Questions tagged [alexander-polynomial]

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8
votes
1answer
394 views

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 ...
6
votes
1answer
433 views

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 ...
6
votes
0answers
114 views

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 ...
3
votes
1answer
178 views

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 ...
2
votes
3answers
261 views

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 $\...
2
votes
0answers
74 views

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 ...
1
vote
0answers
46 views

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 ...