Questions tagged [alexander-polynomial]
The alexander-polynomial tag has no usage guidance.
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Why is the Alexander polynomial a quantum invariant?
When we think of quantum invariants, we usually think of the Jones polynomial or of the coloured HOMFLYPT. But (arguably) the simplest example of a quantum invariant of a knot or link is its Alexander ...
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How are the Conway polynomial and the Alexander polynomial different?
Background story:
I have just come out from a talk by Misha Polyak on generalizations of an arrow-diagram formula for coefficients of the Conway polynomial by Chmutov, Khoury, and Rossi. In it, he ...
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Is there any genus one knot other than $5_2$ with Alexander polynomial $2t^2-3t+2$?
It is known that genus one fibred knots are two trefoils and the figure-eight knot. Is there any characterization of the knot $5_2$? Specifically, is there any other genus one knot that shares the ...
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Are there knots that can be distinguished by the Alexander-Conway polynomial, but not the Alexander polynomial?
On page 9 of Kauffman's Formal Knot theory, Kauffman claims
The Alexander-Conway Polynomial is a true refinement of the Alexander Polynomial. Because it is defined absolutely (rather than up to ...
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Is there a geometric interpretation of the second derivative of the Alexander polynomial at $1$?
For an (oriented) knot in $S^3$ the number $\Gamma(K) := \Delta_K’’(1)$ shows up in a number of places in knot theory, for example the Casson-Walker-Lescop invariant. Here $\Delta_K(t)$ is the ...
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Infinite family of different prime knots with trivial Alexander polynomial
I am looking for infinite families of prime knots that have all Alexander polynomial equals to 1. I wrote "families" (and not "family") since perhaps there are different ...
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HOMFLYPT vs. Jones vs. Alexander polynomial?
I'm searching for examples (perhaps the simplest one?) to show that the HOMFLYPT polynomial is stronger than the Jones and Alexander polynomial, respectively.
Any ideas what is the 1st knot in the ...
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Why does the inverse Alexander polynomial appear in the MMR conjecture?
In an attempt to better understand why the inverse Alexander polynomial appears in the MMR conjecture, I was reading the paper [1] of Bar-Natan and Garoufalidis giving their proof of the conjecture ...
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Non-commutative knot invariants
$\newcommand{\ab}{\mathrm{ab}}$Let $L=K_1\cup \dots \cup K_r$ be a link embedded in a 3-sphere. Here, $K_1,\dots, K_r$ are the component knots of $L$. A prototypical invariant associated with $L$ is ...
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Multivariate Alexander polynomial vs single variable (Conway) Alexander polynomial
I consider the multivariate Alexander polynomial $\Delta(t_1,\ldots,t_n)$ for a $n$-component link (defined using e.g. the Fox derivative).
If we wish to construct a 1-variable polynomial $A(t)$, we ...
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Multivariable vs single variable Alexander polynomial for links?
If we take a $n$-component link $L$, we have the multivariable Alexander polynomial $\Delta(L)(t_1,\ldots,t_n)$. Is there a standard single-variable Alexander polynomial? If yes, is it just euqal to $\...
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General formula for a topologically slice odd pretzel knot
An odd 3-strand pretzel knot $K=P(p,q,r)$ has $\Delta_K(t)=1$ if $pq+pr+qr=1$. This fact, along with a theorem of Fintushel and Stern (every odd 3-pretzel knot with trivial Alexander polynomial is not ...
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Is there a measure of the failure of the Alexander polynomial to distinguish knots?
Has there been any research into something like the ratio of distinct Alexander-indistinguishable knots to total knots (up to some measure of complexity)? This was a random question asked of me by a ...
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Alexander polynomial of the pretzel knot $P(2m+1,2n,2k+1)$
Is there a (closed) formula for the Alexander polynomial of the pretzel knot $P(2m+1,2n,2k+1)$, $m,k\ge 0 , n \ge 1$ ?
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Undirected Alexander polynomial (sort of)
Take the skein relation of the Alexander polynomial: $S^1-S^{-1}-zS^0=0$, where z is the parameter of the Alexander polynomial and $S$ the overcross braid element. "Multiply" the equation with ...
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