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Thomas Fiedler

Publications and source records attributed to Thomas Fiedler.

18 recordsLinked to original sources
arXiv2026

A refined 1-cocycle for regular isotopies and the refined tangle equations

We refine the combinatorial 1-cocycle $\mathbb{L}R_{reg}$ for regular isotopies of long knots to a 1-cocycle with values in the free $\mathbb{Z}[x,x^{-1}]$-module generated by regular isotopy classes of oriented tangles with exactly one signed ordinary double point. We use it to define the refined tangle equations for couples of knot diagrams, where the coefficients are now Laurent polynomials instead of integers. A solution of the tangle equations gives quantitative information about any knot isotopy which relates two given knot diagrams. If the tangle equations have no solution, then the diagrams represent different knots.

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arXiv2025

Quantum equations for knots

This paper contains linear systems of equations which can distinguish knots without knot invariants. Let $M_n$ be the topological moduli space of all n-component string links and such that a fixed projection into the plane is an immersion. If a string link is the product of some string link diagram $T$ and the parallel n-cable of a framed long knot diagram $D$, then there is a canonical arc $push$ in $M_n$, defined by pushing $T$ through the n-cable of $D$. In this paper we apply the combinatorial 1-cocycles from the HOMFLYPT and Kauffman polynomials in $M_n$ with values in the corresponding skein modules to this canonical arc in $M_n$. Some of the 1-cocycles lead to linear systems of equations in the skein modules, for each couple of diagrams $D$ and $D'$. If the system has no solution in the Laurent polynomials then $D$ and $D'$ represent different knots. We give first examples where we distinguish knots without any knot invariants. In particular, we distinguish the knot $9_{42}$ from its mirror image with equations coming from the HOMFLYPT polynomial. Notice that the knot $9_{42}$ and its mirror image share the same HOMFLYPT polynomial. On the other hand, each solution of the system gives rather fine information about any regular isotopy which connects $D$ with $D'$.

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arXiv2025

The tangle-valued 1-cocycle for knots

This paper contains the strongest and at the same time most calculable knot invariant ever. Let $\Theta$ be the topological moduli space of all ordered oriented tangles in 3-space. We construct a non-trivial combinatorial 1-cocycle $\mathbb{L}$ for $\Theta$ that takes its values in $H_0(\Theta;\mathbb{Z})$. The 1-cocycle $\mathbb{L}$ has a very nice property, called the {\em scan-property}: if we slide a tangle $T$ over or under a given crossing $c$ of a fixed tangle $T'$, then the value of $\mathbb{L}$ on this arc $scan(T)$ in $\Theta$ is already an isotopy invariant of $T$. In particular, let $D$ be a framed long knot diagram. We take the product with a fixed long knot diagram $K$ and we consider the 2-cable, with a fixed crossing $c$ in $2K$. $\mathbb{L}(scan(2D))$ gives an element in $H_0(\Theta)$. To this element we associate the {\em set of Alexander vectors}, consisting of the corresponding integer multiples of the one-variable Alexander polynomials of (the standard closures) of all sub-tangles of each of the tangles. We can vary the knots $(K,c)$ and moreover we can iterate our construction by starting now again the $scan$ with the tangles in $\mathbb{L}(scan(2D))$ and so on. The result is the infinite {\em Alexander tree}, which is an isotopy invariant of the knot represented by $D$. {\em As examples we show with just one edge of the Alexander tree that the knot $8_{17}$ and the Conway knot are not invertible!} This makes the Alexander tree a very promising candidate for a complete and "locally" calculable knot invariant, because the tangles in $\mathbb{L}(scan(2D))$ can be drawn with linear complexity and their Alexander polynomials can be calculated with quartic complexity with respect to the number of crossings of $D$.

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arXiv2023

Polynomial invariants which can distinguish the orientations of knots

This paper contains the first knot polynomials which can distinguish the orientations of classical knots and which make no excplicit use of the knot group. But they make extensive use of the meridian and of the longitude in a geometric way. Let $M$ be the topological moduli space of long knots up to regular isotopy, and for any natural number $n > 1$ let $M_n$ be the moduli space of all n-cables $nK$ of framed long knots $K$ which are twisted by a given string link $T$ to close to a knot in the solid torus, with a marked point on the knot at infinity. First we construct integer valued combinatorial 1-cocycles for $M_n$ by using Gauss diagram formulas for finite typ invariants. We observe then that our 1-cocycles allow to fix certain crossings of $nK$ as local parameters of the 1-cocycles. Finally, we transform the local parameter into an unordered set of global parameters by following the crossings in the isotopy. We evaluate now the 1-cocycles on a canonical loop in $M_n$. The outcome are polynomial valued invariants of $K$, where the variables are indexed by finite type invariants and by regular isotopy types of string links $T$.

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arXiv2021

More 1-cocycles for classical knots

Let $M^{reg}$ be the topological moduli space of long knots up to regular isotopy, and for any natural number $n > 1$ let $M^{reg}_n$ be the moduli space of all n-cables of framed long knots which are twisted by a string link to a knot in the solid torus $V^3$ . We upgrade the Vassiliev invariant $v_2$ of a knot to an integer valued combinatorial 1-cocycle for $M^{reg}_n$ by a very simple formula. This 1-cocycle depends on a natural number $a \in \mathbb{Z}\cong H_1(V^3;\mathbb{Z})$ with $0<a<n$ as a parameter and we obtain a polynomial-valued 1-cocycle by taking the Lagrange interpolation polynomial with respect to the parameter. We show that it induces a non-trivial pairing on $H_0(M^{reg}_n) \times H_0(M^{reg})$ already for $n=2$.

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arXiv2019

A refinement of the first Vassiliev invariant can distinguish the orientation of knots

We refine the Polyak-Viro Gauss diagram formula for the Vassiliev invariant of order two in a very simple way for the 2-cable of a framed long knot. Surprisingly, the resulting isotopy invariant of framed knots can detect already the non-invertibility of knots. This makes the natural generalization of our invariant for all n-component string link satellites of a framed long knot to a candidate for distinguishing all classical knots.

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arXiv2019

Singularization of knots and closed braids

We construct the first combinatorial 1-cocycle with values in the $ \mathbb{Z} [x,x^{-1}]$-module of isotopy classes of singular long knots in 3-space with a signed planar double point, and which represents a non trivial cohomology class in the topological moduli space of long knots. It can be interpreted as an invariant with values in a $ \mathbb{Z} [x,x^{-1}]$-module generated by 3-manifolds for each element of infinite order of the mapping class group of the complement of a satellite knot in $S^3$. The 1-cocycle seems to be trivial on all loops for long knots which are not satellites but it is already non trivial on the Fox-Hatcher loop of any composite knot, on the loop which consists of {\em dragging} a trefoil through another trefoil and on the {\em scan-arc} for the 2-cable of the trefoil. The {\em canonical resolution} of the value of the 1-cocycle for dragging a knot through another knot leads to a symmetric bilinear form on the free $ \mathbb{Z} [x,x^{-1}]$-module of all unframed oriented knot types into itself. We conjecture that its radical contains only the trivial knot. Evaluating e.g. the Kauffman-Vogel HOMFLYPT polynomial for singular knots on the value of the 1-cocycle applied to the associated quadratic form, leads to a couple of new 3-variable knot polynomials. We construct also the first non trivial 1-cocycle for those closed positive 4-braids which contain a half-twist. It takes its values in a symmetric power of the $ \mathbb{Z}$-module of isotopy classes of closed positive 4-braids with a double point.

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arXiv2019

Knot polynomials from 1-cocycles

Let $M_n$ be the topological moduli space of all parallel n-cables of long framed oriented knots in 3-space. We construct in a combinatorial way for each natural number $n>1$ a 1-cocycle $R_n$ which represents a non trivial class in $H^1(M_n; \mathbb{Z} [x_1,x_2,...,x_1^{-1},x_2^{-1},...])$, where the number of variables $x_m$ depends on $n$. To each generic point in $M_n$ we associate in a canonical way an arc {\em scan} in $M_n$, such that $R_n(scan)$ is already a polynomial knot invariant. We show that $R_3(scan)$ detects the non-invertibility of the knot $8_{17}$ in a very simple way and without using the knot group. There are two well-known canonical loops in $M_n$ for each parallel n-cable of a long framed knot $K$: Gramain's loop {\em rot} and the Fox-Hatcher loop {\em fh}. The calculation of $R_n$ is of at most quartic complexity for these loops with respect to the number of crossings of $K$ for each fixed $n$. It follows from results of Hatcher that $K$ is not a torus knot if the rational function $R_n(fh(K))/R_n(rot(K))$ is not constant for each $n>1$. $ \oplus_n R_n$ is a natural candidate in order to separate all classes in $H_1(M_1;\mathbb{Q}) \cong H_1(M_n;\mathbb{Q})$, and in particular to distinguish all knot types $π_0(M_1)$.

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arXiv2018

One-cocycle invariants for closed braids

We introduce new polynomial isotopy invariants for closed braids. They are constructed as polynomial valued {\em Gauss diagram 1-cocycles} evaluated on the full rotation of the closed braid $\hat β$ around the core of the corresponding solid torus. They can be calculated with polynomial complexity with respect to the braid length and their derivatives evaluated at $x=1$ are finite type invariants of closed braids. Let the solid torus V be standardly embedded in the 3-sphere and let L be the core of the complementary solid torus $S^3\setminus V$. We give examples which show that a natural refinement of our invariants can detect (even with linear complexity with respect to the braid length if the number of strands is fixed, and with quadratic complexity if it is not fixed) the non-invertibility of the 2-component link $\hat β\cup L\hookrightarrow S^3$, what quantum invariants fail to do.

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arXiv2013

Quantum one-cocycles for knots

We give a method to construct non symmetric solutions of a global tetrahedron equation from solutions of the Yang-Baxter equation. The solution in the HOMFLYPT case gives rise to the first combinatorial quantum 1-cocycle which represents a non trivial cohomology class in the topological moduli space of long knots. We conjecture that the quotient of its values on Hatchers loop and on the rotation around the long axis is related to the simplicial volume of the knot complement in the 3-sphere and we prove this for the figure eight knot. Surprisingly, the formula for the solution in the HOMFLYPT case of the positive global tetrahedron equation gives also a solution in the case of the 2-variable Kauffman invariant. But there is also a second solution giving rise to yet another non trivial quantum 1-cocycle.

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arXiv2010

On homotopies with triple points of classical knots

We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point $p$ of the cylinder is called {\em coherent} if all three branches intersect at $p$ pairwise with the same index. A {\em triple unknotting} of a classical knot $K$ is a homotopy which connects $K$ with the trivial knot and which has as singularities only coherent triple points. We give a new formula for the first Vassiliev invariant $v_2(K)$ by using triple unknottings. As a corollary we obtain a very simple proof of the fact that passing a coherent triple point always changes the knot type. As another corollary we show that there are triple unknottings which are not homotopic as triple unknottings even if we allow more complicated singularities to appear in the homotopy of the homotopy.

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arXiv2009

There are non homotopic framed homotopies of long knots

Let $\mathcal {M}$ be the space of all, including singular, long knots in 3-space and for which a fixed projection into the plane is an immersion. Let $cl(Σ^{(1)}_{iness})$ be the closure of the union of all singular knots in $\mathcal {M}$ with exactly one ordinary double point and such that the two resolutions represent the same (non singular) knot type. We call $Σ^{(1)}_{iness}$ the {\em inessential walls} and we call $\mathcal {M}_{ess} = \mathcal {M} \setminus cl(Σ^{(1)}_{iness})$ the {\em essential diagram space}. We construct a non trivial class in $H^1(\mathcal {M}_{ess}; \mathbb{Z}[A, A^{-1}])$ by an extension of the Kauffman bracket. This implies in particular that there are loops in $\mathcal {M}_{ess}$ which consist of regular isotopies of knots together with crossing changings and which are not contractible in $\mathcal {M}_{ess}$ (leading to the title of the paper). We conjecture that our construction gives rise to a new knot polynomial for knots of unknotting number one.

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arXiv2007

A link polynomial via a vertex-edge-face state model

We construct a 2-variable link polynomial, called $W_L$, for classical links by considering simultaneously the Kauffman state models for the Alexander and for the Jones polynomials. We conjecture that this polynomial is the product of two 1-variable polynomials, one of which is the Alexander polynomial. We refine $W_L$ to an ordered set of 3-variable polynomials for those links in 3-space which contain a Hopf link as a sublink.

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arXiv2007

Knot polynomials via one parameter knot theory

We construct new knot polynomials. Let $V$ be the standard solid torus in 3-space and let $pr$ be its standard projection onto an annulus. Let $M$ be the space of all smooth oriented knots in $V$ such that the restriction of $pr$ is an immersion (e.g. regular diagrams of a classical knot in the complement of its meridian). There is a canonical one dimensional homology class for each connected component of $M$. We construct homomorphisms from the first homology group of $M$ into rings of Laurent polynomials. Each such homomorphism applied to the canonical homology class gives a knot invariant. Let $γ$ be a generic smooth oriented loop in $M$ (i.e. a one parameter family of knot diagrams in the annulus). For finitely many points in $γ$ the corresponding knot diagram has in the projection $pr$ an ordinary triple point or an ordinary auto-tangency. To each such diagram we associate some Laurent polynomial by using extensions of the Kauffman bracket or of the Kauffman state model for the Alexander polynomial. We take then an algebraic sum of these polynomials over all triple points and all autotangencies in $γ$. The resulting polynomial depends only on the homology class of $γ$ if and only if it verifies two sorts of equations: the tetrahedron equations and the cube equations. We have found five different non trivial solutions.

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arXiv2006

Isotopy invariants for closed braids and almost closed braids via loops in stratified spaces

Let $ϕ: S^1\times D^2\to S^1$ be the natural projection. An oriented knot $K\hookrightarrow V = S^1\times D^2$ is called an almost closed braid if the restriction of $ϕ$ to K has exactly two (non-degenerate) critical points (and K is a closed braid if the restriction of $ϕ$ has no critical points at all). We introduce new isotopy invariants for closed braids and almost closed braids in the solid torus V. These invariants refine finite type invariants. They are still calculable with polynomial complexity with respect to the number of crossings of K. Let the solid torus V be standardly embedded in the 3-sphere and let A be the axis of the complementary solid torus $S^3\setminus V$. We give examples which show that our invariants can detect non-invertibility of 2-component links $K\cup A\hookrightarrow S^3$. Notice that all quantum link invariants fail to do so and that it is not known wether there are finite type invariants which can detect non-invertibility of 2-component links.

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arXiv2000

Global knot theory in F^2 x R

We introduce a special class of knots, called global knots, in F^2 x R and we construct new isotopy invariants, called T-invariants, for global knots. Some T-invariants are of finite type but they cannot be extracted from the generalized Kontsevitch integral (which is consequently not the universal invariant of finite type for the restricted class of global knots). We prove that T-invariants separate all global knots of a certain type. As a corollary, we prove the non-invertibility of some links in S^3 without making any use of the link group.

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