Virtually symmetric representations and marked Gauss diagrams - PowerPoint PPT Presentation

Virtually symmetric representations and marked Gauss diagrams Manpreet Singh Indian Institute of Science Education and Research (IISER) Mohali, India Joint work with Valeriy Bardakov and Mikhail Neshchadim Department of Mathematics, IISER Mohali August 19, 2020 1 / 86

Classical links A link is a smooth embedding of finite disjoint circles S 1 in 3 -sphere S 3 . 2 / 86

Classical links A link is a smooth embedding of finite disjoint circles S 1 in 3 -sphere S 3 . Two links L 1 and L 2 are said to be ambient isotopic if there is exist an ambient isotopy H : S 3 × [0 , 1] → S 3 such that H ( L 1 , 0) = L 1 and H ( L 1 , 1) = L 2 . 3 / 86

Link diagrams A link diagram is a generic projection of a link L onto a plane with over- and under-crossing information at double points. 4 / 86

Link diagrams A link diagram is a generic projection of a link L onto a plane with over- and under-crossing information at double points. Figure: Trefoil knot diagram. 5 / 86

Link diagrams A link diagram is a generic projection of a link L onto a plane with over- and under-crossing information at double points. Figure: Trefoil knot diagram. Two link diagrams D 1 and D 2 are said to be equivalent if they are related by a finite sequence of moves shown below, upto planar isotopy: R 1 R 2 R 3 Figure: Reidemeister moves. 6 / 86

Link diagrams A link diagram is a generic projection of a link L onto a plane with over- and under-crossing information at double points. Figure: Trefoil knot diagram. Two link diagrams D 1 and D 2 are said to be equivalent if they are related by a finite sequence of moves shown below, upto planar isotopy: R 1 R 2 R 3 Figure: Reidemeister moves. Theorem (K. Reidemeister) Two links are ambient isotopic iff any diagram of one can be transformed into a diagram of the other by a sequence of Reidemeister moves. 7 / 86

Classical link group Classical link group of link L : Fundamental group of link complement π 1 ( S 3 − L ) and it is a link invariant. 8 / 86

Classical link group Classical link group of link L : Fundamental group of link complement π 1 ( S 3 − L ) and it is a link invariant. x 1 x 3 x 2 Figure: Trefoil knot diagram ( T ) . 9 / 86

Classical link group Classical link group of link L : Fundamental group of link complement π 1 ( S 3 − L ) and it is a link invariant. x 1 x 3 x 2 Figure: Trefoil knot diagram ( T ) . b a a c c = b a 10 / 86

Classical link group Classical link group of link L : Fundamental group of link complement π 1 ( S 3 − L ) and it is a link invariant. x 1 x 3 x 2 Figure: Trefoil knot diagram ( T ) . b a a c c = b a = a − 1 ba. 11 / 86

Classical link group Classical link group of link L : Fundamental group of link complement π 1 ( S 3 − L ) and it is a link invariant. x 1 x 3 x 2 Figure: Trefoil knot diagram ( T ) . b a a c c = b a = a − 1 ba. π 1 ( S 3 − T ) = � x 1 , x 2 , x 3 || x 2 = x x 3 1 , x 3 = x x 1 2 , x 1 = x x 2 3 � . 12 / 86

Virtual links L. H. Kauffman, Virtual knot theory , European J. Combin. 20 (1999), no. 7, 663–690. A virtual link diagram is a generic immersion of finite disjoint oriented circles into a plane where double points are either classical crossings or decorated with a circle around it, called a virtual crossing . Figure: A virtual knot diagram. 13 / 86

Virtual links Two virtual links diagrams are said to be equivalent if one diagram can be transformed into the another diagram by a finite sequence of generalized Reidemeister moves . Generalized Reidemeister moves:= Reidemeister moves + 14 / 86

Virtual links Two virtual links diagrams are said to be equivalent if one diagram can be transformed into the another diagram by a finite sequence of generalized Reidemeister moves . Generalized Reidemeister moves:= Reidemeister moves + the moves shown below. V R 3 V R 1 V R 2 V R 4 Figure: Virtual Reidemeister moves. 15 / 86

Virtual links Two virtual links diagrams are said to be equivalent if one diagram can be transformed into the another diagram by a finite sequence of generalized Reidemeister moves . Generalized Reidemeister moves:= Reidemeister moves + the moves shown below. V R 3 V R 1 V R 2 V R 4 Figure: Virtual Reidemeister moves. An equivalence class of a virtual link diagrams is called a virtual link . 16 / 86

Virtual links Two virtual links diagrams are said to be equivalent if one diagram can be transformed into the another diagram by a finite sequence of generalized Reidemeister moves . Generalized Reidemeister moves:= Reidemeister moves + the moves shown below. V R 3 V R 1 V R 2 V R 4 Figure: Virtual Reidemeister moves. An equivalence class of a virtual link diagrams is called a virtual link . Theorem (L. Kauffman) Virtual links are proper generalization of classical links. 17 / 86

Gauss diagrams A Gauss diagram consists of finite number of disjoint circles oriented anticlockwise with finite number of signed arrows whose head and tail lies on circles. 18 / 86

Gauss diagrams A Gauss diagram consists of finite number of disjoint circles oriented anticlockwise with finite number of signed arrows whose head and tail lies on circles. 19 / 86

Gauss diagrams To each virtual link diagram one can associate a Gauss diagram. 20 / 86

Gauss diagrams To each virtual link diagram one can associate a Gauss diagram. x 1 * 4 x 2 x 4 3 1 x 3 2 Figure: A virtual knot diagram K . 21 / 86

Gauss diagrams To each virtual link diagram one can associate a Gauss diagram. x 1 * 4 x 2 x 4 3 1 x 3 2 Figure: A virtual knot diagram K . Oriented Gauss code for K : 1O − 2U − 1U − 2O − 3O − 4U − 3U − 4O − 22 / 86

Gauss diagrams To each virtual link diagram one can associate a Gauss diagram. x 1 * 4 x 2 x 4 3 1 x 3 2 Figure: A virtual knot diagram K . Oriented Gauss code for K : 1O − 2U − 1U − 2O − 3O − 4U − 3U − 4O − 1U 2U - 2O - 1O * 3O 4O - - 4U 3U Figure: Gauss diagram for the virtual knot diagram K . 23 / 86

Gauss diagrams x 1 * 4 x 2 x 4 3 1 x 3 2 Figure: A virtual knot diagram K . x 2 1U 2U - 2O - 1O x 1 x 3 * 3O 4O - - 4U 3U x 4 Figure: Gauss diagram for the virtual knot diagram K . 24 / 86

Gauss diagrams Two Gauss diagrams are said to be equivalent if one diagram can be changed into the another diagram by a finite sequence of moves as shown below: Figure: Reidemeister moves on Gauss diagrams. 25 / 86

Gauss diagrams Two Gauss diagrams are said to be equivalent if one diagram can be changed into the another diagram by a finite sequence of moves as shown below: Figure: Reidemeister moves on Gauss diagrams. There is one-to-one correspondence between virtual links and equivalence classes of Gauss diagrams. 26 / 86

Virtual link group (L. Kauffman) Let D be a given Gauss diagram, 27 / 86

Virtual link group (L. Kauffman) Let D be a given Gauss diagram, ◮ label the arcs from one arrow head to another arrow head as x 1 , x 2 , . . . , x n . These are our generators for virtual link group G K ( D ) . 28 / 86

Virtual link group (L. Kauffman) Let D be a given Gauss diagram, ◮ label the arcs from one arrow head to another arrow head as x 1 , x 2 , . . . , x n . These are our generators for virtual link group G K ( D ) . ◮ for each arrow add a relation as shown below. b a ǫ c a c = b a ǫ G K ( D ) = � x 1 , x 2 , . . . , x n || one relation for each arrow � . 29 / 86

Example x 2 1U 2U - 2O - 1O x 1 x 3 * 3O 4O - - 4U 3U x 4 x − 1 x − 1 x − 1 x − 1 G K ( D ) := � x 1 , x 2 , x 3 , x 4 || x 2 = x 3 , x 3 = x 1 , x 4 = x 1 , x 1 = x 3 � . 1 2 3 4 30 / 86

Peripheral structure for virtual links using group G K ( D ) 31 / 86

Peripheral structure for virtual links using group G K ( D ) Let D be a Gauss diagram and G K ( D ) be the group associated to it. ◮ Meridian: Take generator corresponding to any of the arcs in a given Gauss diagram, say x . 32 / 86

Peripheral structure for virtual links using group G K ( D ) Let D be a Gauss diagram and G K ( D ) be the group associated to it. ◮ Meridian: Take generator corresponding to any of the arcs in a given Gauss diagram, say x . ◮ Longitude: Start moving from the meridian arc along the circle and write a ǫ when passing the head of on arrow, whose sign is ǫ and tail lies on the arc a , until we reach the meridian arc, and at the end write x − p , where p is so chosen that the longitude is in the commutator subgroup of G K ( D ) . 33 / 86

Peripheral structure for virtual links using group G K ( D ) Let D be a Gauss diagram and G K ( D ) be the group associated to it. ◮ Meridian: Take generator corresponding to any of the arcs in a given Gauss diagram, say x . ◮ Longitude: Start moving from the meridian arc along the circle and write a ǫ when passing the head of on arrow, whose sign is ǫ and tail lies on the arc a , until we reach the meridian arc, and at the end write x − p , where p is so chosen that the longitude is in the commutator subgroup of G K ( D ) . ◮ Peripheral pair: ( m, l ) . 34 / 86