9
votes
Knots having the same Alexander module which are not S-equivalent
If $K$ and $L$ are oriented knots, then $K\# L$ and $K\# {-L}$ have the same Alexander module (namely the direct sum of the Alexander modules of $K$ and $L$). But $\operatorname{sign}(K\# L)=\...
- 2,797
9
votes
Accepted
Knots having the same Alexander module which are not S-equivalent
Knots with Nakanishi index 1 have cyclic Alexander module, but their algebraic unknotting numbers might differ.
Work of Borodzik-Friedl ensures that the algebraic unknotting number equals the minimal ...
- 1,033
8
votes
Accepted
Are there knots that can be distinguished by the Alexander-Conway polynomial, but not the Alexander polynomial?
Let $\overline{L}$ denote the mirror image of a link $L$. The (reduced) Alexander-Conway polynomial $D_L(t)$ of an $n$-component link $L$ satisfies $D_{\overline{L}}(t)=(-1)^{n+1}D_L(t)$. More ...
- 1,033
7
votes
HOMFLYPT vs. Jones vs. Alexander polynomial?
I will give an example that is likely not the simplest one. The example comes from the paper Behavior of knot invariants under genus 2 mutation by Dunfield, Garoufalidis, Shumakovitch, and ...
- 306
6
votes
Accepted
Is there any genus one knot other than $5_2$ with Alexander polynomial $2t^2-3t+2$?
Ian Agol, in the comments says:
Yes, there should be plenty. Think of the Seifert surface for the 5_2
knot as a disk with two strips (1-handles) attached. By tying knots
into the strips (with zero ...
6
votes
Accepted
Is there a geometric interpretation of the second derivative of the Alexander polynomial at $1$?
Given a knot in $S^3$, think of it as an embedding
$$f : S^1 \to S^3.$$
The configuration space of $5$ distinct points in $S^3$ is denoted $C_5(S^3)$, this is a $15$-dimensional manifold and it ...
- 44.3k
6
votes
Infinite family of different prime knots with trivial Alexander polynomial
The P(p,q,r) pretzel knots with p, q, r odd integers have a genus 1 Seifert surface that just consists of two disks connected with three twisted ribbons, with p, q, r twists, respectively. So they ...
- 890
5
votes
Infinite family of different prime knots with trivial Alexander polynomial
Whitehead doubles are one such family of genus one knots, see this MO answer
- 1,617
5
votes
Is there any genus one knot other than $5_2$ with Alexander polynomial $2t^2-3t+2$?
For the prime knot example, the first in the tables is $15n43522$ with diagram:
- 121
4
votes
Is there any genus one knot other than $5_2$ with Alexander polynomial $2t^2-3t+2$?
I finally joined MO this evening to ask an entirely unrelated question, but I thought I might respond to this one as well. The Alexander polynomial does not detect $5_2$, as noted in the previous ...
- 101
4
votes
Multivariate Alexander polynomial vs single variable (Conway) Alexander polynomial
The relation is $\Delta_L(t)\stackrel{.}{=}\Delta_L(t,...,t)(t-1)$ (no need to take the normalised version). One reference is Proposition 7.3.10 of Kawauchi's "survey of knot theory". Arguably, a more ...
- 1,033
3
votes
Multivariable vs single variable Alexander polynomial for links?
Both polynomials arise by considering the first homology of covering spaces of the link exterior $X_L$. If the link $L$ has $n$ components, then $\Delta_L(t_1,...,t_n)$ is extracted from the free ...
- 1,033
3
votes
Is there any genus one knot other than $5_2$ with Alexander polynomial $2t^2-3t+2$?
Maybe a different approach but having the same flavor: Take a (untwisted) Whitehead double of any nontrivial knot $K$. Denote such a double knot $WD(K)$. Construct a satellite knot using $5_2$ knot as ...
- 2,083
2
votes
Accepted
Alexander polynomial of the pretzel knot $P(2m+1,2n,2k+1)$
It seems like the lower half of the Alexander polynomial of the pretzel knot $ P(2m+1,2n,2k+1)$ , up to multiplication by $\pm t^{\alpha}$ , is given by $$ \Delta_{h}(t)= -nt + \sum_{i=2}^{2m+1}(-1)...
- 65
2
votes
Non-commutative knot invariants
Perhaps one can study non-commutative analogs of the Alexander
polynomial which will have significant knot theoretic applications as
well, and perhaps one can compute these in some examples. After all
...
- 1,033
2
votes
Infinite family of different prime knots with trivial Alexander polynomial
Another family of examples is given by the "generalised Kinoshita-Terasaka" knots, here is a picture from Lickorish' "An introduction to Knot Theory".
Here $d$ is assumed to be ...
- 601
2
votes
Multivariable vs single variable Alexander polynomial for links?
I think this is the one you get from HOMFLY polynomial. Recall that for the usual Conway triple $(L_+,L_-,L_0)$ HOMFLY polynomial $P$ has this relation:
$lP(L+)+l^{-1}P(L_-)+mP(L_0)=0$
Setting $l=i$ ...
- 2,205
1
vote
Multivariable vs single variable Alexander polynomial for links?
Yes, there is such a one-variable polynomial. In fact, in my understanding, this came earlier than the multi-variable version.
Gwénaël Masseyeau has some notes on the subject, with references to work ...
- 10.9k
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