From 11a97cecc9767a10f7963da9a81fdb55b4fcec2c Mon Sep 17 00:00:00 2001 From: Maria Marchwicka Date: Fri, 7 Jun 2019 18:57:28 +0200 Subject: [PATCH] sum of concordant knots --- images/concordance_sum.pdf | Bin 0 -> 36217 bytes images/concordance_sum.pdf_tex | 68 +++++ images/concordance_sum.svg | 469 +++++++++++++++++++++++++++++++++ lectures_on_knot_theory.tex | 32 ++- 4 files changed, 566 insertions(+), 3 deletions(-) create mode 100644 images/concordance_sum.pdf create mode 100644 images/concordance_sum.pdf_tex create mode 100644 images/concordance_sum.svg diff --git a/images/concordance_sum.pdf b/images/concordance_sum.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6922a7dcf2c9e89cbee90471d941e7b458d2a5fc GIT binary patch literal 36217 zcmd^o2|!Hk_kY?;DM>^!wAq?wxwC1ZvXn$rghH$KZAyzQB~g)mDcK{kX5aTWOChMYO6rWpBgskM1XzYU~=gKZ0R?zp^2*?695jTUARNO=X+-x#u*GjvdA^8*R#jxCr^kz z%hPne7VWOwxX$uWr?k$_9bWXDa@)1@i_WQp|1D9tiY>oa#NpFCa!N!bv)%~Qn1X#D#+8PawW5k zEb?S-zW|>|SKp1K7zg-K8hm|Qx#AN|D;jW0H$17o8GJ4eB2QYL$g$sC3RmnmokoF8z!X>y zI)eh6panPrZFKhc^a%)^1Sf`5`m-o3bU#I~(`~p{ps#lnwlkJT;bHsCV(D}WY-m6D z&ZNL5I0g5Pa3UIo z!QjxSEQG;fuqa#_kILaNm`onU1M6!W78(Ud^C{Rxv8KV1VbS3P%HWs7TG9R3Q^HL| zdWQ1D(Mx;B37hRHQIXNuO`)A^ohfDmd?yEZ`;HuJgLV;6VWclVEIQKLmruco*YuPq zLD1o9dPo7qu=^4SCX>s;eo7Nd?B1fiqA&st3X2TEdfMQ$VDKmigCTr;E{#j!FeIq) zuRS_EB6e%iIMI`aF^H`-45^v7C;Y_;Cm^4bXB4QTP^?clhr%H+3-l0arI3+@p{{hb zlBd0&D7}7fkiR%O#%`cXOsbnp2YH)b>-TD%xMEn^ElRt~-TZURM;4FX)YiU#|DtBi z!6OxKwuT@3G^21s#eD~U=Dpc5p99kBE&AQdz2YM@VWA{&NamcgqDzCdS zGIxoNR|+lFq_XPtoJ{Hlr6C>(L(ekuLpNLbP;GsNEPIqPq}}$EIek~%bYPetb-AXx zZ{*R3JYO~MGf~Eena)QiTAN-_zN0$k)x0^>sgvE74lT6w)?9pjyN$)-{0n{aFV5kN zI{y4@+u5%DI4^T<=(V?1R&(e?>+8LA-YV>ys4i8bh?ylin*v*t0V0k@?@ERX zO~yuZOVHX$K0cv-ew!C$7j9hRzjnd+ZEGiZ`fqx;=t}$Z3Nm}gE5DF37O1Kc9}7oM zeh@}_FcsFp;r^c3kHHj7SVp3pAhds|Arn7Vq&o2vpr@kHF>Q!7po$awg6dK1i?zKo z#m_UCkD)i=c~KiLeqJdiheg9q&|-_j<}xS+h7`&pKPj32LCXJtX!0m@ z95dpBK^%|oiRKoeX!4a`kq3i87O!SS6Gz2xZJ`J?fomZ&l_r|lC2)u8$3&AsRNaH3 zL%pK{!a^}dVPMiYAS}c)AQZ#6bRn3fp+Bi?^aGlJ%4Tu73_#BCg~sCwJJElj(=gB( zO^?VRK4Ob*g~nsSe{Atf^FR@RD!|RM_?3hp0b~w7C=M1l6ip9~?=rIu3l58P4)^p1 zS3nG9i}r^_`tadVO_^KV=0m&q28QE&n> zeFZyUbYbY>%qZB$|6q?I*e#?F1sgzY#9! zEC!Rpph2{Q$3=h%aK00HVdxFP2OS47xy73WdjviS=8c$Po{5ze-T?+y)Kkc;O)Y}M z`TpoFi2PdOk!D24vdQrywnBF^00>e%PIQl$(}^}T`JDuAf!l;lWB5#dxD>@m5J(0hRCx>(Ac>;}qPiR!9TE^4#t(?XSk-JW1=T??@=Y>8 zK3G9A;#ZHnaV4l?i&3MyGSwpt@*Ml^25urdo7)Cg>*! zrQn=tiz{drB4kx@6-j(XTwVkoF(z4p?}UeT^o)uOKw}{c@HjX$I4f%GOqo=Kh0s9} zve^tOjm_YIX{FN{G%B6R;PF_3dl6_~f{gqEeazreX;yZ0SRJE;BdFshZiK*yBs>45BqazmCh1M;{}0iW7$U?~ zi~toeiHJ%VxoQg0HRnp;3{3>WKWYGeAiDKiR6c=IaE64xs2UH48^Rb%82yBHpb$E2 zMBV<9c?Npkl`kjV5%BT(W&&3PCD77l$Ua4@@Fn#E0~o_5ouSqJ1DL zQv7&n_5r(0OY8&Y*bD8JkSTH7gpSMq!ju9Sg2kZ#fnWhBp+Xn&9Vj_W2LT_5EOGD3 zUM|IEY_>CMj(5`7qr)$ZR<2a7l$9B_U7=bE#f8WU?wH9CQCtWrzY$mEGuaf_1WHiA zi(;`4CWitWbU-;3Hlc?@)Q>{4VU8Ha&cA?cxVy@r;1L2u5~9IW28nL%9nDo`M`*ry zs(on2qtBDejzGyC$c?q}jU9obQEOm+;f zXx*6X=l%;z_RKFi80UFPoq744(z?r(15(CDNGKQ^Frz~HVRw(S9Y!z_FN2~GSXQ&9 zpf`=mr7?Nna^XP;!4G`6#D)bMDj_$M6{ldJ=u8|ZH#bUT{rW9d{>y@>Qey@Jjf`ty zCW``F6H5aHMWWX}?U?H6{+f!Prl!>>l-xh|;8@>PYdh}HPS+f$sWn$IVQRY+&#W;I zsb>ybSuqbQ*0)jFZhl5R>FIKdm1iprrWtZWk5|Rd-1Tr+Ta!Lc=Lu zH0=8Hv^TxzY)6HV0gdj`V_4B6h7-++26O8!yJ(10k0FnTJVHUsghsUjvSGvq4%IsF zq%5w`E+RrG(*%OQ2@^4vAi_ip&4?>Ak!{747am8zuG12nh!EowLf#X(k5X|W(XNCn zG+(rYu9nPSv)*ji*cSmkpUQTQn4z-5XWb^_@`iUB&f2XqZ{7^u*RS6#Z{MzWU1?zv z!xPqD^36J}b9sR4`MSir4rrrgv@hN*7HoHu$GD zrLUUH%zHXdpS)Q(dBX@K`;*JPj^~yXdQeN-w7cGalySYT)|0_L1=cQF)~of8Dr7d~=rH@;{=`&#usU{n*@oJqJLkWD*k5yb15C{~FBFJWX4lT*Gw!ll3_KoxM9nHY>_F(|O1d(C1Yo&lTSJDUQQ zQE(KTR{*bB!Ut>;gI}E6g7Gsh0RI|*qaZj53C~H?=wv-DbSw}A6W!(%JXxI!@G7Jy z31=2L00S5-Q91w~R{PhU{8#0+pau!WgF&j{8}WN@l__|6 z`33BzEnc9xE4Ax_2Qm95>6%g9hdFE<`7CqT1O2kHX~!+DUd|jg-L@)$?~tpsZo;dP^K{R4Zqs&nBVhg@lxFzS$Tm(#|6t-M|_)0QMJLq^}tc3gfgKRBXe zpoj5J_jh(%=iVMW+OYkNdlvVftyqywTh{i+ytMzZrKs-!dRqI(4)T^63Y zm!{R8IxEtjmTcyv)uz;AUMEf`%bDtwqjw88Z$0==)%g(3UjAcMXCA*=QN|k6@mMRx zsJhwK?Ygc!IiRSt{e=;61IM>nTk>l3jQ*RC#lkZOt?TK)%e6i)*YotA1G^LCmtD}e zzpknqJkyO|PhiVW@)P$f@p5?^J7(W* z6W77KF{W3}mrr}-RzC|ZnHO{X0$-o3dsyT^1Ti)xG!5LJ<5)Umy>UfcN{7Lw^SDAD z{wtaGr;=7|ZEY1=i;`AgGGH&j1y{RAvNESSNy}Nm^rg5}mqzrToEiV`5nZa2D{gRj z6c%RY1aTXprufdpc%W!ve6jKyvH-A&4k@m)i4cvLvx3PA5uIUHReT(!p)>3fEkS26 zPb=weN#l>7%p_yOJd7#uIfKK73$b|!OdW!kNz57j%Pa|=sM7M36?N%13r@p z0%`ymD@;Mg%1Kk@w`5+xu#5P3N~8Nv%H@BL?%$V7HiH72bPN6+x%A5sCaeT1ONjyl zz>$=J=HwC+7csePLeAexSivsQ61l_>G0Gn}jz-b!FPJR+yOKB9LYMVtp0U~J%OA^7 z!I)DpUQIxqO=J)O#$k&l4y3xjs1HPj^kl3qJ@jJAWUyiOmqBC0q5y=7(7E91^Vn2` zhOoIDNTgv=xjZ&UI3F&WPa$xcm}S8S2mYcw_&=v^z?fsh{+C2$2&o&)5m@Smk)pf{ zSzTizCJ@J_)D1K*tR<-%7%vdD2j)+UTLCfJ3vxHG>$F6BV6Kk%c?of{Zw035R4$Xo zrLj=6jUXI!T7=4_^LQ}X26;d%7UXLnY!(&FIETxlm?8+9%H*-=@MJi~{vAPT<{z4m zM@JWEnUORn2&rj!$%s;52Z(?=7%-bBQi~ zdVAT#Jp1$Zj?APLLiLNSZtv0LNDmS*n zofYn~T& zPBL-L&Yv`!t#M{a&HBC?n?|kdtSUc3{>7lu*17Yx z{07G~))eR~^xom-zpm?51tXpm!4O&(pfMc+xB^PT6s!R zbW&^GI=3LRo{=Mv)Al3Z3|W*Jzjo%@sHDf%S@Exq#9usAlvUe*-|m1d=a+0ZE2b^$ z_qvDP-F1uZZrv5pp{^=l(|9g&O=H{+qgp8}CG@n2vm#h>0>(;$XT?UJfwhPd6@CH_ z(%BpqO_+A{JMiG2nqP-ChRpy%4$(N^LPW6czZh@-GWs?P2>%p;n@?ci5<{l~ox~MQ zJ^oK-M7kF(Rd0#R3MwP;L8VD#Ed^~&dw$SYCNK(qtl-4qz*HDt{Uxq)a|sLwzLCJ- zsjqgGtc|dNLro4mX;T6NEN>LrH(~hWo5&Ai2_o{t@QJu36B`_XGmTxRCCHCURGvcK z6Ja5#yhpSTAa)~l)xYh%dr)AfbCa{H7cN-frmf>a(W&YlG0Vm2ro!ihqKLB0 znD})|@0o{MeUA9dZA?h8%Go|*@H?;TdGQ}y25i4=HTvqao9B9Y9lLqe)9qMbdiOK? z+ALo0eT%i>=n~t4v97*3jy7q@1DF0YaHOSn%FXB1Dg419xAGnQ&dk4@mOA?7Db}%% zSxP0(_L){^8Smpbemy(-mFlVXa>hG1$)$w8j?%6l<2`J3=bJt{8e93#@@r0C_1v{R z(C=u_4c)EW9jDJt-W4=0Xy_}IGMa^>w_~zyyF+gGck9Irj`b+X3BPjaLdwlF;USJ|g!WLt90E;gr6M zlI7-~I>~ot_8IR>c}-t%JjGo7Zdb0A%b~67Y+5Zp*MHS*@5SBcXE-^nkaKw5E++1& zN6GvA&v6su1~5(4Pdo@51HFUq_Zrk6X1NA0w3Q$aZxERT>09)_-Aq~-Bk>|;`z2tUGmj~m(vYe-zn|wtDCeh!_s5x zy-}Y1jXdPcJSk&N(d;Qnz$o z;(1M-;P9S7{#q=b_Wfmw%0C~t8P{{?*U^l?0ncx+S1ylH_b6~Hez#kvXmI71^Gtbv zui_%}GS+O~xY$e<9 zxi-h&eq7!-WMPQ)0sAzS%dz^0<7IQxWtQEM-JyoCmLdC9kJ>gI<|@b=tQ)#fGFsi7VTb@&^jdr~EtHOK&^T)+Mq~&N8jmhB$f1yWQ z>R(+NGd#_@QT5Qe!n^zZhbYfW$ybtF*|}oKV!qR?)#oN&UYl5Y!*_zw!dJ7DcFm3S z-nKWRQb8sl@L5>QFzPj=idN*2Zz7k_VMpfN#7f;L%}oF5ZHp9VUGjf@-m#r)DxJ}b z-DgKZm8V8GgS_3DIs?LM7VfF=y-!(z7+p^~S9NJeXw>lim-@!en8?14q((;>M&_np zxa~NcuQg{^(%d_W*~iB8*es_MwasIOpG^^Li+la-s-|X)Pf2}8Zxc9y?1==yBy`h&K$!J+MDP2b7=A|}?CT^X`a#?!#@ay_TwLQU!LFwMlt(;NGRCwx{+u-!EE%8Ab^)7f9f zuI+8Ha-Mcg-uzl6{eIyoilr7oIq^@6&&ORk&_C|pw9h+^e%T@Z`J;~a zi~sPb;gEa5+u0KyUI?4LwClyE-S3-yiT;pZb*_E7?V>(q>Pexi?;fO7C0Nh#&_t?J z=j^1+F}{4DOu@RaXr76@`o8fm)#baT`;^&R|FiN!rHb`&ZMy0!O+$}kFBGhInYcz+ zFG$%~*Is_<^`bEC8$nh5XmifqNaZ|KpQ&o6Vp4Izn%2e=d0%Ah<8z{RSewX%t5K1~ zlwK*}k(#u}3v`a~i*Fn-H&M>qzNOLLrf}VQj#dR_Zs-s#Ims}K8YF}md@pWg_E*qX`zZ~RQ&_CJT z@X}Ylrkjb%0nbwV6~-C)t!HKA25L-QRLska**;9`n)Y264ee{Yjs(uEIB2)f?AX_e zM`Yp5yhm8Mid5mK!K%Cz2C{*yji$x zU(t>Qi^7+8OAmKeot3;vy|UiC@9dfRmghF;&awnG|-otZ{8& zdhyQbXJQ?Ow*J`3=0X0n`MHy{bmF^M*T1tKl2kJ;Y+M@i^0BHqwo72`5Q763JO_p4 z=ojoyKb_>5jSP*|m2u2q+24*DzfQYb_LOTOGF!cFD-4ZGUFc^R^nSyMD`{)&G<;+_d(>iN~D`^^zm^WjtAB!LPA!0(7 zV_N&!&I5=M?ngrgv#gyRcbATI zJbdj@tlrmFaeJrhk<}$8?%(HZ1|B8(3C>2Bu8Zn~(K+aZEa&e|Q~N$+Lm2?rn$#ii zwurGq7CHuYp=&1B_Z{d#X}s~gU%*QbO2HG;nH`=O8OjG&oL%tZ2vS#YO3kY{cgLvc zXDYcXKRta~GhII-y5LzoBYWqLh{H7vwFlHwopzo)ajQ1-#pqA*v#DJkWIbxVdTrHb ztCtU|9r9A{o_@S8AaHrLT)FMCidI?UwX!A-f5ucXw&o?aQ}YkFs21p#Hvg=GW6)tW z_2B3&Hcy={blR*IvC>8NF>gV-(hipZ>f|~bzvW}nDUN-%uJhlz^N^W_mhIBi<6ElE zMQfQp+o0SlIc2M{!+zuXIVH=Z(kJ^@c@=cu;9IJ-AvNwsc;n8yT5EQ`9o)y>J2l|^ zkh<2rb*C71*BThTxQMs;`E#2LFSkxA`VRVI-Wv>XRlFQ8TTnjv;G-EwlP-7gQu*i^YJG2B?J3-FYjaDkrWYS~y|GUZo`Pl^4{~+b1>IN}5 zMTbZ*`cJxc_)gxI|HSoz#I0kFD!SwWp+la+w_--bj7nT?#83h>(hW zMASvajo&wm^&QB>~g^}Eqik0lcm^nY~Tcf$zY)!&>v7u>6==Flo*6_1<~GP0V*Mz z;A1b<-eXs4iM_|XdhrtzCiT8G``b(vDFEGUDv!rTQ1$|TqS82Y;R<}Amio1N>knlB z5<`Y~L_{>)WH0A0rD^kO@(&SJWC=kbNDx)ZOeS4W9v3=GoytVpe_m1kSSateC`SUj z;F=NsqEE&xNtTt~-J?pAot)qQ;h}O(* zUV3l38S9!XpJ<$?u=V=dmD5;;8d9d&@sSm1(xwbcx>(^Rd}xVyOo}GOWAsg+3=8aw zD?I6%6i0I?3;h@!q;xIwBVc`p)Gc+|6R=%*f2$#c!l`m|l%}8TI1&cp}$zb3ip7!_X9U>+} zM&p=_DL+#hCL{O}1WfjwO3zX>32vL3*aVvp`~gkU;)2guKwA971cUG`O!fmAk5Fk7 zBBTf;#j;@u44p&evJn<9F_kf$MO$hiRJ;CSVe8ziu@M{+`Rt$LlkZvt%(&1 z7iv(v?(uJOYa#gm{pqQH8LR=H(edKnD44~O9RB5RSPiq$P7JIaRMj^*{=96eBcgO- zguziqEoB!+8><`0qa|9Gkc~+<1BxB{A&nZvQdBH9umV&-0BcyaXK_9DZLo%M4bknz zGAzW;AQh|$>Pr7OSfgV#$c1NaQGq=UtU`sAJGj2YOEe39h?i%^6Ey@>`Bi|B0ZU?G zO(UHF*|lsQKn4bcaZoOUiC7+oOM^6T7R-2Y;q_L6qN6z1{rakRLT6W00vx|k%S=Xd z8tuOh7Z)%MVL_Q5dFZMJJbZlnqg@) zhl^leAx!;eU?Ge?9k0&*cW5{wEMz$2un>_73q>{BVVH@qP>2>$M*a~8Q?Rgtz$sXI zmr%9fr;x6+njDHB^DyzbNa2}G%}1$o|9JE zLsdj&!xVx8zkHSIs=Ppv^u^HOcoi74xUI2VQI1s4rft|;lee9$4bJEODVK-UYZ9|?_l0R)wE7fKljaL3ACVY7J4t=+MnO>}y;gn`- za;a_dKd%w5k9wy#^KW$zJ6XBBFioiowE)wfN`6&J3RVu*=VxS1SL!X!Rs zIOAaOOT>>5RTAJ5VR;0Yc%ni47JE+&o-rx-fddzi7BfB_R@9^aAPR*ZjqedpMWi!f z?i~I#P0I-l22{juno<)naZ|MAS1>U)(+Oo^x!Clw@C`E}^apo_uYnL_;>C^`H)xu( z>Uje_W}JRM!K#&(R(hh5S7Il+a)6Fp>m674o31~z-h4C7`^b396IbHO6erEAGFE%- z^gzj*-58&9V1kn6momexyc_2g)3q%Z<_vqKTX~V+qs(`=VL~lQM!d+*ui${>fi$RJBJ(+p~Xg z%xDX}9iwRHHQSx?@JICaX%=S9wR!%&Y>K)?tZwyj!=(ePo9di?yaS6;sD5Ulqug#WSs6+`6pls)>UGpWNT z?MZqcdL-9l%;@W82jxaoH)xc!3g6MY_~k>#@;fggE9weX^?$E^;L#$=Li3VQwA2lH z2O|QIaObyYw?8*8G%ry-cBx&=*cwhn%9Oj?X53TB4t#i!9d@+c6pxRUs}5d&wKLW% z&os?e|75LtZLhwu9ja|=^&;rRo z=>JnP6iJc)Ynivds`w$yQ-Zi+fjU&t;9HaB8Ce+UGD5TD!Op6#yRO%~+EX;lNo`D8 zIxUT2*ipUV)2%~Ay%v@;oT8#FKU!SBoI88@@~{)DtLj#VPRr|D`e0hlN~Lu*Co~?_ zZtn4suYBxW{iBA@vyx{W*SGFkmAORzM9il!m#R0ho3~X28XTykZ5TY#wf{XIkE0ec zF8Q|>q`vN>pjn({{PJwh^~W-P7j=ha6|`09pFPF%e2T}(MPuB@Cs!g9|U6-hKzGl9R-{ABve9p^( zy%Jv!d|k+k-ZzIWyQh8bQ_n5kYyR2(I{G4KB{#0;8s^Sk33Bc`e5<-^`05&HKR9KX z>Rwgc_|=+UG}gVt)_TS8VKWi0SM5AsR$T4dL(i|GYhjv6Td(ZWAQKLI*T`EeR~1gq zZps$^qN9gerxz*mJM3(Ac*Mpv$9Hjtrv}QjGij~i8SU)A*jBBXZaTZu&Qr^3V#+V1 zCt(XX;tpFG#%y;`ceBS03HzWJMvAQ`#Q|*fsC5;Uq3X6bP zEUdl3?qU&;YO z^CR}s_>mA9!`u;ksnbY-=C%!shIhs@iO5V;q#k7#+elx}sIW*%4{y(a$S}~sbcD*K z^ze_03h!$Mw+M}sB0a8j^0OD;lc>kSb8v{X?%ZvA8(9n>tcpDQT zpO3*qOP-7Oqp{JXC-F8Y0!O+Ay!DsV26mX-#sV)>;(YKH7Si)FnJ8n4Ul-naNA3rN zA@xIFXd-beytI!rmIxayzA1hU1a5}Z21OQ0Z9KH3pZKvXCYmoH(Z)evXC~gp21r0| zLjjjWKQ@|CE71lC;F4_|w8*1GKQ0CW#gFA+8M+c}sFjd77GCg5x-N%>R$UN3mP4fG!#XMUxP(MlPtvB*fcbbCf>$jkog4)T$0a+kn}wr-X}`x z$7Ygb5K8rs`*9d#nsR{nlK01>lg9%Ja*}Au!)D3Eo{a%EO0plgH>6`32&BVET$jN@ zvzjGn#X}tt@ivI%kn9WtQdUX*;HANmV+m7j=o%UEd!|V_md+#Tdj>GRBu|29p>@o} z&j%A=k~|4he&}?lct0MEq+j5&WV%2`u_T|uydh~U8FVgM;70r!3>rz^7~p+N(ixru z#aZIVGU;eCi$oiO&KFCxp>KJWY$NL-212&+40wk)>ALWyXVU$#7$jL>u*k9sZ=5BM z2OFJvlej-Nyp4<0hDMwv`bm)wNWquFS7dnuUL#>MP+=t50_}|Ylj=f39CRG znJ0N{v=F8Ev5>zlIUYPLPKh=Gv!53=gel1bv%V4x*}CEC!NA+^zo&lVNw84&Cn31LlGY!eXgivexaOBop! s7KPS2M12bkp4*4|fuDe$4=?V?kMfL+!q7RB$zZTmjf@5kAEf&K0K+a0CjbBd literal 0 HcmV?d00001 diff --git a/images/concordance_sum.pdf_tex b/images/concordance_sum.pdf_tex new file mode 100644 index 0000000..d856807 --- /dev/null +++ b/images/concordance_sum.pdf_tex @@ -0,0 +1,68 @@ +%% Creator: Inkscape inkscape 0.92.2, www.inkscape.org +%% PDF/EPS/PS + LaTeX output extension by Johan Engelen, 2010 +%% Accompanies image file 'concordance_sum.pdf' (pdf, eps, ps) +%% +%% To include the image in your LaTeX document, write +%% \input{.pdf_tex} +%% instead of +%% \includegraphics{.pdf} +%% To scale the image, write +%% \def\svgwidth{} +%% \input{.pdf_tex} +%% instead of +%% \includegraphics[width=]{.pdf} +%% +%% Images with a different path to the parent latex file can +%% be accessed with the `import' package (which may need to be +%% installed) using +%% \usepackage{import} +%% in the preamble, and then including the image with +%% \import{}{.pdf_tex} +%% Alternatively, one can specify +%% \graphicspath{{/}} +%% +%% For more information, please see info/svg-inkscape on CTAN: +%% http://tug.ctan.org/tex-archive/info/svg-inkscape +%% +\begingroup% + \makeatletter% + \providecommand\color[2][]{% + \errmessage{(Inkscape) Color is used for the text in Inkscape, but the package 'color.sty' is not loaded}% + \renewcommand\color[2][]{}% + }% + \providecommand\transparent[1]{% + \errmessage{(Inkscape) Transparency is used (non-zero) for the text in Inkscape, but the package 'transparent.sty' is not loaded}% + \renewcommand\transparent[1]{}% + }% + \providecommand\rotatebox[2]{#2}% + \ifx\svgwidth\undefined% + \setlength{\unitlength}{2270.54270346bp}% + \ifx\svgscale\undefined% + \relax% + \else% + \setlength{\unitlength}{\unitlength * \real{\svgscale}}% + \fi% + \else% + \setlength{\unitlength}{\svgwidth}% + \fi% + \global\let\svgwidth\undefined% + \global\let\svgscale\undefined% + \makeatother% + \begin{picture}(1,0.35664263)% + \put(0.34523208,0.32026902){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.13180612\unitlength}\raggedright \end{minipage}}}% + \put(0.37403638,0.28673265){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.05553241\unitlength}\raggedright \end{minipage}}}% + \put(-0.24570519,0.33337987){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.07663471\unitlength}\raggedright \end{minipage}}}% + \put(0,0){\includegraphics[width=\unitlength,page=1]{concordance_sum.pdf}}% + \put(0.00755836,0.35865552){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.03301039\unitlength}\raggedright $K_1$\end{minipage}}}% + \put(0.31573503,0.30921292){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.11352441\unitlength}\raggedright $K_1\prime$\end{minipage}}}% + \put(0.11757955,0.35187197){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.15898166\unitlength}\raggedright \shortstack{Annulus $A_1$}\end{minipage}}}% + \put(0.1209483,0.44515339){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.11151923\unitlength}\raggedright \end{minipage}}}% + \put(0,0){\includegraphics[width=\unitlength,page=2]{concordance_sum.pdf}}% + \put(0.00912905,0.16388952){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.03301039\unitlength}\raggedright $K_2$\end{minipage}}}% + \put(0.30238413,0.11444693){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.11352441\unitlength}\raggedright $K_2\prime$\end{minipage}}}% + \put(0.11915025,0.15710597){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.16683513\unitlength}\raggedright \shortstack{Annulus $A_2$}\end{minipage}}}% + \put(0,0){\includegraphics[width=\unitlength,page=3]{concordance_sum.pdf}}% + \put(0.79479654,0.15057288){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.21876087\unitlength}\raggedright $K_1\prime \# K_2\prime$\end{minipage}}}% + \put(0.39316361,0.18038181){\color[rgb]{0,0,0}\makebox(0,0)[lt]{\begin{minipage}{0.07306307\unitlength}\raggedright $K_1 \# K_2$\end{minipage}}}% + \end{picture}% +\endgroup% diff --git a/images/concordance_sum.svg b/images/concordance_sum.svg new file mode 100644 index 0000000..abb8310 --- /dev/null +++ b/images/concordance_sum.svg @@ -0,0 +1,469 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + + + + $K_1$ $K_1\prime$ \shortstack{Annulus $A_1$} + + + + + + + $K_2$ $K_2\prime$ \shortstack{Annulus $A_2$} + + + + + $K_1\prime \# K_2\prime$ $K_1 \# K_2$ + diff --git a/lectures_on_knot_theory.tex b/lectures_on_knot_theory.tex index d759edf..2328b94 100644 --- a/lectures_on_knot_theory.tex +++ b/lectures_on_knot_theory.tex @@ -225,7 +225,7 @@ We smooth all the crossings, so we get a disjoint union of circles on the plane. \end{figure} \noindent -Note: in general the obtained surface doesn't need to be connected, but by taking connected sum of all components we can easily get a connected surface (i.e. we take two disconnected components and cut a disk in each of them: $D_1$ and $D_2$; now we glue both components on the boundaries: $\partial D_1$ and $\partial D_2$. +Note: the obtained surface isn't unique and in general doesn't need to be connected, but by taking connected sum of all components we can easily get a connected surface (i.e. we take two disconnected components and cut a disk in each of them: $D_1$ and $D_2$; now we glue both components on the boundaries: $\partial D_1$ and $\partial D_2$. \begin{figure}[h] \begin{center} @@ -295,6 +295,13 @@ $T(6, 2)$ link: \end{figure} \end{itemize} \end{example} +\begin{fact} +\[ +g_3(\Sigma) = \frac{1}{2} b_1 (\Sigma) = +\frac{1}{2} \dim_{\mathbb{R}}H_1(\Sigma, \mathbb{R}), +\] +where $b_1$ is first Betti number of $\Sigma$. +\end{fact} \subsection{Seifert matrix} Let $L$ be a link and $\Sigma$ be an oriented Seifert surface for $L$. Choose a basis for $H_1(\Sigma, \mathbb{Z})$ consisting of simple closed $\alpha_1, \dots, \alpha_n$. @@ -583,7 +590,9 @@ An oriented knot is called negative amphichiral if the mirror image $m(K)$ if $K Prove that if $K$ is negative amphichiral, then $K \# K$ in $\mathbf{C}$ \end{example} - +% +% +% \section{\hfill\DTMdate{2019-03-18}} \begin{definition} A knot $K$ is called (smoothly) slice if $K$ is smoothly concordant to an unknot. \\ @@ -612,9 +621,19 @@ For any $K$, $K \# m(K)$ is slice. \begin{fact} Concordance is an equivalence relation. \end{fact} -\begin{fact} +\begin{fact}\label{fakt:concordance_connected} If $K_1 \sim {K_1}^{\prime}$ and $K_2 \sim {K_2}^{\prime}$, then $K_1 \# K_2 \sim {K_1}^{\prime} \# {K_2}^{\prime}$. +\begin{figure}[h] +\fontsize{10}{10}\selectfont +\centering{ +\def\svgwidth{\linewidth} +\resizebox{0.8\textwidth}{!}{\input{images/concordance_sum.pdf_tex}} +} +\caption{Sketch for Fakt \ref{fakt:concordance_connected}.} +\label{fig:concordance_sum} +\end{figure} + \end{fact} \begin{fact} $K \# m(K) \sim $ the unknot. @@ -649,6 +668,13 @@ $A \cdot B$ doesn't depend of choice of $A$ and $B$ in their homology classes. \end{proposition} + +\section{\hfill\DTMdate{2019-03-11}} +\begin{definition} +A link $L$ is fibered if there exists a map ${\phi: S^3\setminus L \longleftarrow S^1}$ which is locally trivial fibration. +\end{definition} + + \section{\hfill\DTMdate{2019-04-15}} In other words:\\ Choose a basis $(b_1, ..., b_i)$ \\