We show how the signed evaluations of link polynomials can be used to calculate unknotting numbers. We use the Jones-Rong value of the Brandt-Lickorish-Millett-Ho polynomial Q to calculate the unknotting numbers of 8_{16}, 9_{49} and 6 further new entries in Kawauchi's tables. Another method is developed by applying and extending the linking form criterion of Lickorish. This leads to several conjectured relations between the Jones value of Q and the linking form.
Given a knot in the three-sphere, is it possible to unknot it by performing a single twist, and if so, what are the possible linking numbers of such a twist? We develop obstructions to unknotting using a twist of a specified linking number. The obstructions we describe are built using classical knot invariants, Casson-Gordon invariants, and Heegaard Floer theory.
Given a virtual link diagram $D$, we define its unknotting index $U(D)$ to be minimum among $(m, n)$ tuples, where $m$ stands for the number of crossings virtualized and $n$ stands for the number of classical crossing changes, to obtain a trivial link diagram. By using span of a diagram and linking number of a diagram we provide a lower bound for unknotting index of a virtual link. Then using warping degree of a diagram, we obtain an upper bound. Both these bounds are applied to find unknotting index for virtual links obtained from pretzel links by virtualizing some crossings
Combinatorial two-player games have recently been applied to knot theory. Examples of this include the Knotting-Unknotting Game and the Region Unknotting Game, both of which are played on knot shadows. These are turn-based games played by two players, where each player has a separate goal to achieve in order to win the game. In this paper, we introduce the Linking-Unlinking Game which is played on two-component link shadows. We then present winning strategies for the Linking-Unlinking Game played on all shadows of two-component rational tangle closures and played on a large family of general t…
We compute the unknotting number of two infinite families of pretzel knots, $P(3,1,\dots,1,b)$ (with $b$ positive and odd and an odd number of 1s) and $P(3,3,3c)$ (with $c$ positive and odd). To do this, we extend a technique of Owens using Donaldson's diagonalization theorem, and one of Traczyk using the Jones polynomial, building on work of Lickorish and Millett.
The algebraic unknotting number u_a(K) of a knot K was introduced by Hitoshi Murakami. It equals the minimal number of crossing changes needed to turn K into an Alexander polynomial one knot. In a previous paper the authors used the Blanchfield form of a knot K to define an invariant n(K) and proved that n(K) is a lower bound on u_a(K). They also showed that n(K) subsumes all previous classical lower bounds on the (algebraic) unknotting number. In this paper we prove that n(K)=u_a(K).
We call a knot $K$ a complete Alexander neighbor if every possible Alexander polynomial is realized by a knot one crossing change away from $K$. It is unknown whether there exists a complete Alexander neighbor with nontrivial Alexander polynomial. We eliminate infinite families of knots with nontrivial Alexander polynomial from having this property and discuss possible strategies for unresolved cases. Additionally, we use a condition on determinants of knots one crossing change away from unknotting number one knots to improve KnotInfo's unknotting number data on 11 and 12 crossing knots. Licko…
The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. The algebraic unknotting number is the minimum number of crossing changes needed to transform a knot into an Alexander polynomial-one knot. We work with a generalization of unknotting number due to Mathieu-Domergue, which we call the untwisting number. The untwisting number is the minimum number (over all diagrams of a knot) of right- or left-handed twists on even numbers of strands of a knot, with half of the strands oriented in each direction, necessary to transform that knot…
Oikawa defined an unknotting operation on virtual knots, called a CF-move, and gave a classification of 2-component virtual links up to CF-moves by the virtual linking number and his $n$-invariant. In particular, it was proved that a CF-move characterizes the information contained in the virtual linking number for 2-component odd virtual links. In this paper, we extend this result by classifying odd virtual links and almost odd virtual links with arbitrary number of components up to CF-moves, using the virtual linking number. Moreover, we extend Oikawa's $n$-invariant and introduce two invaria…
Using the recent Gauss diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of the polynomials of Alexander/Conway, Jones, HOMFLY, Brandt-Lickorish-Millett-Ho and Kauffman there are only finitely many positive knots with the same polynomial and no positive knot with trivial polynomial. We also discuss an extension of the Bennequin inequality, showing that t…
For an oriented link diagram D, the warping degree d(D) is the smallest number of crossing changes which are needed to obtain a monotone diagram from D. We show that d(D)+d(-D)+sr(D) is less than or equal to the crossing number of D, where -D denotes the inverse of D and sr(D) denotes the number of components which have at least one self-crossing. Moreover, we give a necessary and sufficient condition for the equality. We also consider the minimal d(D)+d(-D)+sr(D) for all diagrams D. For the warping degree and linking warping degree, we show some relations to the linking number, unknotting num…
In this paper, we derive formulae for the determinant of weaving knots $W(3,n)$ and $W(p,2)$. We calculate the dimension of the first homology group with coefficients in $\mathbb{Z}_3$ of the double cyclic cover of the $3$-sphere $S^3$ branched over $W(3,n)$ and $W(p,2)$ respectively. As a consequence, we obtain a lower bound of the unknotting number of $W(3,n)$ for certain values of $n$.
This paper gives an alternate definition of the Affine Index Polynomial (called the Wriggle Polynomial) using virtual linking numbers and explores applications of this polynomial. In particular, it proves the Cosmetic Crossing Change Conjecture for odd virtual knots and pure virtual knots. It also demonstrates that the polynomial can detect mutations by positive rotation and proves it cannot detect mutations by positive reflection. Finally it exhibits a pair of mutant knots that can be distinguished by a Type 2 Vassiliev Invariant coming from the polynomial.
J.P. Levine showed that the Conway polynomial of a link is a product of two factors: one is the Conway polynomial of a knot which is obtained from the link by banding together the components; and the other is determined by the $\barμ$-invariants of a string link with the link as its closure. We give another description of the latter factor: the determinant of a matrix whose entries are linking pairings in the infinite cyclic covering space of the knot complement, which take values in the quotient field of ${\Bbb Z}[t,t^{-1}]$. In addition, we give a relation between the Taylor expansion of a l…
Topological polymers have various topological types, and they are expressed by graphs. However, the Jones polynomial, we have a difficulty to compute it; computational time is growing exponentially with respect to the crossing number. The simplest Vassiliev invariant is the linking number and thus we will seek a next simple one is as the Milnor's triple linking number. In this paper, we introduce simple Gauss diagram formulas of Vassiliev invariants of Milnor type. These are non-torsion valued, whereas the base-point-free Milnor's triple linking number is usually torsion-valued.
In this paper we introduce a new invariant of virtual knots and links that is non-trivial for infinitely many virtuals, but is trivial on classical knots and links. The invariant is initially be expressed in terms of a relative of the bracket polynomial and then extracted from this polynomial in terms of its exponents, particularly for the case of knots. This analog of the bracket polynomial will be denoted {K} (with curly brackets) and called the binary bracket polynomial. The key to the combinatorics of the invariant is an interpretation of the state sum in terms of 2-colorings of the associ…
The paper concerns the tree invariants of string links, introduced by Kravchenko and Polyak and closely related to the classical Milnor linking numbers also known as $\barμ$--invariants. We prove that, analogously as for $\barμ$--invariants, certain residue classes of tree invariants yield link homotopy invariants of closed links. The proof is arrow diagramatic and provides a more geometric insight into the indeterminacy through certain tree stacking operations. Further, we show that the indeterminacy of tree invariants is consistent with the original Milnor's indeterminacy. For practical purp…
An explicit polynomial in the linking numbers $l_{ij}$ and Milnor's triple linking numbers $μ(rst)$ on six component links is shown to be a well-defined finite type link-homotopy invariant. This solves a problem raised by B. Mellor and D. Thurston. An extension of our construction also produces a finite type link invariant which detects the invertibility for some links.
Define the complete n-complex on N vertices to be the n-skeleton of an (N-1)-simplex. We show that embeddings of sufficiently large complete n-complexes in R^{2n+1} necessarily exhibit complicated linking behaviour, thereby extending known results on embeddings of large complete graphs in R^3 (the case n=1) to higher dimensions. In particular, we prove the existence of links of the following types: r-component links, with the linking pattern of a chain, necklace or keyring; 2-component links with linking number at least lambda in absolute value; and 2-component links with linking number a non-…
We use the idea of expressing a nonoriented link as a sum of all oriented links corresponding to the link to present a short proof of the Lickorish-Millett-Turaev formula for the Kauffman polynomial at $z= -a- a^{-1}$. Our approach explains the observation made by Lickorish and Millett that the formula is the generating function for the linking number of a sublink of the given link with its complementary sublink.
For every spatial embedding of each graph in the Petersen family, it is known that the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2. In this paper, we give an integral lift of this formula in terms of the square of the linking number and the second coefficient of the Conway polynomial.
A well-known identity (Alex+) - (Alex-)=(t^{1/2}-t^{-1/2}) (Alex0) holds for three 1-links L+, L-, and L0 which satisfy a famous local-move-relation. We prove a new local-move-identity for the Z[t,t^{-1}]-Alexander polynomials of 2-links, which is a 2-dimensional analogue of the 1-dimensional one. In the 1-dimensional link case there is a well-known relation between the Alexander-Conway polynomial and the linking number. As its 2-dimensional analogue, we find a relation between the Z[t,t^{-1}]-Alexander polynomials of 2-links and the alinking number of 2-links. We show high dimensional analogu…
In the 1980's Daryl Cooper introduced the notion of a C-complex (or clasp-complex) bounded by a link and explained how to compute signatures and polynomial invariants using a C-complex. Since then this was extended by works of Cimasoni, Florens, Mellor, Melvin, Conway, Toffoli, Friedl, and others to compute other link invariants. Informally a C-complex is a union of surfaces which are allowed to intersect each other in clasps. The purpose of the current paper is to study the minimal number of clasps amongst all C-complexes bounded by a fixed link $L$. This measure of complexity is related to t…
We study relations between the Alexander-Conway polynomial $\nabla_L$ and Milnor higher linking numbers of links from the point of view of finite-type (Vassiliev) invariants. We give a formula for the first non-vanishing coefficient of $\nabla_L$ of an m-component link L all of whose Milnor numbers $μ_{i_1... i_p}$ vanish for $p\le n$. We express this coefficient as a polynomial in Milnor numbers of L. Depending on whether the parity of n is odd or even, the terms in this polynomial correspond either to spanning trees in certain graphs or to decompositions of certain 3-graphs into pairs of spa…
We explain a correspondence between some invariants in the dynamics of color exchange in a 2d coloring problem, which are polynomials of winding numbers, and linking numbers in 3d. One invariant is visualized as linking of lines on a special surface with Arf-Kervaire invariant one, and is interpreted as resulting from an obstruction to transform the surface into its chiral image with special continuous deformations. We also consider additional constraints on the dynamics and see how the surface is modified.
