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Introduction to Knot Theory 25/02/2018 1.4. Operations on knots. We will now discuss certain operations on knots and links. Definition 1.15. The mirror image L! of a knot/linkL is the result of applying toL the reflection (x,y,z) 7→(−x,y,z). The reversed ←− knot/link L isL with orientation of all components reversed. We say ←− thatL is anticheiral ifL∼L ! and cheiral otherwise. IfL∼ L then we say thatL is reversi!le . Example 1.16. "igure#eight is anticheiral (e$ercise) whereas the mir# ror image of a right#handed trefoil is the left#handed one% mirror We will show later that these two knots are not e&uivalent so that the trefoil is a cheiral knot. It is also reversi!le% to see this rotate its diagram !y 1'( degrees with respect to an a$is in the plane of the pro)ection% rotate Definition 1.17. *etL 1 andL 2 !e two links and pick pointsp i ∈L i fori+1, ,. The connected sum L1-L2 ofL 1 andL 2 is the result of cutting the links at the pointsp i and regluing them in an oriented way% p1 p2 L1 L2 L1 L2 1 2 Theorem 1.18. The connected sumL 1-L2 does not depend on the choice ofp 1 andp 2 whenL 1 andL 2 are knots. Sketch of Proof. We will use the following facts from topology. (1) . tame knotK has a regular neigh!orhood that is an em!ed# ding of a solid torush%S 1 ×D 2 →R 3 such thatK(t)+h(t, (). (,) /iven closed !allsO ′ ⊂ O there is an am!ient isotopy that takesO intoO ′ and does not move points outside a given open neigh!orhoodU ofO. (0) There is an isotopy 1 of a solid torus such that it it (e , r) ifr+1, and 1(e , r)+ i(t+s) 1 �(e , r) ifr< 2 for a givens. ′ *etp 1 !e another point onL 1 and 2$ a regular neigh!orhoodV of the knot. 3onsider a !allO that intersectsL 1 in an arc aroundp 1 ′ and which contains knotL 2. 4hrink it into a !allO aroundp 1 that is ′ contained inV then move it alongV to reach the pointp 1 using (0). Then e$pand it !ack using (,). This descri!es an isotopy that takes the connected sum with respect to pointsp 1 andp 2 to the one with ′ respect top 1 andp 2. � Property 1.19. The connected sum is a commutative monoid opera- tion on knots: (1) the unknotU is the neutral element:U-K∼K∼K-U, (,) it is associative:K-(K ′-K′′)∼(K-K ′)-K′′, (0) it commutes:K-K ′ ∼K ′-K. We will see later that - has no inverses% the only inverti!le knot is the unknot. Example 1.20. The connected sum of two right trefoils is the granny knot whereas the connected sum of the left and right trefoil is the s&uare knot% granny knot square knot 3 There is an empirical proof5 that these two knots are not e&uivalent% shoelaces tied as the s&uare knot are more dura!le that those tied as the granny knot. ,. Simple invariants It is a di6cult pro!lem to see whether two knots/links are e&uiva# lent or not. 4uch &uestions can !e often answered !y computing link invariants% certain &uantities that are the same for e&uivalent knots. There are two sources of link invariants% • topological invariants which are computed directly from a link and are preserved under am!ient isotopies and • combinatorial invariants which are computed from a link dia# grams and are preserved under 7eidemeister moves. The former can !e easily de2ned and are usually very powerful !ut very hard to compute. 3ontrary com!inatorial invariants are harder to de2ne (one has to check all 7eidemeister moves) !ut usually easier to compute. The following is an e$ample of a full invariant which takes different values for different knots !ut completely useless in practice% there is no good way to compare such invariants algorithmically. Example 2.1. 9e2ne the knot type :K; ofK as the e&uivalence class of its diagram modulo 7eidemeister moves. In what follows we construct other more tacka!le invariants. ,.1. Crossing an! unknotting num#ers. Example 2.2. 9e2ne the crossing num!er c(L) of a linkL as the min# imal num!er of crossings among its diagrams. It has the following properties% (1)c(L) + ( if and only ifL is an unlink (,)c(K)� 0 for a nontrivial knotK (0) the crossing num!er of a trefoil is 0. It is very easy to 2nd an upper !ound forc(K) (any diagram gives one) !ut very hard to computec(K). 4 Proposition 2.$. The crossing number is subadditive:c(L-L ′)� c(L)<c(L ′) for any linksL andL ′. Proof. 3hoose minimal diagramsD andD ′ of the linksL andL ′. Then D-D′ is a diagram ofL-L ′ withc(L)<c(L ′) crossings.� .dditivity ofc(L) is an open pro!lem% there is neither a proof nor a counter#e$ample. Problem. Is it true thatc(L-L ′) +c(L)<c(L ′)= 3onsider the following move called the crossing change% With this move we can reduce every link diagram to a diagram of an unlink. Proposition 2.%. LetD be a diagram of a linkL. Then there is a diagramD ′ of an unlink obtained fromD by changing some of its crossings. Proof. 3onsider the regular pro)ectionπ(L) of the linkL used to con# structD. It is a union of loops one per component ofL. >ick one loopγ parametri?ed asγ(t)+(x(t), y(t)) witht∈ :(, 1; and consider a particle P moving inR 3 a!oveγ in a descending way e$cept near the endpoints where it goes &uickly upwards i.e.P(t)+(x(t), y(t), z(t)) wherez(t) has the following graph% z(t) t ( 1 ThenP is an unknot (e$ercise% 2nd the isotopy). 7epeat the a!ove for each component ofL putting each of them !elow the previous one. This results in an unlinkL ′ with the same pro)ection asL. @ence 5 their diagramD andD ′ have the same shape !ut they may differ at crossings. � Definition 2.5. The unknotting num!er u(D) of a link diagramD is the smallest num!er of crossing changes re&uired to reduceD to a diagram of an unlink. The unknotting num!er u(L) of a linkL is the minimal unknotting num!er of its diagrams. Example 2.6. The unknotting num!eru(K) + ( if and only ifK is an unlink. Aoth a trefoil and a 2gure#eight knot have unknotting num!er 1% −→ −→ The unknotting num!er of the cin&uefoil is at most ,% −→ −→ ,.,. &inking num#er and 'rithe. 9e2ne the sign sgn(c) of a cross# ingc as follows% < − This convention follows the right-hand rule% !end slightly the 2ngers of your right hand e$cept the thum! which you should make orthogonal to the rest. Thinking of the thum! as the underpass and the other 2n# gers as the overpass they de2ne the orientation of the positive crossing. Definition 2.7. The writhe w(D) of a link diagramD is the sum of signs of its crossings. The linking num!er lk(L1,L2) of componentsL 1 andL 2 of a linkL is half the sum of signs of crossings !etweenL 1 and L2. 6 Example 2.8. w+<0 lk + <1 w+−0 lk +−1 The linking num!er is a link invariant which implies that the two @opf links are not e&uivalentB they are also nontrivially link. Cn the other hand writhe is not a link invariant as its value is not preserved !y the 2rst 7eidemeister move. Proposition 2.9. The linking number lk(L1,L2) is preserved by Rei- demeister moves. The writhew(D) is preserved by the second and third Reidemester move, but not by the rst one: w +w <1 andw +w −1. � � � � � � � � Proof. The second 7eidemester moves creates or destroys two cross# ings of opposite signs whereas the third 7eidemesiter moves preserves the amount of positive and negative crossings. The 2rst 7eidemeis# ter move only changes the num!er of self#crossings so that the linking num!er is not affected. � Definition 2.10. framed knot is a em!edding of an annulusS 1×:(, 1; intoR 3. em!edding of several annuli is a framed link. "ramed knots/links also admits diagrams. Dsually we re&uire that the pro)ection is one#to#one on each section{p}×:(, 1; of the annulusE if so we say that the diagram has a !lack!oard framing. There is a variant of 7eidemeister Theorem for diagrams of framed links% we 7 are still allowed to perform the second and third move !ut the 2rst one is replaced with ←→ Fotice that the other con2guration of kinks already follows from the sec# ond and third move. Example 2.11. It follows from >roposition ,.G that the writhe is an in# variant of framed knots. Aecause the two trefoils have different writhe they are not e&uivalent as framed knots. ,.0. Colorings. Definition 2.12. 0#coloring of a link diagramD is an assignment of a colorEred green or !lueEto each arc ofD such that at each crossing either all three arcs have the same color or all three colors meet. We say thatD is 0#colora!le if it admits a coloring !y all three colors. Example 2.1$. The standard diagrams of a trefoil and unknot are 0#colora!le% whereas the standard diagrams of the unknot and 2gure#eight are not.
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