical description of the considered character variety. Our results are obtained by using special presentations of the knot groups whose relators are palindromes.

Experimental data from Dunfield et al. using random grid diagrams suggest that the genus of a knot grows linearly with respect to the crossing number. Using billiard table diagrams of Chebyshev knots developed by Koseleff and Pecker and a random model of 2-bridge knots via these diagrams developed by the author with Krishnan and then with Even-Zohar and Krishnan, we introduce a further-truncated model of all 2-bridge knots of a given crossing number, almost all counted twice. We present a convenient way to count Seifert circles in this model and use this to compute a lower bound for the average Seifert genus of a 2-bridge knot of a given crossing number.

We consider cyclic branched coverings of a 3-parameter family of rational knots in $S^3$ and study the left orderability of their fundamental groups. We first compute the nonabelian ${\mathrm{\text{SL}}}_2(ℂ)$-character varieties of the rational knots $C(2p,2m,2n+1)$ in the Conway notation, where $p,m,n$ are integers. We then study real points on these varieties and finally use them to determine the left orderability of the fundamental groups of cyclic branched coverings of $C(2p,2m,2n+1)$.

A nontrivial knot is called minimal if its knot group does not surject onto the knot groups of other nontrivial knots. In this paper, we determine the minimality of the rational knots $C(2n+1,2m,2)$ in the Conway notation, where $m\ne 0$ and $n\ne 0,-1$ are integers. When $\left|m\right|\ge 2$, we show that the nonabelian ${\mathrm{\text{SL}}}_2(ℂ)$-character variety of $C(2n+1,2m,2)$ is irreducible and therefore $C(2n+1,2m,2)$ is a minimal knot. The proof of this result is an interesting application of Eisenstein’s irreducibility criterion for polynomials over integral domains.