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> There now follows a large brain dump concerning knot theory...
An interesting read, are there any real world applications currently? If I
knit myself a jumper and then join the two lose ends together, what sort of
know is that? Haha only joking.
OOC did you ever consider doing a PhD in some subject you are interested in?
I get the impression you would really enjoy it and maybe you can do it in
parallel with your current job.
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>> There now follows a large brain dump concerning knot theory...
>
> An interesting read
Thank you. I read that while I was a bored teenager.
[Insert comment here about what a bored teenager *should* be doing.]
My memory is a little rusty by now... o_O
> are there any real world applications currently?
Apparently DNA tends to get tangled and knotted, and there's protein
folding. These are both loosely related to knot relaxation. But
hard-core knot theory itself? No, not a huge number of directly
practical applications.
The subject *is*, however, ripe with humour.
"What's your favourit branch of mathematics?"
"Knot theory."
"Me neither."
> If I knit myself a jumper and then join the two lose ends together, what
> sort of knot is that? Haha only joking.
Smart-arse. :-P
> OOC did you ever consider doing a PhD in some subject you are interested
> in? I get the impression you would really enjoy it and maybe you can do
> it in parallel with your current job.
1. I barely passed my BSc. Studying something even harder would seem unwise.
2. AFAIK, you need an MSc before you can even attempt a PhD. Since I
nearly failed a BSc and an MSc is significantly harder, it seems
unlikely that I could get this. (To say nothing of the minor detail of
it requiring tens of thousands of pounds in course fees and several
years of my time.)
3. I severely doubt that I could actually perform a PhD at the same time
as doing a full-time job.
4. I already have a BSc, and it hasn't opened any doors for me. I
seriously doubt a PhD would be any significant help in this direction.
I could continue, but I think that'll do for now.
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Invisible wrote:
> At one time, it was briefly theorised that maybe the ~105 elements of
> the periodic table were each a little tangle of energy, and different
> kinds of tangling gave rise to different chemical properties. This
> sparked a great deal of interest in knot theory. Later this idea was
> abandoned, and knot theory became unpopular again. But some people still
> study it.
Correct:
http://en.wikipedia.org/wiki/History_of_knot_theory
> It is useful to think of "the knot", which is an invariant, unchanging
> thing, and "projections" of the knot.
Wikipedia claims these are "knot diagrams".
> Aside from stretching and shrinking the string, a knot projection can be
> "changed" in three fundamental ways. These ways are named after a
> mathematician who's name is beyond my ability to spell or pronounce.
> Remember we're talking about knot projections, that is, 2D drawings of a
> particular configuration of a knot.
>
> - A type I move involves taking a strand, making a small loop with it,
> and poking it over the top of another strand. (Or, alternatively, under
> it.) This makes the projection slightly more complicated (there are now
> more crossings), but does not alter the knot itself. The reverse
> process, i.e., drawing a loop back over (or under) a strand, simplifies
> the projection. (Note that a loop *around* a strand cannot be
> so-simplified. Only a loop completely over or completely under.)
>
> - A type II move involves taking a straight strand and twisting it to
> form a loop. You have now added one new crossing to the projection.
> Alternatively, untwisting the strand to eliminate a loop, thus reducing
> the number of crossings. (You can only do this if nothing passes through
> the loop, of course.)
>
> - A type III move involves moving a strand from one side of an unrelated
> crossing to the other.
>
> These three moves, then, alter the 2D projection of a knot without
> actually changing its fundamental structure --- i.e., which knot it is.
Wrong:
http://en.wikipedia.org/wiki/Reidemeister_move
> The details of specific invariants escape me now. Suffice it to say that
> several mathematicians have come up with polynomials that can be
> constructed from a knot projection. The polynomial itself doesn't
> calculate anything interesting, but its formula is derived from the knot
> projection. The algebraic properties of polynomials are used in such a
> way that the three moves change the subterms, but when you simplify it
> all into standard polynomial form, you get the same answer.
>
> I remember one of these polynomials involves recursively splitting the
> knot projection into simpler ones, eventually building a trivial
> polynomial for each part, and then combining these polynomials back
> together according to how the original parts where connected. Shift some
> algebra, and at the end every projection of a given knot comes up with
> the exact same polynomial.
Wrong:
http://en.wikipedia.org/wiki/Alexander_polynomial
http://en.wikipedia.org/wiki/Jones_polynomial
http://en.wikipedia.org/wiki/Alexander-Conway_polynomial
http://en.wikipedia.org/wiki/HOMFLY_polynomial
> One rather entertaining way goes something like this. (I've probably
> screwed up the algorithm; this is from memory.)
>
> - Pick a starting point on the string, and draw an arrow representing a
> direction. Doesn't matter what you pick, but stick to it.
>
> - Trace your way around the knot. Each time you reach a crossing, number
> it, starting from 1. If the strand you're on goes over the top, use a
> positive number. If it goes under, assign a negative number.
>
> - Write down a list of all the pairs of numbers at each crossing.
>
> - Throw away the lowest number in each pair (ignoring sign).
>
> It is possible to completely reconstruct the know from the list you're
> left with.
Wrong:
http://en.wikipedia.org/wiki/Dowker_notation
(Note particularly that my algorithm is wrong, and that the notation is
ambiguous in a precise way.)
> An alternative way to describe knots is by "braid theory".
>
> A "braid" is a series of vertical strands. Initially, they are all
> parallel. If you say "+3", that means that strand 3 and strand 4 swap
> places, with strand 3 going over the top of strand 4. Alternatively,
> "-3" means the same swap, but strand 4 going over the top.
>
> In this way, you can say "-3, +5, +2". This describes a sequence of
> strand swaps, starting from the top and working downwards. Something
> like this:
>
> 1 2 3 4 5 6
> | | | | | |
> | | \ / | |
> | | / | |
> | | / \ | |
> | | | | | |
> | | | | \ /
> | | | | \
> | | | | / \
> | | | | | |
> | \ / | | |
> | \ | | |
> | / \ | | |
> | | | | | |
> 1 2 3 4 5 6
>
> So that's a braid. Now if you imagine taking this and bending it over so
> that the ends at the top connect with the ends at the bottom, this would
> make a closed loop. In fact, in this case, the result would be *several*
> closed loops. The 1 strand would be an unknot, not connected to anything
> else. Strands 5 and 6 would become a single strand, which can then be
> untwizzled to make an unknot. And strands 2, 3 and 4 would be connected;
> off the top of my head, I'm not sure if this would be a nontrivial knot.
>
> Such a collection of possibly-connected knots is called a "link". In
> general, "closing" a braid (i.e., connecting its ends together) produces
> a link. Sometimes the whole link consists of one knot (i.e., one
> continuous strand), and sometimes several knots that can be seperated.
> And occasionally, several knots connected such that you can't seperate
> them without cutting.
>
> The fun part, of course, is the algebraic structure of a braid.
> Sometimes when you move a twist up or down the sequence, it changes the
> resulting link when the braid is closed. And sometimes it doesn't.
> Teasing how the relationships for this can get quite interesting.
>
> Best of all: any possible knot or link can be represented as a braid.
> (Although working out how usually isn't easy.)
Wrong:
http://en.wikipedia.org/wiki/Braid_theory
> Another method for constructing knots is to use "tangles".
>
> A "tangle" is a section of string or strings that have 4 ends. The ends
> are locked in place and can't move, and you can't loop the strands over
> those ends. If you imagine drawing a square with one end bolted to each
> corner of the square and the strands inside not allowed to leav the
> confines of the square, that's roughly what a tangle is.
>
> Again we have an algebra of tangle construction here. (I may well be
> getting some lefts/rights mixed up here, but the ideas are essentially
> correct.)
>
> We start with the "0 tangle". This is where the two top corners are
> linked, and the two bottom corners are linked, and the strands aren't
> tangled up in any way.
>
> Then we have the "1 tangle". This is where you take the 0 tangle and
> swap round the two right ends, such that the strand from the bottom-left
> corner passes over the one from the top-left corner. The "-1 tangle" is
> identical, but twisted the opposite way. (I.e., the top-left thread is
> on top.)
>
> You can "add" two tangles together by placing them side by side, and
> connecting the two right-hand ends of the left tangle to the two
> left-hand ends of the right tangle.
>
> If, for example, you add a 0 tangle to a 0 tangle, you get a new 0
> tangle. If you add a 1 tangle to a -1 tangle, you also get... a tangle
> where one thread moves over the other, and then back again. Performing a
> type-I move, this becomes the 0 tangle again.
>
> So, 0 + 0 = 0 and (+1) + (-1) = 0. That's cute. But if you add a 1
> tangle to a 2 tangle, you get a tangle where the two threads cross over
> each other twice in the same direction - the "2 tangle". (A "-2 tangle"
> is defined similarly, but with the twist in the opposite direction.)
>
> So, an N tangle is the 0 tangle twisted N times clockwise, and a -N
> tangle is twisted N times anticlockwise. (Assuming you look at it from
> the right direction.)
>
> There is also an "infinity tangle", which is like the 0 tangle, rotated
> This involves placing one above the over, and joining the corners in
> that direction instead.
>
> Here, however, we find that tangle algebra doesn't work *quite* like
> number arithmetic; if you multiply the 0 tangle by the 0 tangle, you get
> something that isn't even a tangle; it's like a 0 tangle with a trivial
> knot floating in the middle of it. And a 1 tangle multiplied by a 1
> tangle gives you something that can't be described any other way. (In
> particular, *not* a 1 tangle!)
>
> By pairing up the ends of a tangle (either vertically or horizontally -
> it makes a difference) you can again construct any possible knot. And
> again there's an interesting algebra of operations which are equivilent
> and those which are not.
Wrong:
http://en.wikipedia.org/wiki/Tangle_theory
> In a similar way, you can "add" regular knots. You take two knots, cut
> them both, and join the cut parts. The thing is, depending on exactly
> where you cut them, and how you join the ends up, you can make several
> different knots in any addition operation. So for general knots,
> "adding" isn't very precisely defined. (At least, if you just specify
> two knots and that they be added, the result is not well-defined. You
> need to specify lots of extra info to make it well-defined. Even the
> knot projection might make a difference.)
Wrong:
http://en.wikipedia.org/wiki/Knot_sum
(There are only two possible results from a knot sum, and it doesn't
matter where the join is, only the relative orientation of the two knots.)
> There are other ways of making knots too. The trefoil is a "toriodal
> knot". That is, you can generate it by marking a point on a circle, and
> rotating that circle while sweeping it around a perpendicular circle.
> (In other words, tracing a path on the surface of a torus.) By varying
> the number of rotations of one circle for each rotation of the other,
> you can build various different knots, of which the trefoil is just one.
> (There is an infinite set of parameters that generate any given toriodal
> knot, however.)
Correct:
http://en.wikipedia.org/wiki/Torus_knot
> Related to this are "cable knots". This involves sweeping a circle not
> along a circle but along another knot. Sometimes the result can be
> unravelled to be isomorphic to the original knot; sometimes it can't.
> (In other words, sometimes it's a genuinely new knot.) You can also
> generate links this way by not rotating the circle as it is swept;
> sometimes these links are seperable, and sometimes they aren't.
Wrong:
http://en.wikipedia.org/wiki/Cable_knot
http://en.wikipedia.org/wiki/Satellite_knot
> What about knots in 4D instead of just 3D? What would that be like?
>
> Well, it turns out to be pretty boring, actually. In 4D, and knot
> composed of a 1D strand can actually be completely untangled. That is,
> every knot is equivilent to the trivial knot in 4D. That's not terribly
> interesting.
>
> What you *can* do, however, is construct knots out of a 2D "ribbon"
> rather than a 1D "string". The result is a family of knots that only
> exist in 4D, but it's really *far* too mind-bending to think about.
> (Projecting back into 3D can look pretty though...)
Correct:
http://en.wikipedia.org/wiki/Knot_theory#Higher_dimensions
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> 1. I barely passed my BSc. Studying something even harder would seem
> unwise.
But let me guess, you did well on some parts but badly on other parts? If
you could study only the bits you really enjoyed and were good at, you would
do better.
> 2. AFAIK, you need an MSc before you can even attempt a PhD.
Oh ok, I wasn't aware of that limitation (Engineering degrees are usually
all 4 years so of course this limitation wasn't mentioned to us).
> 3. I severely doubt that I could actually perform a PhD at the same time
> as doing a full-time job.
Why not? You could probably do a PhD instead of posting here :-)
> 4. I already have a BSc, and it hasn't opened any doors for me. I
> seriously doubt a PhD would be any significant help in this direction.
Oh I'm sure it would, today almost everyone has some sort of degree, having
a PhD will make you stand out from the crowd, much like having a degree a
few decades ago did. Besides, you will probably enjoy it, plus it will get
you into an enjoyable job later.
But yes, it could be a bit pricey to fund yourself.
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>> 1. I barely passed my BSc. Studying something even harder would seem
>> unwise.
>
> But let me guess, you did well on some parts but badly on other parts?
> If you could study only the bits you really enjoyed and were good at,
> you would do better.
I very much doubt you can actually do a PhD in "doing cool stuff with a
computer". It's a tad vague, eh?
>> 2. AFAIK, you need an MSc before you can even attempt a PhD.
>
> Oh ok, I wasn't aware of that limitation (Engineering degrees are
> usually all 4 years so of course this limitation wasn't mentioned to us).
My course was 4 years too, but it was only a BSc. I might be wrong about
the MSc requirement, but that's what I heard.
>> 3. I severely doubt that I could actually perform a PhD at the same
>> time as doing a full-time job.
>
> Why not? You could probably do a PhD instead of posting here :-)
LOL! Yeah, right. :-P
Besides, don't you have to, like, spend years searching through the
library to find every piece of work that has ever been written about
your subject, read and memorise it all, and then present a giant summary
of it? Don't you have to trudge across the plains of Tibet to find an
ancient sage to consult on the works on the Ancient Masters to see if
they have anything relevant to add? I don't think I could do that from
my desk at work.
>> 4. I already have a BSc, and it hasn't opened any doors for me. I
>> seriously doubt a PhD would be any significant help in this direction.
>
> Oh I'm sure it would, today almost everyone has some sort of degree,
> having a PhD will make you stand out from the crowd, much like having a
> degree a few decades ago did.
Meh. I doubt it. It seems everybody just asks "how many years' coding
experience do you have?" and "what are your customer service skills like?"
> Besides, you will probably enjoy it, plus
> it will get you into an enjoyable job later.
The former, perhaps, the latter, unlikely. (So those jobs still exist?)
> But yes, it could be a bit pricey to fund yourself.
Er, yes.
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And lo On Tue, 17 Feb 2009 11:45:40 -0000, Invisible <voi### [at] dev null> did
spake thusly:
> There now follows a large brain dump concerning knot theory...
Yay!
> At one time, it was briefly theorised that maybe the ~105 elements of
> the periodic table were each a little tangle of energy, and different
> kinds of tangling gave rise to different chemical properties.
>
> (With String Theory, the idea seems to be coming back somewhat. But I
> digress...)
Yeah interesting how some ideas can come back from the grave.
> So what actually *is* knot theory? Well, it's the study of mathematical
> "knots". As you might expect, these abstract entities have properties
> similar to but not quite the same as a real knot in a piece of string.
Also perhaps worth pointing out it's only one area of study in topology.
<snip>
> One rather entertaining way goes something like this. (I've probably
> screwed up the algorithm; this is from memory.)
>
> - Pick a starting point on the string, and draw an arrow representing a
> direction. Doesn't matter what you pick, but stick to it.
>
> - Trace your way around the knot. Each time you reach a crossing, number
> it, starting from 1. If the strand you're on goes over the top, use a
> positive number. If it goes under, assign a negative number.
Okay let's try the trefoil knot.
> - Write down a list of all the pairs of numbers at each crossing.
-1, 4
2, -5
-3, 6
> - Throw away the lowest number in each pair (ignoring sign).
Leaves 4,-5,6. Hmmm? Okay let's try that again following Dowker notation.
1,4
2,5
3,6
As this is an alternating knot, no changes in signs required.
Write out the odd numbers with corresponding entry beneath
1, 3, 5
4, 6, 2
Throw away the top numbers to leave 4,6,2.
> An alternative way to describe knots is by "braid theory".
>
> A "braid" is a series of vertical strands. Initially, they are all
> parallel. If you say "+3", that means that strand 3 and strand 4 swap
> places, with strand 3 going over the top of strand 4. Alternatively,
> "-3" means the same swap, but strand 4 going over the top.
>
> In this way, you can say "-3, +5, +2". This describes a sequence of
> strand swaps, starting from the top and working downwards. Something
> like this:
>
> 1 2 3 4 5 6
> | | | | | |
> | | \ / | |
> | | / | |
> | | / \ | |
> | | | | | |
> | | | | \ /
> | | | | \
> | | | | / \
> | | | | | |
> | \ / | | |
> | \ | | |
> | / \ | | |
> | | | | | |
> 1 2 3 4 5 6
>
> So that's a braid. Now if you imagine taking this and bending it over so
> that the ends at the top connect with the ends at the bottom, this would
> make a closed loop. In fact, in this case, the result would be *several*
> closed loops. The 1 strand would be an unknot, not connected to anything
> else. Strands 5 and 6 would become a single strand, which can then be
> untwizzled to make an unknot. And strands 2, 3 and 4 would be connected;
> off the top of my head, I'm not sure if this would be a nontrivial knot.
Trivial, It's a rubber-band twisted twice.
> This has been another broadcast brought to you by an under-employed
> computer science graduate, for the benefit of similarly over-interested
> souls. TTFN!
Interesting, polish it up and stick it on your blog.
--
Phil Cook
--
I once tried to be apathetic, but I just couldn't be bothered
http://flipc.blogspot.com
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Phil Cook v2 wrote:
> Yay!
See, that's what I'm talkin bout! :-D
> Also perhaps worth pointing out it's only one area of study in topology.
Yeah, but topology is friggin *weird*. Knot theory makes actual sense.
> Okay let's try the trefoil knot.
>
>> - Write down a list of all the pairs of numbers at each crossing.
>
> -1, 4
> 2, -5
> -3, 6
>
>> - Throw away the lowest number in each pair (ignoring sign).
>
> Leaves 4,-5,6. Hmmm? Okay let's try that again following Dowker notation.
>
> 1,4
> 2,5
> 3,6
>
> As this is an alternating knot, no changes in signs required.
>
> Write out the odd numbers with corresponding entry beneath
>
> 1, 3, 5
> 4, 6, 2
>
> Throw away the top numbers to leave 4,6,2.
See my other reply. I've got the algorithm wrong.
>> An alternative way to describe knots is by "braid theory".
>>
>> A "braid" is a series of vertical strands. Initially, they are all
>> parallel. If you say "+3", that means that strand 3 and strand 4 swap
>> places, with strand 3 going over the top of strand 4. Alternatively,
>> "-3" means the same swap, but strand 4 going over the top.
>>
>> In this way, you can say "-3, +5, +2". This describes a sequence of
>> strand swaps, starting from the top and working downwards. Something
>> like this:
>>
>> 1 2 3 4 5 6
>> | | | | | |
>> | | \ / | |
>> | | / | |
>> | | / \ | |
>> | | | | | |
>> | | | | \ /
>> | | | | \
>> | | | | / \
>> | | | | | |
>> | \ / | | |
>> | \ | | |
>> | / \ | | |
>> | | | | | |
>> 1 2 3 4 5 6
>>
>> So that's a braid. Now if you imagine taking this and bending it over
>> so that the ends at the top connect with the ends at the bottom, this
>> would make a closed loop. In fact, in this case, the result would be
>> *several* closed loops. The 1 strand would be an unknot, not connected
>> to anything else. Strands 5 and 6 would become a single strand, which
>> can then be untwizzled to make an unknot. And strands 2, 3 and 4 would
>> be connected; off the top of my head, I'm not sure if this would be a
>> nontrivial knot.
>
> Trivial, It's a rubber-band twisted twice.
Probably. Actually, wait - there are only 2 crossings. No nontrivial
knot has that few. Yes, it's definitely trivial. *sigh* Rusty...
>> This has been another broadcast brought to you by an under-employed
>> computer science graduate, for the benefit of similarly
>> over-interested souls. TTFN!
>
> Interesting, polish it up and stick it on your blog.
Now, see, when I spend ages writing something like this, I kinda want
people to go "hey, that's interesting. I had no idea this crap even
existed!" But typically they go "OK, that's nice dear".
I just wish I could find a place where the stuff I know would actually
impress people... *sigh*
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> I very much doubt you can actually do a PhD in "doing cool stuff with a
> computer".
Of course you can, just substitute "cool stuff" for a subject that you
actually find cool.
> Besides, don't you have to, like, spend years searching through the
> library to find every piece of work that has ever been written about your
> subject, read and memorise it all, and then present a giant summary of it?
> Don't you have to trudge across the plains of Tibet to find an ancient
> sage to consult on the works on the Ancient Masters to see if they have
> anything relevant to add? I don't think I could do that from my desk at
> work.
Depends on the subject of course, but nowadays I think most journals and
other academic resources are available online. For a computing related PhD
I would imagine most of your time will be spent at the computer.
> Meh. I doubt it. It seems everybody just asks "how many years' coding
> experience do you have?" and "what are your customer service skills like?"
If you have a PhD you are not going to be applying for those sorts of jobs,
and more importantly companies are not going to expect you to be a code
monkey 24/7 when you are much more capable than that.
Here's an interesting CV:
http://www.geisswerks.com/ryan/resume_ryan_geiss.doc
See, just write something cool *and actually finish it* and then everyone
wants to employ you!
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>> I very much doubt you can actually do a PhD in "doing cool stuff with
>> a computer".
>
> Of course you can, just substitute "cool stuff" for a subject that you
> actually find cool.
Such as...?
There's lots of stuff that interests me, but none of it is exactly "new"
or "revolutionary".
> Depends on the subject of course, but nowadays I think most journals and
> other academic resources are available online. For a computing related
> PhD I would imagine most of your time will be spent at the computer.
Well, I guess it depends on what precisely you wanted to tackle. Either
way, I suck at research, so...
>> Meh. I doubt it. It seems everybody just asks "how many years' coding
>> experience do you have?" and "what are your customer service skills
>> like?"
>
> If you have a PhD you are not going to be applying for those sorts of
> jobs, and more importantly companies are not going to expect you to be a
> code monkey 24/7 when you are much more capable than that.
Yes, but does anybody actually employ PhDs? Most of the ones I know of
still hang around universities...
> Here's an interesting CV:
>
> http://www.geisswerks.com/ryan/resume_ryan_geiss.doc
>
> See, just write something cool *and actually finish it* and then
> everyone wants to employ you!
"Seeking: Collaboration with a group of smart, creative people, in order
to produce work of practical or aesthetic value to mankind."
...OK, that's pretty special, right there. o_O
But then, we *are* talking about the legendary Ryan Geiss... (I bet he's
Polish or something! All the best programmers are, for some reason.)
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>> Of course you can, just substitute "cool stuff" for a subject that you
>> actually find cool.
>
> Such as...?
>
> There's lots of stuff that interests me, but none of it is exactly "new"
> or "revolutionary".
People are doing PhDs in lots of things that you seem to be interested in,
quick Google results:
http://www.liv.ac.uk/~su14/knotgroup.html
http://www.ph.tn.tudelft.nl/PRInfo/reports/msg00512.html
http://research.microsoft.com/en-us/groups/vision/default.aspx
Just try googling "PhD "+thing you are interested in. You will be surprised
what is on offer. I was surprised to hear from my 4th year project
supervisor that if I wanted I could carry on my project "how brake pedal
feedback in vehicles affects driver response" as a PhD, I would have thought
such a specialist topic to be of no interest to anyone, but now I see how
companies would be very interested.
> Well, I guess it depends on what precisely you wanted to tackle. Either
> way, I suck at research, so...
You seem to have demonstrated otherwise here, frequently you seem to have
taken ideas and material and then expanded upon it yourself. It doesn't
matter if you were unaware that someone else had already done the same, if
you were doing a real PhD you would search a bit more thoroughly before
starting work.
> Yes, but does anybody actually employ PhDs? Most of the ones I know of
> still hang around universities...
Over half the people working at my employer in Oxford have PhDs, it's mainly
a research lab (my department is the exception, we deal with developing
technology for specific customers). Surely a lot of people at your place
have PhDs too? I think a lot of people who hand around Universities do so
because they want to (they just enjoy academia) rather than because they
can't find a job outside.
People with the PhDs are the ones inventing new things at the concept stage,
then the Engineers like me get to work out how to actually make it into a
product :-)
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