John Horton Conway

John Horton Conwaydied4月11日的Covid-19。他82岁。在社会疏远措施中,对抗冠状病毒大流行,一个常见的避免是“生活继续”。但有时它没有。

Conway是普林斯顿大学的Emeritus教授。在数学家中,他以宽度和聪明而闻名,以及他的个性和似乎无限的好奇心。在Mathscinet中,他的论文中有一点是数字理论,大约六分之一的阶段理论,以及凸面或离散几何形状的十分之一。其余部分在MSC中分散约20个其他类。Conway设法在其他20个地区做出持久的贡献,例如他在代数拓扑和结理论中的工作,他在他身上命名为他:亚历山大康威多项式。同时,在幕后,康威经常贡献谜题,游戏和想法Martin Gardner,谁会在他身上写下他们着名专栏科学的美国人

In the Mathematical Reviews database, Conway has73 coauthors。很多人都很有名,但很多人都没有。Conway似乎受到好奇的推动,而不是与人们合作时的声誉。

Many people know of Conway because of theGame of Life(还here)。像许多新代理程序员一样,生活是我编程的第一件事之一。在我的情况下,它在使用打卡的Hewlett-Packard机器上的Fortran。这是一个伟大的项目,因为它也是一个非常简单的生物系统模型。对于年轻的数学专业,这个例子是值得注意的,因为到目前为止的课程中的所有模型和应用程序都是基于微积分。这显然不是。

很多人也知道康威作为令人印象深刻和经典的书籍Winning Ways for your Mathematical Plays。(可悲的是,其他两个同轴师,Elwyn Berlekamp.理查德·盖伊那also died in the past year or so.) The beginning and end of our review of the first edition of the book quickly tell you the truth about it: “The two volumes are crammed to the brim with information, colored illustrations and examples … The thrust of the book lies in the direction of formulating exact or suboptimal polynomial strategies for very broad classes of combinatorial games. It is likely to remain an eminent leader in this field for many years to come.” His在球形包装上预订尼尔斯洛安是another classic.

Conway was one of the authors of the monumental一种tlas of Finite Groups。When I first heard about the Atlas, I thought, “That’s crazy.” I turned out to be half right, it was crazy brilliant. The core of the book is a compilation of the字符表of all the finite simple groups known at the time. By some miracle, the authors were able to convince the publisher to produce the book in a large format (42 x 31.6 x 2.9 cm), which was helpful for bigger groups with bigger tables. It was bulky to carry and had a tendency to bend and to curl at the edges. Our reviewer,罗伯特格里斯,建议表可以在磁带上提供。[注意:见http://brauer.maths.qmul.ac.uk/Atlas/.]地图集显然是一个灵感谎言团体和陈述的图谱,它在线完全。

Conway had a knack for naming things, as in his famous paper “Monstrous moonshine” with Simon Norton. (The complete review is below.) The paper conjectures remarkable correspondences between conjugacy classes of the finite simple group called the Monster and congruence subgroups of the modular group, PSL(2,$\mathbb{Z}$). The conjecture was proved inMR1172696.经过Richard Borcherds,谁是博士学位。康威学生。

康威的影响很广,就像他的着名一样。康威的死后不久,很快的帖子很快就开始涌现,几个由数学家只知道康威的人,但对他们少量与他的互动感到强烈回忆。以下是只有其中一些的链接。

Siobhan Roberts写了一个很好的康威概况这Guardian2015年。她还发表了一个引人入胜的传记,标题为Genius at play

这re is a quote from Isaac Asimov about science:在科学中听到最令人兴奋的短语,将新发现的那个是那个新发现,不是“尤伊卡!”(我找到了它!)但“这很有趣......”John Conway是该实施例。


MR0654501
Berlekamp, Elwyn R.;Conway, John H.;盖伊,理查德K.
获胜您的数学戏剧的方式。卷。1。
Games in general.学术出版社,Inc。[Harcourt Braces Jovanovich,Publishers],伦敦 - 纽约,1982.XXXI.+426+xipp. ISBN: 0-12-091150-7; 0-12-091101-9
90dxx(05-02 90-02)

MR0654502
Berlekamp, Elwyn R.;Conway, John H.;盖伊,理查德K.
获胜您的数学戏剧的方式。卷。2。
特别是游戏。学术出版社,Inc。[Harcourt Braces Jovanovich,Publishers],伦敦 - 纽约,1982年PP。一世-xxxiii.和429–850 and一世-xix。ISBN:0-12-091152-3;0-12-091102-7
90dxx(05-02 90-02)

两个卷用信息,彩色的插图和示例填充到边缘,并且这里可以仅指示每个章节中包含的主要主题。前13章章节构成了第1卷,题为“一般游戏”。
第一章:党人游戏的概念,说明by means of a number of examples, especially Blue-Red Hackenbush, which plays a fundamental role in partizan games, analogous to that played by Nim in impartial games: In a blue-red string figure, Left removes any blue edge and all edges not connected to ground anymore. Right moves similarly on red edges. The player first unable to move loses.
第2章:与Partizan Games一起使用的工具,例如简单原理:游戏的价值$(l | r)$,如果是一个数字,是最简单的号码$(l,r)$。正,负,零和模糊位置。比赛总和的概念。公正游戏:绿色哈彭班(任何一个玩家可以删除绿色边缘),Nim,Nimbers(公正游戏的值)。
第3章:NIM,MEX规则,Sprague-Grundy理论的变化:每一个公正的游戏都只是一个虚假的Nim堆。{从计算的角度来看,公正游戏之间可能存在相当大的差异。因此,没有人知道Wythoff游戏的总和是否存在多项式时间策略,而Nim是琐碎的多项式。}在第3章的第二部分,注意力转移回Partizan游戏:可逆移动,主导地位,主导地位小模糊游戏的大小。
第4章:返回公正游戏:$ p $.$ n $-positions. Octal games such as Kayles and Dawson’s Kayles. It is conjectured that Grundy’s game (divide any pile of tokens into two unequal piles) is ultimately periodic.
第5章:更多关于Partizan Games。交换机和数字的总和:在交换机中移动$(x|y)\ (x\geq y)$和largest temperature$ {\ textstyle \ frac 1 {2}}(x-y)$。热门游戏,小游戏。例子:霸气,蟾蜍和青蛙。
第6章:对选项不一定是数字的热门游戏的更深分析。平均值和停止值。温度计。公平而易激情的比赛。
第7章:关于Hackenbush的全部。结肠原理,平价和融合原理分析绿色荷兰武器。蓝红色Hackenbush总是一个数字,即使对于红木家具的子集也可能很难找到。寻找红木床的价值是NP-HARD。分析Hackenbush Hotchpotch - 这可能涉及所有三种颜色的原子量。转基身游戏的临时调查。对于国际象棋的exptime-pleantenate,请参阅审稿人和D. lichtenstein的文章[J.组合。理论系列。一种31(1981),199-214;MR0629595]。This also implies the Pspace-hardness of chess.
Chapter 8: All small games, remote stars, computing atomic weights for analysing Hackenbush.
第9章:Partizan和公正游戏的加入(在每个组件游戏中移动)分析,正常和蛋清。遥感函数。
Chapter 10: Analysis of the union (move in any number of component games) of partizan games, normal play. (Smith’s result for impartial unions is cited in Chapter 11.) Analysis of urgent unions (the game ends as soon as its first component does) of partizan games, normal and misère play.
Chapter 11: Games with infinitely many positions but only finitely many moves: infinite ordinal numbers. Games which may not end: loopy partizan games [see also the reviewer and U. Tassa, Math. Proc. Cambridge Philos. Soc.92(1982),193-204]。Loopy Hackenbush。
Chapter 12: Loopy impartial games: to win need remoteness in addition to$ p,n $labeling. Entailing move games such as the following: either split a stack of coins into two smaller ones, or remove the top coin from a stack. In the latter case, the opponent has to move in the same stack.
第13章:蛋纪利的微妙分析。“属”的概念以及如何通过诺亚的方舟定理来帮助脱落驯服,恢复甚至一些不安的游戏。
剩下的章节14-25分为第2卷,题为“特别是游戏”。
Chapter 14: Games played by turning coins. Connection with Nim-multiplication.
第15章:通过移动硬币在条带上播放的游戏,如银元,安东尼瓜猴,讽刺,西蒙米,克洛特(一种与不平等桩的尼姆的形式),Kotzig的Nim。有界尼姆,摩尔的$ \ text {nim} _k $$d$- 尼姆。
第16章:点和箱的各种次优策略以及与Kayles和Dawson Kayles的连接。圆点和盒子是np-hard。
第17章:通过曲线加入两个斑点的游戏,满足各种条件,例如卢卡斯游戏(包括ère演奏)。豆芽。
第18章:Sylver Coinage分析(名称一个整数不是先前命名整数的线性组合)。另见盖伊的研究问题[amer。数学。每月83(1976), 634–637]. The chapter also includes Chomp (arithmetic and geometric versions) and Zig-Zag. These (together with von Neumann’s Hackenbush mentioned at the end of Chapter 17) are special cases of poset games.
Chapter 19: Games played on a chessboard with a King and Go stones (Kinggo) or a duke and Go stones (Dukego). The Angel and the Square-Eater. Wolves-and-sheep and variations thereof.
第20章:狐狸和鹅的分析:四个“鹅”在棋盘上向上移动,试图捕获一个像国王在跳棋中移动的“狐狸”。{这个游戏,用数字上的两种类型的令牌播放,是pspace-hard。}
Chapter 21: The French Military Game: A game of pursuit similar to Fox and Geese.
第22章:TIC-TAC-TOE和类似游戏。Go-Moku和Hales-Jewett配对策略[A.W. Hales和R. I. Jewett,Trans。amer。数学。SOC。106.(1963),222-229;MR0143712]。十六进制和香农交换游戏。Phutball。
Chapter 23: Analysis of peg solitaire games. Beasley’s proof that 18 moves are necessary for central peg solitaire. Variations of peg solitaire.
Chapter 24: Puzzles with cubes, puzzles with wire and string, the Tower of Hanoi and ternary numbers, the 15 puzzle. Analysis of the Hungarian cube puzzle, tactics for solving other “Hungarian” puzzles. Examples of other puzzles: paradoxical pennies, paradoxical dice, magic squares. The chapter ends with some calendar-theoretic computations, including the dates of Easter and Rosh Hashanah.
Chapter 25: The “game” of Life. The main result is a reduction of difficult mathematical problems such as Fermat’s Last Theorem to the predictability problem of the final fate of an initial life pattern. This is done by computer simulation with appropriate life patterns.
Each chapter ends with a section called “Extras”, where underlying principles or additional details are given. Instead of formal proofs, short convincing arguments or examples are provided. This tends to increase considerably the amount of material packed in the 850 pages of the book. Together with the wit, humour and originality of approach, it also increases the readability or apparent readability. To really understand and prove everything in the book, not to mention to attempt solutions of the many questions inspired on every page of the book, will engage many people for many years.
一种s the authors state, the book is not an encyclopedia, since there are many games, theories and puzzles not included in it. In fact, there are certain directions not pursued in the book, such as transpolynomiality of games or questions of undecidability or computability of strategies. The thrust of the book lies in the direction of formulating exact or suboptimal polynomial strategies for very broad classes of combinatorial games. It is likely to remain an eminent leader in this field for many years to come.

Reviewed byAviezri S. Fraenkel.


MR0554399.
Conway,J. H.;Norton,S. P.
怪物的月光。
公牛。伦敦数学。SOC。11.(1979),不。3,308–339.
20D08 (10D12)

这种极具信息丰富的纸张细节在模块化组的模块化组和“怪物”有限组的模块组和子群的亚组之间的许多,显然系统的,巧合。(另见J. G. Thompson的伴随论文[20029年20030年以下]。)
这authors begin with a brief history of the increasing observations of such coincidences, prior to their own work. Section 2 then describes the main conjectures. The authors give a correspondence between conjugacy classes of the Monster and congruence subgroups of the modular group—roughly, an element of order$ n $在怪物中将对应于上面的子组$\Gamma_0(n)$。Expand a hauptmodul for each such subgroup by Fourier coefficients in powers of$q=e^{2\pi i\tau}$。对于固定的$ k $, 这$ q ^ k $-coefficients of these functions provide a class function, which is conjectured to be an actual character of the Monster (the$ k $TH“HEAD字符”)。第3-7节讨论了对应的技术细节,包括课程之间的关系以及与某些怪物元素的水蛭格子的关系。第8节介绍了函数和头字符之间的头字符中的“复制”公式,以及其他“扩展”和“压缩”。第9节描述了怪物以外的群体的类似工作(“Moonshine”)。在第10节中,作者询问为什么只出现一致类型的Genus-0子组 - 并提出所有这些亚组的确定。(显然是300-400,其中171次与怪物本身有关。)
本文包括表格:怪物的不可缩小度(来自由费舍,起居石和索恩确定的字符表);在Irrefucibles方面,几个头特征的样本分解;怪物的类列表,指示与模块化组的子组的对应关系;前10个头部字符完整;以及关于上述类别的各种其他信息和上述公式。
Since the appearance of the paper, R. Griess has constructed the Monster [“The friendly giant”, Invent. Math., to appear]. The head-character conjecture can be verified by a finite but lengthy computation—and this has been essentially completed by A. O. L. Atkin, P. Fong, and the reviewer. An actual module affording the head characters, at least for the centralizer in the Monster of a 2-central involution, has been constructed by V. G. Kac [Proc. Nat. Acad. Sci. U.S.A.77(1980),没有。9(1),5048-5049];Griess现在正在努力将行动扩展到完整的怪物组。
随后的工作似乎仅提供了对与(无限离散)Genus-0分析数字理论的亚组的诸多问题的众多问题。

Reviewed byStephen D. Smith


MR0827219
Conway,J. H.(4-CAMB);柯蒂斯,R.T.(4-CAMB);Norton,S. P.(4-CAMB);帕克,R. A.(4-CAMB);威尔逊,R. A.(4-CAMB)
有限群体的图谱。
Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray.牛津大学出版社,Eynsham,1985.XXXI.v+252 pp. ISBN: 0-19-853199-0
20d05(20-02)

最后,关于许多有限简单组的字符表和相关信息的官方集合已经出现在书形式中。这些信息对于有限群体理论的专家非常重要,并且该体积包含整齐地提出了非专科学家可以欣赏的教学材料。多年来,作者在非常高的水平上使用了材料。它被经验重新编写和精致。在1979年的1979年的第1979年的有限群体会议上,Simon Norton会议携带一袋破烂的打印输出和字符表,以处理有关简单群体的紧急问题。现在,我们都可以拥有这种快速访问的力量,但以级别的格式!
有限群体的“经典”字符表$G$是经过definition a$ k \ times k $复数号的矩阵,其行被索引$ k $不可减少的字符,其列由索引$ k $conjugacy classes; of course, it is not unique because there is no generally accepted way to order the index sets, though the principal character (corresponding to the trivial homomorphism$ g \ to {\ rm gl}(1,{\ bf c}))$是always listed first. The$(i,j)$入口是$ \ chi_i(g_j)$,值的价值$i$关于代表的不可挽回的性格$ j $Th Comugence类,这个代数数总是如$d\ |g_j|$th roots of unity, where$d=\chi_i(1)$是这degree of$ \ chi_i $
过去的25年里的努力对有限扩散进行分类e simple groups created a greater need to have numerical and combinatorial information about the known groups. The occasional tables produced by R. Brauer or J. S. Frame or J. Todd years ago were followed by a flood of tables in the 1960s and 1970s. Generally, these were distributed informally, often with no name or source written on them and always without proof. Referring to a character table in a research article was awkward at times. The general theory of Brauer gave many arithmetic conditions on the character table which in “easy” cases allowed one to fill in many blank entries for the table of a particular group. This was not always the case. For instance, David Hunt’s work on the tables for the Fischer3美元-Transposition团体采取了特别长的时间,涉及广泛的计算机工作和对具有已知字符表的子组的衔接限制表的研究。
总之,五位作者收集了这一早期和未发表的工作中的一些,然后大大延伸并将其放在适合容易现代应用的表格中。
该书组织如下:(i)介绍和解释(28页),(ii)字符表(235页),(iii)补充表(6页),(iv)参考(8页)和索引(1页)。
(I): Sections 1, 2 and 3 contain a rapid introduction to the families of finite simple groups. It is clear and telegraphic in style and not intended for someone who is looking for full discussions and constructions.
Sections 4 through 7 discuss the multiplier, automorphism groups, isoclinism and the group extension theory which is relevant to interpreting the blocks (and broken-edge blocks) in the tables, notation for conjugacy classes, algebraic numbers and algebraic conjugates of these two concepts. We comment on the tables themselves in (II). The authors’ notations for algebraic integers are very successful for character tables, e.g.,$z=z_N=\exp(2\pi i/N)$$ b_n = \ frac12 \ sum ^ {n-1} _ {t = 1} z ^ {t ^ 2} $$ c_n = \ frac13 \ sum ^ {n-1} _ {t = 1} z ^ {t ^ 3} $(for$N\equiv 1$(mod 3)), etc.
博览会的一个故障是作者使用术语和符号而不解释,然后稍后定义它们。在上面的定义序列中,对于$z_N$$b_N$$c_N,\cdots$一个发现“$n_2$”, but not a definition until further down the column. The notation$ ^ * k $用于7.3节,但没有提示用于寻找定义的位置。如果包含的符号和定义索引以帮助在中间阅读的读者来帮助。
作者讨论了简单群体的几个现有符号系统。系统中的部分一种tlasmake the reviewer uncomfortable.
最明显的物品是使用“o”的简单组合因子$ n $- 二维正交组$\epsilon$超过$ {\ bf f} _q $。在其他系统中,这个小组将是${\rm P}\Omega^\epsilon(n,q)$或者one of$ d_m(q)$$ ^ 2d_m(q)$(什么时候$ n = 2m $) 或者$ b_m(q)$when$n=2m+1$。作者拒绝了这些符号,因为它们希望所有这些简单组的基本名称的一个字母。
第二条评论是关于分配给零星群体的名称;见表1,第VIII页。组理论家通常使用的原则是在发现者之后命名零星组,并使用与这些名称相关的符号。这是康威组的有时例外情况(由...表示$.0,\;.1,\;.2$$ .3美元自1968年以来但是$ {\ rm co} _0,\;{\ rm co} _1,\; {\ rm co} _2 $, 和$ {\ rm co} _3 $在此体积中),费 - 群体(表示)$ m(22),\; m(23)$$M(24)’$最初,但后来的$ {\ rm fi} _ {22},\; {\ rm fi} _ {23} $$ {\ rm fi}'_ {24} $)和1973年11月由费舍尔和审计员发现的怪物;该一种tlassymbols are$ m $.那FG and$ f_1 $) and the Baby Monster (the$\{3,4\} ^+$1973年早些时候由Fischer发现的植物组;这一种tlassymbols are$B$$ f_2 $) and the Harada group (called the Harada-Norton group in the一种tlas;这一种tlassymbols are HN and$ f_5 $)。
制度$F$与下标有几个不错的group-theoretic features. However, there seems to be no natural systems covering all sporadics. Why not keep the names and remember the history, at least? Perhaps later developments will suggest a good solution.
Finally some comments about notation for other finite groups. Several recommendations in 5.2 really are at variance with general usage. The authors mention$C_m$对于循环的订单组$ m $.but not$ {\ bf z} _m $! Their term “diagonal product”$A\triangle B$被称为回调或纤维产品。基本组的最常见的符号是$p^{1+2n}$或者$ p_ \ epsilon ^ {1 + 2n} $。Since notation for an extension$A\cdot B$沿着升序读取左右读取,写入更适合$(a \ times b)\ frac12 $$ \ frac12(a \ times b)$
(ii):在第6节中讨论了各个表的组织。有关良好的示例,请参阅第XXIV页面。让$G$是简单的群体。表格块与每个块对应于表单的扩展$ m.g.a $那where$ m $.是Schur乘法器的循环商$a$是外部万态体群的循环子组;因为有原因为什么这些案件足以(几乎),见6.5和6.6。
在块的左侧是向下运行的字符列表$(\ chi_1 = 1,\ chi_2,\ chi_3,\ cdots)$及其指标(0,$ + $或者$ - $由于角色不是真实的,由真正的代表提供,或者是真实的,但没有得到真正的代表所带来的)。横跨顶部是一个有几行关于列的信息(由共轭类索引,$C_i,\;i=1,\cdots,k$)。这experience of the last 25 years has shown the importance of enriching the traditional “classic” character table to include power maps (i.e., for$n\in{\bf Z}$那which classes contain the$ n $th powers of elements from a fixed class), factorizations (i.e. if$ g \以c_i $$\pi$是一组素质和$ g = g_ \ pi g _ {\ pi'} $是这unique commuting factorization of$g$在to a$\pi$-element and a$ \ pi'$-element, which$C_j$包含$ g_ \ pi $), and so on. A simple application of this information, which is not possible to execute with a strictly classical table, is to find the dimension of the space of cubic invariants on a module$ v $affording the character$ \ chi $。对称张量立方体的角色$ v $$g\mapsto \frac16\{\chi(g)^3+3\chi(g)\chi(g^2)+2\chi(g)^3\}$所以它的内在产品具有琐碎的特征$G$给出答案。
这difficulty of getting these blocks correct increases generally according to the sequence$m=1$$a=1$;$a=1$;$ m,a $arbitrary. Indeed the authors acknowledge errors which turned up as the book went to press (see page xxxii, bottom). How the notations extend across the several upward and downward extensions is articulated well.
(iii):最后一部分一种tlastext consists of three tables and a list of references. (1) Partitions and classes of characters for$ s_n $那useful, say, in working out particular invariants of the group in question. (2) Involvement of sporadic groups in one another (the single “?” in this一种tlas表现在被声称是“$ - $“在R. A. Wilson的最近工作中)。(3)超过250个简单群体的订单,基础10的订单和分解形式和SCUR乘法器和外部万态体组。
(iv)参考书目仅限于(i)关于有限简单群体的家庭的一些非常一般的作品和(ii)26个散发群中的每一个的冗长物品清单。
Absolute初学者的调查文章(无证据)值得一提,可以进入(i),例如,由R. Carter的纸张[J.伦敦数学。SOC。40(1965), 193–240;MR0174655.] for groups of Lie type and a paper by the reviewer [in数学和物理中的顶点运算符(Berkeley, Calif., 1983), 217–229, Springer, New York, 1985;MR0781380] for sporadic groups. Also, references for Schur multiplier and automorphism groups would be of general interest.
数字信息表对于错误是臭名昭着的,它确实需要比较;例如,MCLaughlin组的顺序在D. Gorenstein的第136页上给出了错误的Finite simple groups[Plenum,纽约,1982;MR0698782]。在Higman-SIMS组之后,$G$那was discovered in 1968, it was deduced that$G$必须有亚组$K\leq H\leq G$$H\cong {\rm PSU}(3,5)$$ k \ cong {\ rm alt} _7 $。当然,人物$G$必须明确地限制到特征$ k $$ h $但是字符表然后在手工产生矛盾!找到表中的错误。
应该是研究人员,迫切需要证明定理,相信一种tlas?问题类似于接受有限简单组的分类。两种努力都受到广泛尊重,两者都在高层工作以实现目标,但已承认存在错误。在这两种情况下,集团理论社区认为,环境计划中可能只需要局部调整,以处理错误。所以,答案是:“是的,但是$\ldots$”.
只有纯粹主义者将把他或她重新转回完成的任何一个完成。为了取得进展,我们必须接受它们基本正确但要注意一段时间并尽可能地寻找备用论点。在正式写入争论时,人们可以将它们视为公理。
Norton已显示自发布以来发现的错误列表。一个是不合适的字符表!值得一提的是,聊天何何最近发现了最大的$ 7 $-local subgroup of the Monster not on the一种tlaslist. There may be a problem with the list of maximal subgroups for${\rm Co} _1$
{审核人的言论:审稿人对少数情况下奖学金的不正确感到失望(尽管XXXII第8.5.1节的免责声明)。没有完全证明怪物字符表的正确性(虽然没有怀疑)。(a)共轭类的测定需要足够了解子组中的元素的胶合剂${\bf M}$形式$2^{1+24}\cdot{\rm Co}_1$;这authors guessed the basic information, then proceeded. (b) The existence of the irreducible character of degree 196883 was taken as a hypothesis (196883 is the smallest number which could be the degree of a nonprincipal character); a proof that such a character exists was claimed by Norton in 1981 but no manuscript has appeared, and its relationship with (a) has not been explicitly stated; existence of such a character is necessary to complete the program devised by J. G. Thompson [Bull. London Math. Soc.11.(1979),没有。3,340-346;MR0554400.] for proving uniqueness of${\bf M}$
{It would have been helpful to have some recent references, e.g. to the reviewer’s recent work on code loops. The reviewer understands that future editions will contain no new references.
{The book is attractive in appearance. The cover is a cherry red with white writing on stiff cardboard. The authors’ names form a neat matrix listed vertically in alphabetical order (which agrees with their respective ages, apparently), each with two initials and a 6-letter last name. The price is extremely fair. The authors are to be commended for their influence on the price and for getting the publisher to replace the originally intended soft binding.
{这本书对于大多数公文包来说都是很大的大。在审稿人的副本上绑定的绑定变形并干扰了易于关闭和开放的书,以平躺在桌子上。由于与结合的斗争,附近的页面的边缘已经开始受到困扰。一个想法是使磁带上的表格,可能是打算计算机计算的用户的重大努力。
{数学社区(和物理社区)应该感谢造物者一种tlasfor their extremely fine service. An appreciation and use of the finite simple groups might be expected to spread noticeably faster as a result.}

Reviewed byR. L. Griess


MR0258014
Conway,J. H.
一种n enumeration of knots and links, and some of their algebraic properties.1970Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967)第329-358页Pergamon, Oxford
55.20

In this essential paper (i) a new efficient notation for describing specific knots is expounded, (ii) identities are reported which reflect the behaviour of knot invariants on changing some structure elements coded in the notation, (iii) lists of all prime knots up to 11 crossings and of all prime links up to 10 crossings are given in this notation, a census which checks (and corrects) and enlarges the existing tables still based on Tait’s, Little’s and Kirkman’s work before 1900. The ideas are presented here in expository style, whereas a more technical paper with more complete presentation of the subject is promised.
第一部分阐述了新的符号。这是基于选择边缘连接的4价平面图,最简单的是,标有1 *,看起来像一个8;实际上,在表的范围内只需要八种这样的图形。如果选择的图形的每个节点替换为“纠结”,则一个人得到一个规准的a($ \ mu $-component) knot. A “tangle” consists of a normed projection of strings such that there are four free ends pointing to the four compass directions. A few operations are defined on tangles, such that the notation and classes of “integral”, “rational” and “algebraic” tangles can be defined recursively starting with the specific tangle marked by 1. That way the symbol for a knot indicating the graph and the substituted algebraic tangles contains some structural elements of the knot. As H. F. Trotter points out [抽象代数中的计算问题(Proc。Conf。,牛津,1967),PP。359-364,Pergamon,牛津,1970年;MR0258015],这种符号似乎是“用于计算机表示的手工和(可能对一些微小修改)的最佳”。
这n the author describes the interplay between knot equivalences and elements figuring in the notation; for rational tangles substituted into the graph 1*, there are remarkable connections with continued fractions. The next sections contain remarks concerning Alexander polynomial, Minkowski unit and signature; some identities are reported which allowed short computations but which have wider applications, in part not yet fully explored. The last section gives the inferences to draw from the new lists to the open problem of whether every slice knot is a ribbon knot.
这lists (computed by hand) given as an appendix include the 1-component knots up to 8 crossings with symmetries, signature, Minkowski unit, determinant and polynomial, up to 10 crossings with symmetries, determinant and polynomial, and an enumeration of all 11-crossing knots, alternating and nonalternating; furthermore, the links up to 8 crossings with linking numbers, symmetries, signature, Minkowski unit, determinant and polynomial, up to 9 crossings with linking numbers and polynomial, and an enumeration of the 10-crossing links. {Remark: the column heading on p. 345 should read$ \ delta ^ 0 $代替$ \ sigma ^ 0 $。}

Reviewed byH. E. DeBrunner.

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关于Edward Dunne.

我是数学审查的执行编辑。此前,我是17年的AMS册计划的编辑。在为AMS工作之前,我在牛福德大学和俄克拉荷马州立大学工作的学术职业生涯。1990 - 91年,我为海德堡的斯特林斯 - Verlag工作。我的博士学位。来自哈佛。我收到了世界一流的文科教育,作为圣克拉拉大学的本科生。
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