An invariant of free links valued in free groups V.O.Manturov and S.Kim* V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 1 / 51
1 4 2 3 Basic definitions and notations Definition 1.1 By a framed 4-graph with endpoints we mean a graph satisfying the followings: every vertex is a 4-valent vertex except for 2 n vertices for some 1 n ∈ N ∪ { 0 } which are 1-valent vertices. for each 4-valent vertex we fix a way of splitting of the four emanating 2 half-edges into two pairs of edges called (formally) opposite . Figure: { 1 , 3 } and { 2 , 4 } are pairs of ‘opposite’ V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 2 / 51
R 1 R 0 An intersection of different edges of the framed 4-graph with endpoints in interior points are called a virtual crossing and it is denoted by an intersection inside a circle. Figure: A framed 4-graph with endpoints V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 3 / 51
Definition 1.2 By a unicursal component of a framed 4-graph with endpoints we mean an equivalence class on the set of edges of the graph: Two edges e , e ′ are equivalent if there exists a collection of edges e = e 1 , · · · , e k = e ′ and a collection of 4-valent vertices v 1 , · · · , v k − 1 (some of them may coincide) of the graph such that edges e i , e i + 1 are opposite to each other at the vertex v i . By a unicursal circle of a framed 4-graph with endpoints we mean a unicursal component such that v 1 = v k − 1 . V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 4 / 51
R 1 R 0 Figure: A component of a framed 4-graph with endpoints V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 5 / 51
R 1 R 0 Figure: A component of a framed 4-graph with endpoints V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 6 / 51
R 1 R 0 Figure: A component of a framed 4-graph with endpoints V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 7 / 51
R 1 R 0 Figure: A component of a framed 4-graph with endpoints V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 8 / 51
R 1 R 0 Figure: Framed 4-graph with endpoints with 4 (unicursal) components V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 9 / 51
Definition 1.3 A virtual tangle diagram is a generic immersion of a framed 4-graph with endpoints in R × I with each 4-valent vertex endowed with a classical crossing structure and every 1-valent vertex is in R × { 0 } or R × { 1 } . A virtual (link) diagram is a virtual tangle diagram without endpoints. A virtual tangle is an equivalence class of virtual tangle diagrams by usual Reidemeister moves and the detour move . (1) (2) (3) Figure: Reidemeister moves Figure: Detour move V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 10 / 51
RM1 RM3 RM2 Definition 1.4 A free tangle is an equivalence class of framed 4-graphs with endpoints modulo Reidemeister moves for free diagrams. A free link is a free tangle without endpoints. By an n − n free tangle we mean a free tangle with points { p 0 1 , · · · , p 0 n } in R × { 0 } and { p 1 1 , · · · , p 1 n } in R × { 1 } such that each component has end points p 0 i and p 1 i . Figure: Reidemeister moves for free diagrams V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 11 / 51
T T 3-3 free tangle Free tangle T 1 T 2 T 3 T 4 3 1 T 2 Definition 1.5 If the components of a free tangle are numbered and the numbers of components preserve under Reidemeister moves for free diagrams, then the free tangle is enumerated. Figure: Free tangle diagrams V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 12 / 51
Remark 1.6 Every notion( n − n tangle, link, knot, braid, etc.) is just a partial case of tangles. On the other hand, free tangles( n − n tangles, links, knots, etc.) are equivalence classes of virtual tangles modulo the following relations. Figure: Crossing change and virtualization V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 13 / 51
Motivation In [3], by V.O.Manturov, the following principle was established: Proposition 2.1 If a diagram K is complicated enough then it realizes itself. In other words, if K is complicated enough and K and K ′ are equivalent, then K ⊂ K ′ . This proposition is proved by parity bracket which is valued in group generated by diagrams. V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 14 / 51
Remark 2.2 This effects very similarly to the case of free groups: if the word abcba in Z ∗ Z ∗ Z is irreducible then it appears in any word equivalent to it (for example, abaa − 1 b − 1 bcbc 3 c − 3 a ) . In [4],[5], by V.O.Manturov and I.M.Nikonov, consider homomorphism from pure braid group PB n to a group presentation G k n . consider homomorphism from G k n to Z ∗ N 2 . That is, there is a homomorphism from PB n to Z ∗ N 2 . But this is NOT arranged for the case of general tangles. V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 15 / 51
Remark 2.2 This effects very similarly to the case of free groups: if the word abcba in Z ∗ Z ∗ Z is irreducible then it appears in any word equivalent to it (for example, abaa − 1 b − 1 bcbc 3 c − 3 a ) . In [4],[5], by V.O.Manturov and I.M.Nikonov, consider homomorphism from pure braid group PB n to a group presentation G k n . consider homomorphism from G k n to Z ∗ N 2 . That is, there is a homomorphism from PB n to Z ∗ N 2 . But this is NOT arranged for the case of general tangles. V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 15 / 51
Remark 2.2 This effects very similarly to the case of free groups: if the word abcba in Z ∗ Z ∗ Z is irreducible then it appears in any word equivalent to it (for example, abaa − 1 b − 1 bcbc 3 c − 3 a ) . In [4],[5], by V.O.Manturov and I.M.Nikonov, consider homomorphism from pure braid group PB n to a group presentation G k n . consider homomorphism from G k n to Z ∗ N 2 . That is, there is a homomorphism from PB n to Z ∗ N 2 . But this is NOT arranged for the case of general tangles. V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 15 / 51
Goal : Construct an invariant of free n − n tangles and links valued in free groups. ◦ Distinctions between general tangles and braids: n − n tangles do not possess a group structure: there are no inverse elements. the existence of pure crossings (between a component and itself). Figure: Free n − n tangle diagram with a reverse arc V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 16 / 51
k k i p s i p e i k k Adjusted goal : Construct an invariant of free n − n tangles and links without pure crossings valued in free groups. c 1 c 2 c 3 c 4 Figure: Crossings with respect to a component V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 17 / 51
Statement If two diagrams with no pure crossings are equivalent then they are equivalent by a sequence of moves where no intermediate diagram has a pure crossing. Proposition 2.3 ([2], M.Goussarov, M.Polyak, O.Viro) If two classical links are equivalent as virtual links, then they are equivalent as classical links. Proposition 2.4 ([1], R.Fenn, R.Rimanyi, C.Rourke) If two classical braids are equivalent as tangles, then they are equivalent as braids. V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 18 / 51
Statement If two diagrams with no pure crossings are equivalent then they are equivalent by a sequence of moves where no intermediate diagram has a pure crossing. Proposition 2.3 ([2], M.Goussarov, M.Polyak, O.Viro) If two classical links are equivalent as virtual links, then they are equivalent as classical links. Proposition 2.4 ([1], R.Fenn, R.Rimanyi, C.Rourke) If two classical braids are equivalent as tangles, then they are equivalent as braids. V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 18 / 51
In section 3 we will prove the following statement. If two diagrams with no pure crossings are equivalent then they are equivalent by a sequence of moves where no intermediate diagram has a pure crossing. In section 4 we will construct invariant valued in Z ∗ N for free n − n tangles 2 and free links without pure crossings. V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 19 / 51
i j L L Tangles without pure crossings Definition 3.1 Let L = L 1 ∪ · · · ∪ L n be an enumerated free tangle diagram. A crossing c is called a crossing of type ( i , j ) if two arcs of c are part of L i and L j respectively. A classical crossing c of type ( i , i ) for some i ∈ { 1 , 2 , · · · n } is called a pure crossing . Figure: A crossing of type ( i , j ) V.O.Manturov S.Kim (BMSTU) An invariant of free links valued in free groups August 24. 2015 20 / 51
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