On the additivity of the Thurston--Bennequin invariant of Legendrian knots
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SAT写作素材积累:托马斯杰弗逊In the thick of party conflict in 1800, Thomas Jefferson wrote in a private letter, "I have sworn upon the altar of God eternal hostility against every form of tyranny over the mind of man."This powerful advocate of liberty was born in 1743 in Albemarle County, Virginia, inheriting from his father, a planter and surveyor, some 5,000 acres of land, and from his mother, a Randolph, high social standing. He studied at the College of William and Mary, then read law.In 1772 he married Martha Wayles Skelton, a widow, and took her to live in his partly constructed mountaintop home, Monticello.Freckled and sandy-haired, rather tall and awkward, Jefferson was eloquent as a correspondent, but he was no public speaker. In the Virginia House of Burgesses and the Continental Congress, he contributed his pen rather than his voice to the patriot cause. As the "silent member" of the Congress, Jefferson, at 33, drafted the Declaration of Independence. In years following he labored to make its words a reality in Virginia. Most notably, he wrote a bill establishing religious freedom, enacted in 1786.Jefferson succeeded Benjamin Franklin as minister to France in 1785. His sympathy for the French Revolution led him into conflict with Alexander Hamilton when Jefferson was Secretary of State in President Washington's Cabinet. He resigned in 1793.Sharp political conflict developed, and two separate parties, the Federalists and the Democratic-Republicans, began to form. Jefferson gradually assumed leadership of the Republicans, who sympathized with the revolutionary cause in France. Attacking Federalist policies, he opposed a strong centralized Government and championed the rights of states.As a reluctant candidate for President in 1796, Jefferson came within three votes of election. Through a flaw in the Constitution, he became Vice President, although an opponent of President Adams. In 1800 the defect caused a more serious problem. Republican electors, attempting to name both a President and a Vice President from their own party, cast a tie vote between Jefferson and Aaron Burr. The House of Representatives settled the tie. Hamilton, disliking both Jefferson and Burr, nevertheless urged Jefferson's election.When Jefferson assumed the Presidency, the crisis in France had passed. He slashed Army and Navy expenditures, cut the budget, eliminated the tax on whiskey so unpopular in the West, yet reduced the national debt by a third. He also sent a naval squadron to fight the Barbary pirates, who were harassing American commerce in the Mediterranean. Further, although the Constitution made no provision for the acquisition of new land, Jefferson suppressed his qualms over constitutionality when he had the opportunity to acquire the Louisiana Territory from Napoleon in 1803.During Jefferson's second term, he was increasingly preoccupied with keeping the Nation from involvement in the Napoleonic wars, though both England and France interfered with the neutral rights of American merchantmen. Jefferson's attempted solution, an embargo upon American shipping, worked badly and was unpopular.Jefferson retired to Monticello to ponder such projects as his grand designs for the University of Virginia. A French nobleman observed that he had placed his house and his mind "on an elevated situation, from which he might contemplate the universe."He died on July 4, 1826.。
托马斯杰夫逊的思想英语作文Thomas Jefferson was an influential figure in American history and his thoughts and ideas have had a lasting impact on the United States. As the principal author of the Declaration of Independence, Jefferson was a proponent of individual rights, freedom, and democracy. His thoughts on government, education, and the role of the United States in the world continue to be studied and debated.One of Jefferson's most famous ideas was his belief in the separation of church and state. He argued that the government should not interfere in religious matters, and that individuals should be free to practice their religion without fear of persecution. This idea was enshrined in the First Amendment to the United States Constitution, which guarantees freedom of religion.Jefferson also had strong views on the role of government in society. He believed in a limited government that would protect the rights of individuals without becoming too powerful or intrusive. He was a strong advocate for states' rights and believed that the federal government should have limited powers. This idea has had a lasting impact on American political thoughtand has influenced debates over the size and scope of government.In addition to his political ideas, Jefferson was also a strong advocate for education. He believed that an educated citizenry was essential for the success of a democracy, and he was a strong supporter of public education. Jefferson's ideas on education have had a lasting impact on the United States, and his vision of a well-educated population continues to be an important goal for American society.Finally, Jefferson had strong views on the role of the United States in the world. He believed in a policy of non-intervention and non-interference in the affairs of other nations, and he was skeptical of foreign entanglements. This idea has had a lasting impact on American foreign policy, and debates over the United States' role in the world continue to be influenced by Jefferson's ideas.Overall, Thomas Jefferson's thoughts and ideas have had a lasting impact on the United States. His beliefs in individual rights, limited government, education, and non-intervention continue to shape American society and politics. His ideas continue to be studied and debated, and his legacy as a founding father and influential thinker is secure.托马斯·杰斐逊是美国历史上一个有影响力的人物,他的思想和观念对美国产生了深远的影响。
Door closer, are you?1 The next time you're deciding between rival options, one which is primary and the other which is secondary, ask yourself this question: What would Xiang Yu do?2 Xiang Yu was a Chinese imperial general in the third century BC who took his troops across the Zhang River on a raid into enemyterritory. To his troops' astonishment, he ordered their cooking pots crushed and their sailing ships burned.3 He explained that he was imposing on them a necessity for attaining victory over their opponents. What he said was surelymotivating, but it wasn't really appreciated by many of his loyal soldiers as they watched their vessels go up in flames. Butthe genius of General Xiang Yu's conviction would be validated both on the battlefield and in modern social science research. General Xiang Yu was a rare exception to the norm, a veteran leader who was highly respected for his many conquests and who achieved the summit of success.4 He is featured in Dan Ariely's enlightening new publication, Predictably Irrational, a fascinating investigation of seemingly irrational human behavior, such as the tendency for keeping multiple options open. Most people can't marshal the will for painful choices, not even students at the Massachusetts Institute of Technology (MIT), where Dr. Ariely teaches behavioral economics. In an experiment that investigated decision-making, hundreds of students couldn't bear to let their options vanish, even though it was clear they wouldprofit from doing so.5 The experiment revolved around a game that eliminated the excuses we usually have for refusing to let go. In the real world, we can always say, "It's goodto preserve our options." Want a good example? A teenager is exhausted from soccer, ballet, piano, and Chinese lessons, but her parents won't stop any one of them because they might come in handy some day!6 In the experiment sessions, students played a computer game that provided cash behind three doors appearing on the screen. The rule was the more money you earned, the better player you were, given a total of 100 clicks. Every time the students opened a door by clicking on it, they would use up one click but wouldn't get any money. However, each subsequent click on that door would earn afluctuating sum of money, with one door always revealing more money than the others. The important part of the rule was each door switch, though having no cash value, would also use up one of the 100 clicks. Therefore, the winning strategy was to quickly check all the doors and keep clicking on the one with the seemingly highest rewards.7 While playing the game, students noticed a modified visual element: Any door left un-clicked for a short while would shrink in size and vanish. Since they already understood the game, they should have ignored the vanishing doors. Nevertheless, they hurried to click on the lesser doors before they vanished, trying to keep them open. As a result, they wasted so many clicks rushing back to the vanishing doors that they lost money in the end. Why were the students so attached to the lesserdoors? They would probably protestthat they were clinging to the doors to keep future options open, but, according to Dr. Ariely, that isn't the true factor.8 Instead of the excuse to maintain future options open, underneath it all the students' desire was to avoid the immediate, thoughtemporary, pain of watching options close. "Closing a door on an option is experienced as a loss, and people are willing to pay a big price to avoid the emotion of loss," Dr. Ariely says. In the experiment, the price was easily measured in lost cash. In life, the corresponding costs are often less obvious such as wasted time or missed opportunities.9 "Sometimes these doors are closing too slowly for us to see them vanishing," Dr. Ariely writes. "We may work more hours at our jobs without realizing that the childhood of our sons and daughters is slipping away."10 So, what can be done to restore balance in our lives? One answer, Dr. Ariely says, is to implement more prohibitions on overbooking. We can work to reduce options on our own, delegating tasks to others and even giving away ideas for others to pursue.He points to marriage as an example, "In marriage, we create a situation where we promise ourselves not to keep options open. We close doors and announce to others we've closed doors."11 Since conducting the door experiment, Dr. Ariely says he has made a conscious effort to lessen his load. He urges the rest of us to resign fromcommittees, prune holiday card lists, rethink hobbies and remember the lessons of door closers like Xiang Yu.12 In other words, Dr. Ariely is encouraging us to discard those things that seem to have outward merit in favor of those things that actually enrich our lives. We are naturally prejudiced to believe that more is better, but Dr. Ariely's research providesa dose of reality that strongly suggests otherwise.13 What price do we pay for trying to have more and more in life? What pleasure and satisfaction can be derived from focusing our energy and attention in a more concentrated fashion? Surely, we will have our respective answers.14 Consider these important questions: Will we have more by always increasing options or will we have more with fewer, carefully chosen options? What doors should we close in order to allow the right windows of opportunity and happiness to open?。
有关贝林厄姆的英语作文Benjamin Graham, known as the "father of value investing," was a highly influential economist, investor, and professor. His investment philosophy focused on the intrinsic value of a stock and the importance of margin of safety.贝林厄姆,被誉为“价值投资之父”,是一位极具影响力的经济学家、投资者和教授。
他的投资理念注重股票的内在价值以及安全边际的重要性。
Graham's famous book "The Intelligent Investor" has been a guiding light for many investors, emphasizing the importance of thorough analysis, disciplined behavior, and a long-term perspective.格雷厄姆著名的著作《聪明的投资者》为许多投资者指明了方向,强调了彻底的分析、纪律的行为和长期的视角的重要性。
One of Graham's key principles is the concept of Mr. Market, which illustrates the irrational behavior of the stock market. He advised investors to take advantage of Mr. Market's mood swings and not to be swayed by short-term market fluctuations.贝林厄姆的一个关键原则是“市场先生”的概念,这说明了股市的非理性行为。
1. Virtue is indeed must be self-centered.正确的行动是,确实也是以自我为中心的By right action, we mean it must help promote personal interest.2.... (Poverty) was a product of their excessive fecundity... 贫困是他们过度生育的结果。
The poverty of the poor was caused by their having too many children.3. ...the rich were not responsible for either its creation or its amelioration. 富人不应该为产生贫穷和解决贫穷而承担责任。
The rich were not to blame for the existence of poverty so they should not be asked to undertake t he task of solving the problem.4. It is merely the working out of a law of nature and a law of God.这是自然规律和上帝的意志在起作用。
It is only the result or effect of the law of the survival of the fittest applied to nature of to human society.5. It declined in popularity, and references to its acquired a condemnatory tone.遭到了普遍的质疑,人们提及它都带有谴责的口吻。
People began to reject Social Darwinism because it seemed to glorify brutal force and oppose tre asured values of sympathy, love and friendship. Therefore, when it was mentioned, it was usually t he target of criticism.6. ...the search for a way of getting the poor off our conscience was not at an end; it was only suspended.寻求不为穷人的存在而内疚的办法,这种尝试并没有结束,而只是曾经中断过一段时间。
2016英语二第二篇内容如下:Text2(1) Biologists estimate that as many as 2 million lesser prairie chickens---a kind of bird living on stretching grasslands—once lent red to the often grey landscape of the midwestern and southwestern United States. But just some 22,000 birds remain today, occupying about 16% of the species’historic range.(2) The crash was a major reason the U.S Fish and Wildlife Service (USFWS)decided to formally list the bird as threatened. “The lesser prairie chicken is in a desperate situation,”said USFWS Director Daniel Ashe. Some environmentalists, however, were disappointed. They had pushed the agency to designate the bird as “endangered,”a status that gives federal officials greater regulatory power to crack down on threats. But Ashe and others argued that the “threatened”tag gave the federal government flexibility to try out new, potentially less confrontational conservations approaches. In particular, they called for forging closer collaborations with western state governments, which are often uneasy with federal action and with the private landowners who control an estimated 95% of the prairie chicken’s habitat.(3) Under the plan, for example, the agency said it would not prosecute landowner or businesses that unintentionally kill, harm, or disturb the bird, as long as they had signed a range—wide managementplan to restore prairie chicken habitat. Negotiated by USFWS and the states, the plan requires individuals and businesses that damage habitat as part of their operations to pay into a fund to replace every acre destroyed with 2 new acres of suitable habitat. The fund will also be used to compensate landowners who set aside habitat, USFWS also set an interim goal of restoring prairie chicken populations to an annual average of 67,000 birds over the next 10 years. And it gives the Western Association of Fish and Wildlife Agencies (WAFWA), a coalition of state agencies, the job of monitoring progress. Overall, the idea is to let “states”remain in the driver’s seat for managing the species,”Ashe said.(4) Not everyone buys the win-win rhetoric Some Congress members are trying to block the plan, and at least a dozen industry groups, four states, and three environmental groups are challenging it in federal court. Not surprisingly, industry groups and states generally argue it goes too far; environmentalists say it doesn’t go far enough “The federal government is giving responsibility for managing the bird to the same industries that are pushing it to extinction,”says biologist Jay Lininger.26. The major reason for listing the lesser prairie as threatened is____.[A] its drastically decreased population[B] the underestimate of the grassland acreage[C] a desperate appeal from some biologists[D] the insistence of private landowners27.The “threatened”tag disappointed some environmentalists in that it____.[A] was a give-in to governmental pressure[B] would involve fewer agencies in action[C] granted less federal regulatory power[D] went against conservation policies28.It can be learned from Paragraph3 that unintentional harm-doers will not be prosecuted if they_____.[A] agree to pay a sum for compensation[B] volunteer to set up an equally big habitat[C] offer to support the WAFWA monitoring job[D] promise to raise funds for USFWS operations29.According to Ashe,the leading role in managing the species in_____.[A] the federal government[B] the wildlife agencies[C] the landowners[D] the states30.Jay Lininger would most likely support_____.[A] industry groups[B] the win-win rhetoric[C] environmental groups[D] the plan under challenge。
反杜林论的英文书名"Anti-Dühring" is a significant work by Friedrich Engels, which serves as a critical examination and refutation of the philosophical, economic, and political theories put forth by Eugen Dühring, a German socialist. The book, originally published in 1878, is not only a critique of Dühring's ideas but also an elaboration of Marxist theory. It is divided into three parts: "Philosophy," "Political Economy," and "Socialism," each addressing different aspects of Dühring's work and Engels' counterarguments.The English title of "Anti-Dühring" remains the same in translation, as it is a proper noun referring to the book itself. However, it is important to note that the title is often accompanied by a subtitle or explanatory text toprovide context for English-speaking audiences. For instance, it might be presented as "Anti-Dühring: Herr EugenDühring's Revolution in Science" to reflect the full titleof Engels' original work, which is "Herrn Eugen Dühring's Umwälzung der Wissenschaft."In the English-speaking world, "Anti-Dühring" is recognized as a foundational text in the development ofMarxist theory, offering insights into the dialectical materialism and historical materialism that form the basis of Marxist philosophy. Engels' work has been influential in shaping the intellectual discourse within socialist and communist movements, and it continues to be studied for itsphilosophical depth and historical significance.The structure of the book is logical and methodical, with Engels systematically dismantling Dühring's arguments and presenting a coherent Marxist alternative. The language used is precise and scholarly, reflecting the academic nature of the debate. Despite its density and the complexity of the subject matter, "Anti-Dühring" remains an essential read for those interested in the history of socialist thought and the evolution of Marxist theory.In summary, "Anti-Dühring" is a pivotal work byFriedrich Engels that engages with and refutes the theories of Eugen Dühring. The English tit le of the book is a direct translation of the original German title, and it is often accompanied by a subtitle for clarity. The book is structured into three parts, each focusing on a different aspect ofDühring's theories, and it is written in a scholarly and precise manner. "Anti-Dühring" is a key text for understanding the development of Marxist thought and the critical dialogue within the socialist movement.。
涌现优于权威英文原文"Emergence Trumps Authority"In today's rapidly changing and interconnected world, the concept of emergence is gaining increasing attention as a more effective way to tackle complex problems and drive innovation. Emergence refers to the phenomenon where new and unexpected patterns, properties, or behaviors emerge from the interactions of simpler elements within a system. This stands in stark contrast to the traditional top-down approach of authority, where decisions and solutions are handed down from a single source of power.The main advantage of emergence over authority is its ability to harness the collective intelligence and creativity of a group. Instead of relying on the expertise of a few individuals at the top, emergence draws on the diverse perspectives and experiences of many. This leads to more robust and innovative solutions, as well as greater buy-in and support from those involved in the process.Furthermore, emergence is better suited to navigate the complexities and uncertainties of modern challenges. With the pace of change accelerating and the interdependencies of various systems becoming more evident, no single authority figure can possibly possess all the knowledge and insight needed to address the diverse and evolving issues we face. In contrast, emergence allows for a more organic and adaptive approach, where solutions can emerge and evolve over time as new information and perspectives come to light.Additionally, emergence encourages participation and empowerment, as individuals feel a sense of ownership and responsibility for the outcomes of the collective efforts. This can lead to increased motivation, collaboration, and resilience within the group, as well as a greater sense of satisfaction and fulfillment for all involved.While authority certainly has its time and place, especially in situations requiring clear direction and decisive action, the benefits of emergence cannot be overlooked. By recognizing and harnessing the power of emergence, organizations, communities, and individuals can better adapt to the complexities and uncertainties of our modern world and drive more effective and sustainable solutions. Ultimately, "Emergence Trumps Authority."。
能源类How Fast Can Obama Fix U.S.Environment Policy 奥巴马能多快修复好美国的环境政策?——P55For eight years the U.S. has been seen as a global outlier on climate issues.Now,with just 12 months to go until the world decides on a new Kyoto protocol,it is catch-up time for president-elect Barack Obama.What can he reasonably achieve in that time?Kyoto protocol《京都议定书》,全称《联合国气候变化框架公约的京都议定书》。
美国曾与1998年签署了《京都议定书》,但2001年3月布什政府拒绝批准《京都议定书》One of his first steps at home is likely be the adoption of cap-and-trade legislation. ①this would set the foundation for carbon trading: as in the European Union,the amount of greenhouse gases that industries can lawfully emit would be limited (the “cap”),and companies would have to either buy emission permits or find ways of reducing their emissions.Some states and companies have voluntarily come together to set up their own carbon-emissions market,but the Bush administration has always held that a federal system would harm the economycarbon-emissions 碳排放There seems little doubt that the nation will eventually adopt a cap-and-trade bill and to this end legislators on Capitol Hill have been preparing the ground for several years.Obama is among those who has backed a federal cap-and-trade bill in 2007,and earlier this years,one version failed to get through the Senate.C apitol Hill美国国会Senate n.参议院No rushHow quickly a new bill could be passed is the subject of much speculation. Jonathan Lash, president of the World Resources Institute says it is feasible that a cap-and-trade bill could be adopted before the end of 2009.feasible adj.能行的②Others are more cautious, Senator Jeff Bingaman, a New Mexico democrat who has been active on climate change and who chairs the Energy and Natural Resources Committee, thinks 2010 is more likely.Bingaman says energy legislation focusing on developing alternative energy sources and improving energy efficiency is likely to be passed before climate-change legislation.energy legislation能源效率“I think the reality is that it may take more than the first year to get it all done,” says Bingaman. Once energy legislation passes, “I think our prospects for moving ahead and seriously considering and enacting cap-and-trade legislation is improved,” he adds.Eileen Claussen, president of the Pew Center on Global Climate Change, agrees with Bingaman that a US cap-and-trade law is “much more likely in 2010”.Bingaman cites the complexity of the enterprise and the current financial crisis as factors that could slow the process down in Congress. In addition, there are rumours of alternative Republican-led legislation that could further delay matters.cite vt.引用Rumour n. 谣言Out in the coldObama is also left in the difficult situation of having to watch the out-going Bush administration attend the annual UN climate summit in Poznań, Poland, next month.At last year’s summit in Bali, after two weeks of intense negotiations and sleepless nights,delegates agreed in the eleventh hour to adopt a successor to the Kyoto protocol in December 2009. The Poznań summit is hop ed to pave the way for this agreement.There is a campaign underway for Obama to go to Poznań, but few believe he will. Instead, he is expected to send an observer.All this means expectations for radically different US climate leadership will be piled high on Obamas desk when he takes over the Oval Office on 20 January, 2009.Oval Office美国总统办公室At a conference on carbon trading that took place on Wednesday, an environmental advisor to Obama sought to reassure business and policy-making experts. “The presi dent-elect will move quickly on climate change,” said Jason Grumet, Obama’s lead energy and environment campaign adviser and possible choice for the new energy secretary.reassure vt.使放心“My suggestion to all of you is to enjoy the holiday season and rest up because I think it’s going to be a very, very busy 2009,” he saidHe offered no specifics and answered no questions. He noted that the US has operated “a federal climate programme with mandatory elements, for many, many years now.”mandatory adj.强制的what:(1)Obama prepares to fix U.S. environment policy.why:(3-15)the Bush administration has always held that a federal systemwould harm the economy.One version failed to get through the Senate.Bingaman cites the complexity of the enterprise and the current financialcrisis.Obama is left in the difficult situation.how:(2,16)Be the adoption of cap-and-trade legislation.the U.S. operated a federal climate programme with mandatory elements.与能源话题类似的还有:P61《多个州准备通过法律途径阻止奥巴马的环境政策》P73《美环保署告诫英国石油公司使用低毒化学物质》P76《日本灾难重重》P101《为了受气候威胁的太平洋岛民,澳大利亚需要美国式绿卡》经济类India Undermines Its Own Economy印度削弱自身经济——P2 When Indian Prime Minister Manmohan Singh last visited the White House in 2009, he heralded“a moment of great opportunity” for the two countries, calling on both sides to work together to “harness the immense potential of our talented and enterprising people and support each other’s growth and prosperity.”heralded vt.预示……的来临Yet, as Singh prepares for his visit to Washingt on this week, India’s trade and investment policies threaten to undermine, rather than harness, this potential.With the United States still recovering from a recession and the continued slowdown in India ’ s economic growth, the potential value of a deepe ning partnership has only increased.From my personal experience in India over the last two decades, I have witnessed firsthand the progress that has been made to solidify the relationship between India and the U.S.Between 2000 and 2011, revenue from U.S. exports to India increased seven-fold, and India’s exports to the U.S. more than tripled as the trade and investment relationship gatheredmomentum.seven-fold n.七倍gather momentum 方兴未艾Yet, four years after Singh’s initial visit, the potential of the relationship still remains largely unfulfilled. Surprisingly, India remains only the 13th largest trading partner of the United States, even though it may soon become the world’s third largest economy. The United States trades more with Taiwan and Korea than with India, despite their being only a tiny fraction of India’s size.Recent actions taken by the Indian government have only impeded further progress in the relationship. India has resorted to “compulsory licensing” to appropriate foreign firms’ intellectual property in violation of international trade norms. It has overridden, revoked or infringed upon approximately a dozen pharmaceutical patents held by foreign firms since 2012 alone.impede vt.阻碍compulsory licensing 强制许可,是指不经专利权人同意,直接允许其他单位或个人实施其发明创造的一种许可方式,又称非自愿许可in violation of 违反Override vt.推翻infringe vt. 侵犯Pharmaceutical adj.制药学的And its industrial policy expressly calls for such measures in other sectors as well. These actions create an atmosphere of distrust that will only discourage new investment in India.India has also mandated local content requirements, charged exorbitant tariffs at the border in certain sectors and maintained onerous market entry barriers and foreign equity caps.exorbitant tariff 高额关税onerous adj.繁重的;麻烦的barrier 贸易壁垒equity cap 持股上限One headline aptly captured India’s ambivalence towards easing market access: “The License Raj is Dead. Long Live the License Raj.” These barriers hinder investment in banking, financial services, insurance, retail, telecommunications and many other sectors.ambivalence n.举棋不定This year, India will experience only 5 percent growth—the lowest in a decade. Foreign investment has fallen by nearly two-thirds in the past year, and the rupee has plummeted.rupee n.(货币)卢布Plummet vt.暴跌Although short-term protectionism may appeal to domestic constituencies as the campaign begins for next year’s elections in India, history has proven that such policies are inconsistent with a dynamic economy that generates wealth over the long-term. And given the state of its economy, this is hardly the time for India to alienate foreign investors.Certainly the U.S. can do its part, too, by implementing sorely needed reforms to our immigration policy and ensuring that highly-skilled workers can continue to learn from and contribute to our economy through the H-1B visa program.H-1B visa美国签证的一种,即特殊专业人员/临时工作签证(Specialty Occupations/Temporary Worker visas)But frankly there is far more to be done on the Indian side to ensure fairness in our economic relationship. Respecting intellectual property rights and beginning to remove restrictive barriers to trade and investment will be essential first steps along this path.President Obama should use Prime Minister Singh’s upcoming visit to raise concern that the relationship ’ s “immense potential” that was heralded four years ago remains unfulfilled.While important progress has been made on the strategic front, India’s recent economic policies are hurting the relationship. Mr. Singh’s visit represents a new moment of opportunity—we can only hope that the opportunity is seized more effectively than four years ago.What(1): India’s trade and investment policies threaten to undermineSymptoms(2-6): 1.trade volume is small2.India policies impeded further progress in the relationship3.India will experience the lowest point of economic growth How(7-8): 1.India2.President Obama与经济类话题类似的还有:P8《寻找下一个谷歌》P11《美国经济》P15《英国经济:问题部分》P26《地下交通:预算超支》P48《报业之殇》。
a rX iv:mat h /1323v1[mat h.GT]5M ar21ON THE ADDITIVITY OF THE THURSTON–BENNEQUIN INVARIANT OF LEGENDRIAN KNOTS ICHIRO TORISU Abstract.In this article,we consider the maximal value of the Thurston–Bennequin invariant of Legendrian knots which topolog-ically represent a fixed knot type in the standard contact 3-space and we prove a formula of the value under the connected sum op-eration of knots. 1.Introduction The standard contact structure ξ0on 3-space R 3={(x,y,z )}is the plane field on R 3given by the kernel of the 1-form dz −ydx .A Legen-drian knot K in the contact manifold (R 3,ξ0)is a knot which is every-where tangent to the contact structure ξ0.The Thurston–Bennequin invariant tb (K )of a Legendrian knot K in (R 3,ξ0)is the linking num-ber of K and a knot K ′which is obtained by moving K slightly along the vector field ∂Date :February 26,2001.1991Mathematics Subject Classification.Primary 57R17;Secondary 57M25.Key words and phrases.Legendrian knot,Thurston–Bennequin invariant,con-nencted sum.12ICHIRO TORISUFigure1.tb=+#-#-#-1/2 # of cuspsFigure2.2.FrontsLet K be a Legendrian knot in(R3,ξ0=ker(dz−ydx)).Then a diagram(i.e.projection)of K in xz-plane is called front as in Figure 1.A front does not have vertical tangents;generically,its only singu-larities are transverse double points and semicubical cusps.Note that the number of the cusps is even.Since y=∂zON THE ADDITIVITY OF THE THURSTON–BENNEQUIN INVARIANT3Figure3.Proposition2.1.For two topological knots k1and k2,we have mtb(k1♯k2)≥mtb(k1)+mtb(k2)+1.Proof.Let K1and K2be Legendrian knots whose topological types arek1and k2,respectively and mtb(k1)=tb(K1)and mtb(k2)=tb(K2). We also regard K1and K2as fronts.Further we can assume that K1∩K2=∅and K1(resp.K2)lies in the left(resp.right)region of xz-plane,i.e.{(x,z)|x<0}(resp.{(x,z)|x>0})as in Figure3. Then we connect K1and K2by joining a right cusp of K1and a left cusp of K2as in Figure4.This procedure produces a Legendrian knot whose topological type is k1♯k2and Thurston–Bennequin invariant is mtb(k1)+mtb(k2)+1.This completes the proof.3.Preliminaries from contact topologyIn this section,we recall some basic notions and theorems from recent 3-dimensional contact topology.Further,we may assume the reader is familiar with convex surface theory started by E.Giroux in[6].For details and proofs,see[2],[3],[6],[7],[8].Letξn=ker(sin(2πnz)dx+cos(2πnz)dy)be the contact structure on a solid torus V={(x,y,z)∈R3z|x2+y2≤ǫ},where n∈Z−{0}and R3z is R3modulo z→z+1. The characteristic foliation on an embedded surface in a contact3-manifold is the singular foliation defined by the intersection of thecontact structure and the surface.The set of tangents ofξn to∂V forms4ICHIRO TORISUFigure4.a disjoint union of two simple closed curves on∂V,which are called Legendrian divides.Legendrian divides are leaves of the characteristic foliation on∂V.The next lemma is proved by a standard Darboux-type argument.Lemma3.1.For any Legendrian knot K in(R3,ξ0),there exists a sufficiently small neighborhood N(K)such that(N(K),K,ξ0)is iso-morphic to(V,{(0,0,z)},ξn)for some n.Note that in Lemma3.1,if K is topologically trivial,then n=tb(K). As∂V is a convex surface(i.e.has a contact vectorfield transverse to∂V),the following lemma can be proved by convex surface theory. Lemma3.2.Let T be any embedded torus in(R3,ξ0)and W a solid torus bounded by T.Suppose the characteristic foliation on T is diffeo-morphic to that on∂V and identifying these,the Legendrian divides on T are isotopic to the core curve of W through an isotopy in W.Then (W,ξ0)is isomorphic to(V,ξn)for some n.The following theorem on the classification of topologically trivial Legendrian knots due to Y.Eliashberg–M.Fraser[2]is also needed for the proof of Theorem1.1.Theorem3.3.Any topologically trivial Legendrian knot is Legendrian isotopic to one of standard forms expressed as fronts in Figure5.ON THE ADDITIVITY OF THE THURSTON–BENNEQUIN INVARIANT5cuspsdouble pointsFigure5.B1B2Figure6.4.Proof of Theorem1.1By Propositon2.1,it is sufficient to show the converse inequality. SupposeˆK is a Legendrian knot in(R3,ξ0)whose topological type is the connected sum of k1and k2and its Thurston–Bennequin invariant is maximal.By Lemma3.1,there exists a neighbourhood N(ˆK)ofˆK such that(N(ˆK),ξ0)is isomorphc to(V,ξn)for some n.Let B1and B2 be3-balls in R3such that B1(resp.B2)splitsˆK into the component corresponding to k1(resp.k2)and B1∩B2=∅(Figure6).6ICHIRO TORISUFurther,by convex surface theory,we can assume that(i)∂B1and ∂B2are convex and(ii)∂B1∩∂N(ˆK)and∂B2∩∂N(ˆK)are Legendrian knots on∂B1and∂B2,respectively and(iii)each dividing set on∂B i (i.e.the subset of∂B i consisting of tangents ofξ0and a contact vector field defining the convex surface)intersects∂B i∩N(ˆK)as a diameter of the disk.Then by Edge-Rounding Lemma due to K.Honda in[7],we have a solid torus W such that(i)W equals B1∪B2∪N(ˆK)except small neighbourhoods of∂B1∩∂N(ˆK)and∂B2∩∂N(ˆK)and(ii)∂W is a convex surface whose characteristic foliation is diffeomorphic to that of∂V.By Lemma3.2,it follows that(W,ξ0)is isomophic to(V,ξn) for some n.And notice that W is unknotted in R3and hence the core curve K of W which is Legendrian is also unknotted.Further,by a standard argument,we can assume that K agrees withˆK in the region of N(ˆK)−(B1∪B2).So by Theorem3.3,K is Legendrian isotopic to one of standard forms in Figure5.Therefore W is also identified with a small neighbourhood of that of the standard form.Further, by a homogeneous property of V and a parallel translation of W,we can assume that a region of W corresponding to B1(resp.B2)lies in {(x,y,z)|x<0}(resp.{(x,y,z)|x>0}).Then,identifyingˆK with its front,we can divideˆK along a vertical line into Legendrian knots K1and K2corresponding to k1and k2,respectively as the converse procedure in the proof of Proposition2.1.Counting the Thurston-Bennequin invariant of K1and K2,we have tb(ˆK)=mtb(k1♯k2)=tb(K1)+tb(K2)+1.Therefore mtb(k1♯k2)≤mtb(k1)+mtb(k2)+1.This completes the proof of the main theorem.References1.D.Bennequin,Entrelacements et´e quations de Pfaff,Ast´e risque107-108(1983),87–161.2.Y.Eliashberg and M.Fraser,Classification of topologically trivial Legendrianknots,in Geometry,topology and dynamics(Montreal,PQ,1995)CRM Proc.Lecture Notes15(1998),17–51.3.J.Etnyre and K.Honda,Knots and contact geometry,preprint.4.J.Etnyre and K.Honda,in preparation.5.D.Fuchs and S.Tabachnikov,Invariants of Legendrian and transverse knotsin the standard contact space,Topology36(1997),1025–1053.6.E.Giroux,Convexit´e en topologie de contact,Comment.Math.Helvetici66(1991),637–677.7.K.Honda,On the classification of tight contact structures I,Geom.Topol.4(2000),309–368.ON THE ADDITIVITY OF THE THURSTON–BENNEQUIN INVARIANT7 8.Y.Kanda,On the Thurston–Bennequin invariant of Legendrian knots and nonexactness of Bennequin’s inequality,Invent.Math.133(1998),227–242.9.S.Tabachnikov,Estimates for the Bennequin number of Legendrian links fromstate models for knot polynomials,Math.Research Letters,4(1997),143–156.10.T.Tanaka,Maximal Bennequin numbers and Kauffman polynomials of positivelinks,Proc.Amer.Math.Soc.,127(1999),3427–3432.Department of Computer Science and Engineering,F aculty of En-gineering and Resource Science,Akita University,Tegata,Akita010-8502,JapanE-mail address:torisu@math.akita-u.ac.jp。