(CNN) As scary as the shooting down of a Russian warplane was -- especially by Turkey, a NATO member -- signs emerged Friday that some leaders involved are working to ease tensions between the two countries.

Turkish Prime Minister Ahmet Davutoglu, often characterized as the friendly face of his government, published a conciliatory essay in the Times of London.

Davutoglu said the shooting down by Turkey this week "was not -- and is not -- an act against a specific country."

Turkey requests meeting with Russia

"While the measures to defend our territory will remain in place, Turkey will work with Russia and our allies to calm tensions," Davutoglu wrote.

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If Turkey and Russia are at odds, the essay said, the winner will be the terrorist group ISIS, also known as Daesh.

"This is the time to stand firm against Daesh," Davotoglu said. "Collective action that harnesses the varying strengths of the US, the EU, Russia, Turkey and others can, and will, turn the tide."

And beyond that, the Kremlin said that Turkish President Recep Tayyip Erdogan had requested a meeting with Russian counterpart Vladimir Putin on Monday on the sidelines of the climate change conference in Paris.

Kremlin spokesman Dmitry Peskov said Putin "was informed of this request" but gave no word on whether it would be accepted.

Peskov confirmed that Erdogan tried to reach Putin the day that the Russian warplane was shot down "probably 7-8 hours after the incident, not earlier."

Davutoglu's tone and Erdogan's request for a personal meeting stand in stark contrast to previous bellicosity displayed by Putin and Erdogan.

The two had engaged in days of chest-thumping rhetoric, with each demanding the other apologize. And Russia has started to implement de facto economic sanctions to punish Turkey.

Turkey will not apologize for downing the Russian fighter jet that Turkey says violated its airspace near the Syrian border, Erdogan said Thursday in an exclusive CNN interview.

"I think if there is a party that needs to apologize, it is not us," Erdogan said. "Those who violated our airspace are the ones who need to apologize. Our pilots and our armed forces, they simply fulfilled their duties, which consisted of responding to ... violations of the rules of engagement. I think this is the essence."

He went a step further in a meeting with community leaders in Ankara.

"If the same violation occurs today, Turkey has to react the same way," he said.

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Turkey has said it shot down the Russian warplane Tuesday only after it ignored several warnings and entered Turkish airspace.

Erdogan said Friday that Turkey didn't initially know the nationality of the warplane shot down.

"Turkey did not shoot down Russian plane on purpose. This was only an automatic reaction to a border breach. This is an exercise of the rules of engagement. The nationality of these planes who are flying towards our border were unknown despite the warnings," Erdogan said. "It is impossible to know this at that time."

Russia calls it an 'ambush'

Russia has contested the claim, and rescued co-pilot Capt. Konstantin Murakhtin told state media that "there were no warnings -- not via the radio, not visually." Foreign Minister Sergey Lavrov has said the downing of the Russian jet appeared to be "a planned provocation."

Russia's Defense Ministry has published on its website further details of what it says happened prior Tuesday's incident.

The ministry described the attack as an "ambush," presenting monitoring data from radar stations it said "shows the operation was planned beforehand."

The ministry asserted that the Russian Su24 warplane would have been in sight of Turkish radar for a total of 34 minutes before it was shot down. But, the ministry said, the time needed for a Turkish F-16 to fly from its base at Diyarbakir to the point at which it fired the missile would be 46 minutes.

The ministry said other data "proves" the F-16 was in Syrian air space and shows the Russian bomber "did not cross the Turkish border."

The online statement drew attention to the video footage that emerged of the incident and asserted that whoever filmed it must have known in advance the right time and place to record the incident.

Tensions in the Middle East have escalated, with Erdogan accusing Russia of deceit and Moscow announcing that it would deploy anti-aircraft missiles to Syria. A post on the Russian Defense Ministry's Facebook page showed an S-400 missile system being unloaded from a Russian cargo plane.

The countries' militaries suspended their channels of cooperation, the Russian Defense Ministry said.

And Lavrov said that Russia has decided to suspend its visa-free system with Turkey on January 1.

Lavrov made the announcement after complaining to reporters that Turkey was not cooperating in efforts to keep fighters and would-be fighters, including Russian citizens, from traveling through Turkey -- presumably to and from Syria. He also complained that Turkey was deporting Russians to countries that are not friendly to Russia without giving Russia proper notice.

He said that 17 times this year, Russia sent Turkey information about Russian citizens suspected of involvement with terrorism. All 17 appeals received no response, he said.

Out come the economic weapons

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Economic weapons have also been unsheathed. Russian Prime Minister Dmitry Medvedev has directed government ministers to draw up economic measures against Turkey.

Russia's Agriculture Ministry announced that it was strengthening controls over food and agriculture imports from Turkey. A gas pipeline and the Akkuyu nuclear power plant, in which Russia and Turkey have jointly invested, could also be targeted, Economic Development Minister Alexey Ulyukaev said in a tweet.

In addition, Russia's state-run consumer protection body said it had concerns about the quality and safety of children's clothing, furniture and cleaning products originating from Turkey.

Some Russian tour operators have said they will be curtailing travel to Turkey, a top destination for Russian vacationers.