Three new EU polls have been released tonight, but only one of them was entirely conducted after the murder of Jo Cox. So let's start with that one, which was a phone poll carried out by Survation -
Should the United Kingdom remain a member of the European Union or leave the European Union?
Remain 45% (+3)
Leave 42% (-3)
The problem here is that the changes could be consistent with either a genuine swing to Remain after Ms Cox's death, or with illusory margin of error 'noise'. The last poll from the same firm recorded a hefty swing to Leave, and also put Leave in easily their best ever position in a Survation phone poll, so even in the normal course of events it wouldn't be at all surprising to see a modest reversion to the mean in the subsequent poll. At the end of the day, it's still the second-best ever showing for Leave in a Survation phone poll, and it's still a statistical tie - ie. due to the standard margin of error, it's not possible to say which side is in the lead (even assuming the poll's methodology is basically sound).
UPDATE : This is the peril of trying to write a blogpost on the Saturday night before a referendum. I was about to discuss this evening's YouGov poll showing a reduced Leave lead, only to discover as I checked back on Twitter that a second YouGov poll has been published in the last few minutes, and it was conducted on Thursday and Friday - which may or may not mean that it was wholly conducted after Ms Cox's death, depending on what time the poll commenced on Thursday.
Remain 44% (+2)
Leave 43% (-1)
That perhaps tips the balance in favour of there having been genuine movement towards Remain very recently, although Peter Kellner's declaration that there is only a 10% chance that the apparent changes could be caused by normal sampling variation sounds like mumbo-jumbo to me. YouGov last showed a one-point Remain lead less than two weeks ago.
The best news for Leave comes in the new Opinium poll, which was conducted between Tuesday and Friday - and therefore presumably must have included a significant number of interviews both before and after Ms Cox's death (although given the nature of online polling, it's likely that the majority were before).
Remain 44% (n/c)
Leave 44% (+2)
Probably the most that can be said about those numbers is that they contradict the impression given by the first of tonight's two YouGov polls that there was movement back towards Remain even before the tragedy on Thursday.
To summarise what we've learnt, then : it's possible and perhaps likely that Thursday's events may have caused a swing back to Remain, but there isn't enough information to say that for certain yet, and if it has happened, it doesn't look like the swing has been on a big enough scale to put the referendum out of Leave's reach. The most recent phone and online polls both point towards a statistical tie - although I suspect we'll now see substantial backing for Remain on the betting markets due to the assumption (which could easily be proved wrong) that the closing days of a referendum campaign are bound to see movement towards the status quo.
Bear in mind there is a potential danger for Remain that the support they may have won back in the last 48 hours could prove to be soft, and that the voters who have switched sides may change their minds again once the news agenda returns to the main issue.
UPDATE II : It turns out that only two-thirds of the second YouGov poll was conducted after the murder of Ms Cox, so that adds to the uncertainty and leaves Survation as still the only poll entirely conducted after the tragedy. I'm slightly confused by the YouGov datasets - the actual hard numbers appear to show a wafer-thin Remain lead of just 50.2% to 49.8% after weighting and with Don't Knows excluded, and yet we're told that the percentages after rounding are Remain 51%, Leave 49%. I've looked and looked and scratched my head, and I can't see or think of any obvious explanation for that discrepancy, but perhaps I'm missing something.
It also still appears to be the case that YouGov aren't weighting or filtering by turnout, which may mean that Leave's true strength is being slightly underestimated, because Leave voters continue to be more likely to say they are absolutely certain to make it to the polling station. Admittedly, that factor could be offset by the undecideds, who appear set to break more for Remain (although how many of those people will actually vote is open to question).