robert planque- the role of knot theory in dna research

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Page 1: Robert Planque- The role of knot theory in DNA research

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Page 19: Robert Planque- The role of knot theory in DNA research

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Page 20: Robert Planque- The role of knot theory in DNA research

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Page 21: Robert Planque- The role of knot theory in DNA research

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Page 32: Robert Planque- The role of knot theory in DNA research

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Page 71: Robert Planque- The role of knot theory in DNA research

¤.� B f E �� f �� � g g ÷ � û$ü � ë ¤.�B f E � f �� � g g ÷ � û�÷ � ÷ ° � å a° � û ¤ � ÷ � û ù¤.�B f E �³ f �� � g g ÷ � û$ü � ë ¤.�B f E ³ f �� � g g ÷ � û�÷ � ÷ ° � å a° � û ¤ � ÷ � û ù¤.� B f E �³ f  e� � g g ÷ � û$ü � ë ¤.�B f E ³ f  e� � g g ÷ � û�÷ � ÷ ° � å a° � û ¤ � ÷ � û ùî'ï�ð`ðBñ s�uUp�i � �Xj y�j o��Ue f �Xp g�vX��y�p e p � �(i uU�XcUw9j cUwd uUp�e p i f9yvUj cU� d p g�i e fXo o �

j cUw�÷ ²$uUj i u�j o��9� ²�� �,o���f9o j d j �Xp�vX���Ue p �Uj f9hUo��Xe w9hUy�p c9d o Å,o j cUi p�d uUp�e p i fXy�vUj cU�9d j f9cj c(°,hUp o d j fXc(j o!�2fXo j d j �Xp û�j c9d fj d o�cUp w9�9d j �Xp$i f9hUcXd p e �U�Xe d�fXc�d uUpxf9cUpxuU�9cUg�ÅU�XcUg(j cd uUpj c9�Xp e o p � ��e p i fXy�vUj cU�9d p g�i e fXo o j cUw(f9c�d uUp�f9d uUp e `�È!��gUfXj cUwd uUj o�²�p�q2cUg�d uUpd ²!f�e p °XhUj e p g]i � fXo p g]vUe �9j gUo � ÷ A2êé ÷ âUû ûx�XcUg � ÷ âUû `"s�uUj o��9e wXhUy�p cXd�j o��2fXo o j vU� p�j c�X� �2d uUe p p�i �Xo p o$f9�"e p i f9yvUj cU� d j fXc(�9d��(Å����9cUg��$` ��$�U�Xe d�� e f9y ²$e j d j cUw�gUf ²$c§cUp ²�o �9p j c]e p � �9d j f9cUox²!p�wX�9j c§� j d d � p�j c,� fXe y�� d j fXc

hUo j cUw�vUe �Xj gUo�o j cUi pd uUp(vUe �9j gUo � ÷ A ó[ ÷ âUû û��9cUg � ÷ A ê[ ÷ âUûx�Xe p�cUf9d�e p � � d p g�²$uUp c]j di f9y�p o!d fd uUp��XcUf9d o�d uUp ���Xe p�y��9gUpx� e f9y�`

b d�o p p y�o�d uU�9d�o j y��U� �$�Xi d hU�X� � �$i f9y��Uh,d j cUw!d uUp!�9f9cUp o���f9� �UcUf9y�j �X� � e f9y�o i e � d i u�Ų�f9e �,j cUw��XfXhUe!²�� ��hU�(d fy�fXe pxi fXy��U� j i �9d p g�_ � vUe j gUw9px�XcUf9d o�o p p y�o�d uUpxfXcU� �(�X� �d p e cU� d j �Xp9`�b ��²!p�� j y�j d$fXhUe o p � �Xp oxd f(d uUpi �Xo p�d uU�9dxd uUpo hUvUo d e �9d phUcU�,cUf d o�uU� �Xpe p i fXy�vUj cU�9d j f9c�o j d p o�j c�gUj e p i d"e p �2p � d o�²�p!i �Xc�y��9�9p!o hUe p"d uU�9d�d uUp�i e f9o o j cUwxy��9gUpvX����e p i fXy�vUj cU�9d j f9c�j o��X� ²!� �Uo���f9o j d j �Xp��9cUg�i �9c�d uXhUo$v�p�hUo p g�� fXe�e p gUhUi d j fXc(d f�,cUf d o�²$uUj i u�uU� �Xpxd f�v2p�hUo p g�� f9e!d uUp�o �9p j c�e p � � d j fXc�`

À pxi �Xc�o p d$hU���wXp cUp e �9��o i uUp y�pxd fi fXy��Uh,d p$d uUp��9fXcUp o���f9� �UcUf9y�j �X�2f ���Xc9��,cUf dx� f9e y�p g�j c�gUj o d e j vUh,d j �Xp�e p i f9yvUj cU� d j fXc��Ue f,i p o o p o$²$j d u�d uUp�y�p d uUf,g�f ��e p �i hUe e p cUi p�e p � � d j fXcUo�� e fXy�d uUp�q2p � g¤f9�$i f9yvUj cU� d fXe j i o ` À p(o d e p o od uUp(� �9i dd uU� dd uUj oj o�fXcU� �§�9�9� j g§� fXe�e p i fXy�vUj cU�9d j f9cUo�f9�$gUj e p i d�e p ��p �9d o�²$uUj i u¤j o�d uUp(i �9o p�j cs�cUr9� e p o f9� �9�Xo p§÷ À �9o o p e y��9c�o��!fXÆ Æ �9e p � � j$a �9�X�Xû `�b c¤d uUj oi �Xo p(²�p�i �9c[�9� ²�� �,o�Xo o hUy�pxd uU� dx���f9o j d j �Xp�e p i f9yvUj cU� d j fXc��Uj p � gUo����2fXo j d j �Xpxi e f9o o j cUwU`s�f�XgUgUe p o o!d uUpx�Ue f9vU� p y¿f9��i �X� i hU� �9d j cUwxd uUp��9fXcUp o��2fX� �,cUfXy�j �9�U� f9e!�9cX��_ � vUe j gUw9p�,cUf d�y��XgUp�vX�[gUj o d e j vUh,d j �Xp�e p i fXy�vUj cU�9d j f9c[�Ue f,i p o o p o(²!p�i uUf,f9o p��]o hUvUo d e � d p�,cUf d�âüXútÄ ä ù � � � ù Ä ë ¢ ó ä ��`2s�uUp c�â(i �Xc�v2p�e p i f9yvUj cU� d p g�v9���Xi d j fXcUo!fXc(d uUpe p w9j fXcUo��(Å:�;�XcUg��$ÅX�9cUg��Ue f,gUhUi d o"²$j � �,v2p�i fXc,� f9e y�s�uUp f9e p y�},` a9`Xs�uUp�o p d hU��f9�d uUp�o i uUp y�p�j o�d fxe p gUhUi p!d uUp��XcUf9d�hUo j cUwxo �9p j ce p � � d j fXcUo�o hUi ud uU� d"²$uUfX� p!e p w9j fXcUo� ë ° ó ä ù�|��X±��v£��Xe pgUp � p d p g]j c§p �Xp e �§o d p ��`�s�uUj o�j o��2fXo o j vU� po j cUi p²�p�y�� ��Xo o hUy�p$d uU�9d!d uUpxi e fXo o j cUw9o�j c�d uUp o pxe p w9j fXcUo!�9e px��f9o j d j �Xpx�9cUg�uUp cUi pxy�� �(v2pxe p ��U� �Xi p g�vX�d uUp e p$cUp w9�9d j �Xp$i fXhUc9d p e �U�9e d!d fx� fXe y¿â ê fXcd uUp$f9cUp$uU�XcUg��XcUg�v9�d ²!f

�Xa

Page 72: Robert Planque- The role of knot theory in DNA research

�U�Xe �9� � p ��� j cUp o�d f(fXv,d �9j c�â [ ` À p�²$j � ��o p p�d uU�9dxj c�d uUp�e p gUhUi d j f9c�f ��p �Xp e ��e p w9j fXc� ë ° ó ä ²�p�wXp d��$e p i hUe e p cUi p!e p � �9d j f9c�` À p"²$j � �XcUf ²]y��X�9p"d uUp o p!j gUp �Xo�y�f9e p��Ue p i j o p `À pxo d �9e d�²$j d u(�Xc(�Xi d j fXc(f9c �(` À p$q2e o d$o p d$hU��cUp ²;o �9p j c�e p � �9d j f9cUo�� f9e!d uUp

�Ue f,gUhUi d$�,cUf d o `�É,p d�¤�¡ \ ¢ ÷ � û�üZ¤�£ � ê \ ¡ � � ¡ ¤ ¤ ¤ ¡ � ¥ � ¡ " ¡ Y ä ¡ � ¥ ê " ¡ � ¥ � ¡ ¤ ¤ ¤ ¡ � � ��� ¦ ÷ � û `q J K2Z"K2¡ O I O K2H[��r E,Q� `� "�âküLú\Ä ä ù � � � ù Ä ë ¢ ó ä �H~ ��" } ��u y#~ u " z (�" ������� "��B! " }x�! z � $ "�z � �#� (�" u�� � z�" } ��z � $ � &�~ ! ��(�" � x|&�� x�� w w � x�~ ��|�" ��! u " u�(���x�a�z � $ � &�~ ! ��(�" ! � �(�"Bz � � ! � ����� � } � ��" } ��u ��� ! ��z � w (�" ! � �#u`� � z*" } �*� ��� " u�A ó[ ÷ � � âUû%(���x�A ê[ ÷ � � âUû%(�z �¤ E �³ f �� � g ÷ � û�ü � ë ¤ E ³ f �� � g ÷ � û�÷ � ÷ ° � å a° � û ¤ � ÷ � û¤ E ³ f �� � g ÷ � û�ü � ë ¤¡ ë ¢ ÷ � û�÷� ÷ ° � å a° � û ¤¡ ä ¢ ÷ � û �î'ï�ð`ðBñ È!�§s�uUp fXe p y�},` a²�p��XcUf ²¿d uU� d�d uUp��,cUf d o�j c�°,hUp o d j fXc§�Xe pf9�!d uUp

� fXe yA.Y[ ÷ � � âUû"ü�útÄ ä ù � � � ù Ä ° ê2ä ù ¢9ù �a9ù Ä ° 客9ù Ä ° ó ä ù � � � ù Ä ë ¢ ó ä � �

ÉXd �Xe d j cUw²$j d u(d uUp�i �Xo pxd uUp�e p i f9yvUj cU� d j fXc�j o���f9o j d j �Xpx²�p�i uUf,f9o p�d uUp�e p i fXy�vUj �cU�9d p g�i e f9o o j cUw$�Xo�d uUp!f9cUp�hUo p g�� f9e�d uUp�o �9p j c�e p � � d j fXc�` È!��o p d d j cUwxâ ó üUA ó[ ÷ � � âUû²�p�wXp d

â ê ü�A ê[ ÷ � � âUû ùâ [ ü¿â �b ��²�p�²��9cXd�d f�o p d�hU����cUp ²�o �9p j c(e p � � d j fXc�� f9e�A ê[ ÷ � � âUû�²�p$uU� �Xp�d f�i fXcUo j gUp e�(gUj Â�p e p c9d�i e f9o o j cUw�d uU�Xc�d uUp�fXcUp�hUo p g�� f9e�A ó[ ÷ � � âUû�o j cUi p�d uUj oxj o$cUf9dx��f9o j d j �Xp�Xc9�Uy�f9e p9`�È�h,dxd uUp e p��9e p�U� p c9d ��f9d uUp e o�� p � d Ä�²!p�o d �9e d�o �Uo d p y��9d j i �X� � ��²$j d u�d uUp

��f9o j d j �Xp�i e f9o o j cUwXoxj cE� ä Å�v2p w9j cUcUj cUw�²$j d u�d uUp�fXh,d p ex� p � dxf9cUp9`�s�uUp c�vX��o p d d j cUwâ ó üUA ê[ ÷ � � âUû�²!p�w9p dâ ê üvú�Ä ä å[_Xù Ä ë ù � � � ù Ä ° ê2ä ù ¢9ù �a9ù Ä ° 客9ù Ä ° ó ä ù � � � ù Ä ë ¢ ó ä ��ùâ[xüvú�Ä ä åka9ù Ä ë ù � � � ù Ä ° ê2ä ù ¢9ù �a9ù Ä ° 客9ù Ä ° ó ä ù � � � ù Ä ë ¢ ó ä � �

s�uUj o�i fXy��U� p d p o!d uUp��Ue f,f9� ` �s�uUp o po �9p j c�e p � � d j fXcUo$� p d oxhUoxw9j �Xp��(�Ue f9��f9o j d j fXc�fXc�d uUp��9f9cUp ox�2fX� �,cUfXy�j �9�

f9�"�XcUf9d o�A ó[ ÷ � � âUû!�9cUg|A ê[ ÷ � � âUû$`�9_

Page 73: Robert Planque- The role of knot theory in DNA research

q J K2Z"K2¡ O I O K2H[��r E,P� `� "�â�~ �*(x_� ~ z ! x �������� "��B}�! $ }�! u*z � $ � &�~ ! � (�" � x�~ �$||" �B! u " u(���x � z � $ � &�~ ! � (�" ! � ��(�"���� � } � ���*�'}#(�{��¤ E �³ f �� � g ÷ � û$ü � ë ¤ E ³ f �� � g ÷ � û`÷ � ÷ ° � å a° � û ¤ � ÷ � û¤ E ³ f �� � g ÷ � û$ü¿â äe§ �_ ÷ ° � å a° � û�÷©¨ � ÷ ° � å a° � û�÷¤} � ë ª � ÷

â ë.§ �_ ÷ ° � å a° � û�婨 � ÷ ° � å a° � û�÷¤} � ë ª � �`} � z �$â ä (�� xâ ë (�z ��y��#! � y � w ��x�� " � z &|! ��� x�~ �|¤¡ [ ¢ (���x|¤¡ ä ¢ �î'ï�ð`ðBñ�À j d u(d uUp�gUp q2cUj d j fXc(f9��¤�¡ \ ¢ �Xv2f �Xp�²!p�y�� ��o p dxhU���e p � � d j fXc

¤¡ \ ¢ ÷ � û�ü � ë ¤�¡ \ êUë ¢ ÷ � û�÷ � ÷ ° � å a° � û ¤¡ \ êUä ¢ ÷ � û ù�||�G�|�tÄ ä å[a �s�uUj o�j o���o p i fXcUg�fXe gUp e�e p i hUe e p cUi p!e p � �9d j f9c$²$j d u�i f,p J�i j p cXd o � ë��XcUg � ÷ ° � å ä« ¬ û²$uUj i u��9gUy�j d o���i fXcUi e p d p�o fX� h,d j f9c(� f9hUcUg�j c(�Xc9��d p �,d v2f,fX�(f9c�i f9yvUj cU� d fXe j i o Ä

¤�¡ \ ¢ ÷ � û�ü;â ä � \ ÷[â ë � \²$uUp e p � �XcUg � �Xe po f9� h,d j fXcUoxf ��d uUp�p °,hU� d j fXc�ø�ë!å � ÷ ° � å ä« ¬ û ø�å � ë�ü�|U`�b cd uUj o$i �Xo pxd uUp o p�e f,f d o��Xe p�p °,hU�9��d f

� ü �_ ÷ ° � å a° � û�÷©¨ � ÷ ° � å a° � û�÷¤} � ë ù� ü �_ ÷ ° � å a° � û�婨 � ÷ ° � å a° � û�÷¤} � ë �

s�uUpâ ä �9cUg�â ë �Xe p�hUcUj °,hUp � ��gUp d p e y�j cUp g�vX�(d uUp�j cUj d j �X��i fXcUgUj d j f9cUo Å2j ` p `2d uUpq2e o dx�2fX� �,cUfXy�j �9� o!f9��d uUp�o p °,hUp cUi p Å�vX��o fX� �,j cUw�d uUp�o p dxf �"p °XhU�9d j f9cUo¤�¡ [ ¢ ü�â ä ÷[â ë¤�¡ ä ¢ ü�â ä � ÷[â ë �

s�uUj o$i fXy��U� p d p o!d uUp��Ue f,f � ` �� r

Page 74: Robert Planque- The role of knot theory in DNA research

À p�uU� �Xp�cUf ² e p gUhUi p g[d uUp�i f9y��Uh,d �9d j f9c[f ��¤¡ \ ¢ d f§d uU�9d(f ��¤¡ [ ¢ �XcUgt¤¡ ä ¢ `s�uUp o p�XcUf9d o$�Xe p�f �"d uUp�� f9e y úzaXù õ�ù  ä ù � � � ù  � ���9cUg�y�p�v�p�o p p c��9oxe p �Xi d j fXc�Ue f,gUhUi d o$f �"��o hUvUo d e � d p��XcUf9d�ú  ä ù � � � ù  � ��²$j d u��Xi d j fXc(��p e � fXe y�p g��9d��(`À pd u,hUo��Ue f,i p p g]gUj o i hUo o j cUw��9i d j f9cUo�� dB�(`"�$cU�X� f9wXf9hUo � ��d f�d uUp(�9�U�Ue fX�9i u

�Xv2f �Xp²!p�o d � d pd uUpo �9p j c§e p � � d j fXcUo$� f9e$d uUp�,cUf d o�y��XgUpv9��d uUj ox�9i d j f9c�j c�d uUp� fX� � f ²$j cUw��Ue fX�2fXo j d j f9c�`q J K2Z"K2¡ O I O K2H[��r EX�^ `� "�âküLú\Ä ä ù � � � ù Ä ë ¢ ó ä �H~ ��" } ��u y#~ u " z (�" ������� "��B! " }x�! z � $ "�z � � � (�" u�� � z'" } ��z � $ � &�~ ! ��(�" � x|&�� x�� w w � x�~ ��õU" ��! u " u�(�� x�a�z � $ � &�~ ! ��(�" ! � �(�"�z � � ! � ����� � } � ��" } �'u ��� ! ��z � w (�" ! � ��u�� � zBA óé ÷ âUû�(�� x%A�êé ÷ âUû�(�z � �

¤ E �� f � g ÷ � û$ü � ë ¤ E � f � g ÷ � û`÷ � ÷ ° � å a° � û ¤ � ÷ � û¤ E � f � g ÷ � û$ü � ë ¤ E � � f � g ÷ � û�÷ � ÷ ° � å a° � û ¤ � ÷ � ûî'ï�ð`ðBñ È���s�uUp f9e p y�}U` a�²!p��XcUf ²;d uU� d$d uUp��Ue f,gUhUi d$�,cUf d o$�Xe p

ú;aXù õ�ù Ä ä ù � � � ù Ä ë ¢ ó ä �fXe

ú;a9ù õ(åka9ù Ä ä ù � � � ù Ä ë ¢ ó ä ��ùgUp �2p cUgUj cUw�f9c�d uUp�o j w9c�f9��d uUp�e p i f9yvUj cU� d j fXc�` Æ s�uUp c�j ��²�pxd �X�9p

â ó üUA óé ÷ âUû�ü�ú;a9ù õ�ù Ä ä ù � � � ù Ä ë ¢ ó ä �²�p§o p p§j y�y�p gUj �9d p � �¤d uU�9d�â ê üHA�êé ÷ âUû(�XcUg;â�[¤ü â2`$n2fXe�d uUp§o p i fXcUg�o �9p j ce p � � d j fXck²!p]y�� ��gUe � ²��[�Uj i d hUe p§f ��A2êé ÷ âUû(�XcUg;o p p§d uU� d�j d � o�p °,hUj �9�9� p c9d�d fú;aXù õxå�aXù Ä ä ù � � � ù Ä ë ¢ ó ä ��`,�$f ²kd uUp��Ue f,f9��j o�i fXy��U� p d p g�v9��e p wX�9e gUj cUw*A óé ê2ä ÷ âUû�Xo$â ó �XcUg�e p �2p � d j cUwd uUpxq2e o d$�Xe w9hUy�p c9d ` �s�uUp��9f9cUp o$�2fX� �,cUfXy�j �9��f �"���,cUf d$²$uUj i u�uU�Xo�v�p p c�e p i f9yvUj cU� d p g�� d���y�� �

cUf ² v2p�i f9y��Uh,d p g[v9�[yhU� d j �U� p(e p gUhUi d j fXcUo�f9�$d uUp�õ�i e f9o o j cUwXo�j cD� ë `!�$o(�9cp �,�Xy��U� p²�p�i f9y��Uh,d p�d uUp(�9fXcUp o�2fX� �,cUfXy�j �9�"f9�!d uUp�q2e o de p �9i d j f9c]�Ue f,gUhUi df9�d uUp�hUcU�,cUf d `�s�uUp�e p o hU� dxj o!� f9hUcUg�j c�d uUp�cUp �Xdxì�e fX�2fXo j d j f9c�`­ g>r n p z o m bÞxWb d u m p m a p z u { n p c�m l { b o®C°¯ _ ± ² ² ²:± ¯ } ³´�_ ± µ�± �¶D d c�{�C¯ _ ± ² ² ²e± ¯ } ³�´�_ ± µ�· � ± �¸D ~ z s®s a b m bDd o b${ p c bGn p e7w u b s b u q�d c�d u p � p z�mE~ q-o b � d o { l c �s a b eGd m�CE� ± µ�± ¯ } ³�´�_ ± ² ² ²e± ¯ _ DEd c�{�CE� ± µ�· � ± ¯ } ³�´�_ ± ² ² ²e± ¯ _ DAo b m w�b n s l | b u q �

� }

Page 75: Robert Planque- The role of knot theory in DNA research

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Page 80: Robert Planque- The role of knot theory in DNA research

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Page 91: Robert Planque- The role of knot theory in DNA research

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