Introduction to Knot Theory 18/02/2018

Introduction to Knot Theory 18/02/2018

Introduction to Knot Theory 18/02/2018 1. Knots and knot diagrams 1.1. Embeddings and isotopies. LetX andY be Hausdorff topo- logical spaces. Definition 1.1. A continuous mapf:Y→X is an embedding if it is injective and closed. Recall that every functorf:Y→X is closed ifY is compact. Definition 1.2. An embedding Ψ:Y×I→X×I is an isotopy if there is a continuous mapψ:Y×I→X such that Ψ y, t!" ψ y, t!, t!. #e say that embeddingsf,g:Y→X are isotopic if there e$ists an isotopy Ψ:Y×I→X×I% such thatf y!"Ψ y, &! andg y!"Ψ y, 1!. #e 'ill often 'rite Ψt for the mapy 7→Ψ y, t! 'ith ($edt. Definition 1.3. )mbeddingsf,g:Y→X are ambient isotopic if there is an isotopy Ψ:X×I→X×I% such that * 0 " idX andg"Ψ 1 ◦f. +uch * is called an ambient isotopy. ,ot every isotopy can be e$tended to an ambient isotopy. -n particular% there are embeddings that are isotopic% but not ambient isotopic. 1... Knots and links. Definition 1.4. A /not is an embeddingK:S 1 →R 3. #e consider oriented and non-oriented /nots. 0'o /notsK 1 andK 2 are e1uivalent% 'rittenK 1 ∼K 2% if they are ambient isotopic. An embedding ofn disjoint circlesL:S 1 ⊔...⊔S 1 →R 3 is called a lin/. Example 1.5. un/not right trefoil left trefoil (gure-eight 1 2 t'o unlin/ Hopf lin/ 2orromean rings #hitehead lin/ Remark 1.6. 0he 'ord %%ambient3 in 4e(nition 1.5 is important: all /nots are isotopic as embeddings. -ndeed% one can ta/e a ball that contains almost the 'hole /not% and shin/ it to a point: −→ −→ 0here are pathological /nots% such that the follo'ing one: #e 'ant to avoid such /nots. Definition 1.7. A /not6lin/ is tame if it is e1uivalent to a union of (nitely many intervals such /nots6lin/s are called polyhedral or 7L!. 8ther'ise% a /not6lin/ is 'ild. Example 1.8. 9nots and lin/s from )$ample 1.: are tame. ;or e$- ample: ∼ ∼ -n order to avoid 'ild /nots% one can also restrict to smooth embed- dings and smooth isotopies. -n such case% there is no need for ambient isotopies: it is /no'n that smooth embeddings are smoothly isotopic if and only if they are smoothly ambient isotopic. 1.<. Knot diagrams. LetK be a /not of a lin/. 3 Definition 1.9. A projectionπ:R 3 →H onto a planeH is regular if every point fromπ K! is covered by at most t'o points ofK% there are only (nitely many double points% and at each of them the t'o arcs ofK crosses themselves: regular not regular heorem 1.1". Every tame knot/link admits a regular projection. Sketch of Proof. LetK be a tame /not6lin/. Assume (rst that it is polygonal% i.e.K"I 1 ∪ · · · ∪I r% 'here eachI i is a straight segment. =onsider the follo'ing sets: •V 1 is the set of directions of segmentsI i% •V 2 is the set of directions of vectors 'ith origin at a verte$ of K and pointing to a point ofK% •V 3 is the set of directions of lines that intersectK three times. 'here by a direction 'e understand a line going through the origin of 3 R .V 1 is a (nite union of lines%V 2 is contained in a (nite union of planes% 'hereasV 3 is a union of surfaces. Hence% the complement of V :"V 1 ∪V 2 ∪V 3 is dense% in particular not empty% and every point from it is a direction of a regular projection. -n the general case% letK ′ be a polygonal /not6lin/ e1uivalent toK% and 'hich is contained in anǫ-neighborhood ofK for some smallǫ> &. 0hen regular projection ofK are given by points from the complement ofV that are in distance at leastǫ fromV. � Definition 1.11. A diagram of a /not6lin/K is the imageπ K! ofK under a regular projection% 'here at each double point the part ofK closer toH is bro/en. -t follo's from 0heorem 1.1& that tame /nots and lin/s have diagrams. Example 1.12. 0he pictures from )$ample 1.: are /not6lin/ dia- grams. 4 0he main problem of /not theory is to determine 'hether t'o dia- grams represent the same /not6lin/. ;or instance% all the follo'ing are diagrams of the un/not: 0here are certain transformations% called the Reidemeister moves% that change the diagram but preserves the represented /not6lin/: ←−−−→ ←−−−→ R1 R2 ←−−−→ R3 heorem 1.13 Reidemeister!. Link diagramsD 1 andD 2 represent equivalent links if and only ifD 2 can be obtained fromD 1 through finitely many Reidemeister moves. Sketch of Proof. LetL 1 andL 2 be lin/s 'ith diagramsD 1 andD 2 projected on the same planeH. #e may assume thatL 1 andL 2 are polygonal: every tame lin/L is e1uivalent to a polygonal oneL ′% such that their diagrams are ambient isotopic. #e no' use the fact that polygonal lin/s inR 3 are ambient isotopic if and only if they are related by a se1uence of >-move: #hen the moves are su?ciently small% their projections correspond to Reidemeister moves. ;or instance: 5 � Example 1.14. -t is not al'ays an easy tas/ to chec/ if t'o dia- grams represent e1uivalent lin/s. 8ne of the most famous e$amples is the 7er/o pair: -t 'as conjectured for a long time that these diagrams represent differ- ent /nots% until a se1uence of Reidemeister moves have been found by 9eneth 7er/o in 1@A<..

View Full Text

Details

  • File Type
    pdf
  • Upload Time
    -
  • Content Languages
    English
  • Upload User
    Anonymous/Not logged-in
  • File Pages
    5 Page
  • File Size
    -

Download

Channel Download Status
Express Download Enable

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

  • Not to be reproduced or distributed without explicit permission.
  • Not used for commercial purposes outside of approved use cases.
  • Not used to infringe on the rights of the original creators.
  • If you believe any content infringes your copyright, please contact us immediately.

Support

For help with questions, suggestions, or problems, please contact us