An Introduction to Mathematical Knots

Transcription

1 An Introduction to Mathematical Knots Nick Brettell Postgrad talk, 2011 Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 1 / 18

2 Outline 1 Introduction to knots Knots and Links History Applications Ambient isotopy 2 Knot invariants Knot polynomials 3 Some problems in knot theory Distinctness Chirality Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 2 / 18

3 Introduction to knots Knots and Links What is a (mathematical) knot? A knot is a closed curve in 3-space that doesn t intersect itself The simplest knot is called the unknot A knot diagram is a projection of a knot onto a plane Example Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 3 / 18

4 Introduction to knots Knots and Links A link is a collection of (possibly) intertwined knots Each knot in the link is called a component Example Knot theory is a branch of geometric topology that studies knots and links Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 4 / 18

5 Introduction to knots History History (1867) Lord Kelvin theorised that atoms were knots of swirling vortices in the aether Peter Tait began listing unique knots in the belief that they corresponded to chemical elements (1887) Disproved Topologists started investigating knots in early 20th century Max Dehn, J. W. Alexander, Kurt Reidemeister (1961) Wolfgang Haken found an algorithm for determining if a knot is the unknot (1984) Vaughan Jones discovered the Jones polynomial (1991) Tait flyping conjecture proved by Thistlethwaite and Menasco Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 5 / 18

6 Introduction to knots History History (1867) Lord Kelvin theorised that atoms were knots of swirling vortices in the aether Peter Tait began listing unique knots in the belief that they corresponded to chemical elements (1887) Disproved Topologists started investigating knots in early 20th century Max Dehn, J. W. Alexander, Kurt Reidemeister (1961) Wolfgang Haken found an algorithm for determining if a knot is the unknot (1984) Vaughan Jones discovered the Jones polynomial (1991) Tait flyping conjecture proved by Thistlethwaite and Menasco Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 5 / 18

7 Introduction to knots History History (1867) Lord Kelvin theorised that atoms were knots of swirling vortices in the aether Peter Tait began listing unique knots in the belief that they corresponded to chemical elements (1887) Disproved Topologists started investigating knots in early 20th century Max Dehn, J. W. Alexander, Kurt Reidemeister (1961) Wolfgang Haken found an algorithm for determining if a knot is the unknot (1984) Vaughan Jones discovered the Jones polynomial (1991) Tait flyping conjecture proved by Thistlethwaite and Menasco Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 5 / 18

8 Introduction to knots History History (1867) Lord Kelvin theorised that atoms were knots of swirling vortices in the aether Peter Tait began listing unique knots in the belief that they corresponded to chemical elements (1887) Disproved Topologists started investigating knots in early 20th century Max Dehn, J. W. Alexander, Kurt Reidemeister (1961) Wolfgang Haken found an algorithm for determining if a knot is the unknot (1984) Vaughan Jones discovered the Jones polynomial (1991) Tait flyping conjecture proved by Thistlethwaite and Menasco Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 5 / 18

9 Introduction to knots History History (1867) Lord Kelvin theorised that atoms were knots of swirling vortices in the aether Peter Tait began listing unique knots in the belief that they corresponded to chemical elements (1887) Disproved Topologists started investigating knots in early 20th century Max Dehn, J. W. Alexander, Kurt Reidemeister (1961) Wolfgang Haken found an algorithm for determining if a knot is the unknot (1984) Vaughan Jones discovered the Jones polynomial (1991) Tait flyping conjecture proved by Thistlethwaite and Menasco Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 5 / 18

10 Introduction to knots Applications Applications Chemistry, Biology, Physics Identifying stereoisomers Knotted molecules, DNA Statistical mechanics Some problems in knot theory What knots are there? Is a knot the unknot? Is a knot chiral (different to its mirror image)? Given two knot diagrams, are they the same knot? Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 6 / 18

11 Introduction to knots Applications Applications Chemistry, Biology, Physics Identifying stereoisomers Knotted molecules, DNA Statistical mechanics Some problems in knot theory What knots are there? Is a knot the unknot? Is a knot chiral (different to its mirror image)? Given two knot diagrams, are they the same knot? Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 6 / 18

12 Introduction to knots Applications One of these knots is not like the other Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 7 / 18

13 Introduction to knots Some basics of knot theory Ambient isotopy Example An ambient isotopy is a continuous deformation without breaking the knot If two knot diagrams are ambient isotopic, we can get from one to the other by one of three Reidemeister moves Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 8 / 18

14 Introduction to knots Some basics of knot theory Ambient isotopy Example An ambient isotopy is a continuous deformation without breaking the knot If two knot diagrams are ambient isotopic, we can get from one to the other by one of three Reidemeister moves Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 8 / 18

15 Introduction to knots The Reidemeister moves Ambient isotopy Figure: Type I Figure: Type II Figure: Type III Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 9 / 18

16 Knot invariants Knot invariants Example A knot invariant takes the same value under ambient isotopy A knot is tricolourable if each strand can be given one of three colours, at least two colours are used, and at each crossing, the three incident strands are all different colours, or all the same Something is a knot invariant if it doesn t change under the three Reidemeister moves If two knots differ with regards to some knot invariant, they are different knots (i.e. not ambient isotopic) Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 10 / 18

17 Knot invariants Knot invariants Example A knot invariant takes the same value under ambient isotopy A knot is tricolourable if each strand can be given one of three colours, at least two colours are used, and at each crossing, the three incident strands are all different colours, or all the same Something is a knot invariant if it doesn t change under the three Reidemeister moves If two knots differ with regards to some knot invariant, they are different knots (i.e. not ambient isotopic) Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 10 / 18

18 Knot invariants Knot invariants Example A knot invariant takes the same value under ambient isotopy A knot is tricolourable if each strand can be given one of three colours, at least two colours are used, and at each crossing, the three incident strands are all different colours, or all the same Something is a knot invariant if it doesn t change under the three Reidemeister moves If two knots differ with regards to some knot invariant, they are different knots (i.e. not ambient isotopic) Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 10 / 18

19 Knot invariants Knot polynomials Knot polynomials A knot polynomial is a knot invariant that takes the form of a polynomial The polynomial encodes information about the knot If we compute the polynomials of two knot diagrams and they are different, they represent different knots (i.e. not ambient isotopic) The first knot polynomial was the Alexander polynomial (1923) Conway came up with a skein relation for this polynomial (1960s) Vaughan Jones discovered the Jones polynomial (1984) This led to a flurry of research on knot polynomials, including the bracket polynomial (1987) and HOMFLY polynomial (1985) Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 11 / 18

20 Knot invariants Knot polynomials Knot polynomials A knot polynomial is a knot invariant that takes the form of a polynomial The polynomial encodes information about the knot If we compute the polynomials of two knot diagrams and they are different, they represent different knots (i.e. not ambient isotopic) The first knot polynomial was the Alexander polynomial (1923) Conway came up with a skein relation for this polynomial (1960s) Vaughan Jones discovered the Jones polynomial (1984) This led to a flurry of research on knot polynomials, including the bracket polynomial (1987) and HOMFLY polynomial (1985) Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 11 / 18

21 Knot invariants Knot polynomials The Jones polynomial A knot polynomial that is invariant for oriented links Discovered by Vaughan Jones (1983) while researching von Neumann algebras Awarded Fields medal (1990) Can (sometimes) distinguish between a knot and its mirror image Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 12 / 18

22 Skein relations Knot invariants Knot polynomials A skein relation is a linear relation between the knot polynomial of three links differing only at a specific crossing It can be used to calculate a knot polynomial A skein relation for ( the Jones ) polynomial ( is: ) (t 1/2 t 1/2 )V Also V (unknot) = 1 Example ( ) V = t 2 V = t 1 V tv ( ) ( ) ( ) ( ) t 1 2 t 3 2 V Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 13 / 18

23 Knot invariants Knot polynomials One of these knots is not like the other, take two Example V (t) = t 1 + t 3 t 4 V (t) = t + t 3 + t 4 V (t) = t 1 + t 3 t 4 Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 14 / 18

24 Knot invariants Knot polynomials One of these knots is not like the other, take two Example V (t) = t 1 + t 3 t 4 V (t) = t + t 3 + t 4 V (t) = t 1 + t 3 t 4 Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 14 / 18

25 Knot invariants Knot polynomials One of these knots is not like the other, take two Example V (t) = t 1 + t 3 t 4 V (t) = t + t 3 + t 4 V (t) = t 1 + t 3 t 4 Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 14 / 18

26 Some problems in knot theory Distinctness Recognising distinct knots So we can use knot polynomials to identify when knots are distinct? Sometimes, not always The Jones polynomial is pretty good at identifying distinct knots The HOMFLY polynomial is better Still not a complete invariant Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 15 / 18

27 Some problems in knot theory Distinctness Recognising distinct knots So we can use knot polynomials to identify when knots are distinct? Sometimes, not always The Jones polynomial is pretty good at identifying distinct knots The HOMFLY polynomial is better Still not a complete invariant Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 15 / 18

28 Some problems in knot theory Chirality Identifying if a knot is chiral Example Recall a knot is chiral if its mirror image is a different knot Otherwise it is amphicheiral If the Jones polynomial of a knot is V (t), its mirror image s polynomial is V (t 1 ). If a knot is amphicheiral, then V (t) = V (t 1 ) (palindromic) The right-handed trefoil knot s polynomial is V (t) = t 1 + t 3 t 4, whereas the left-handed version is V (t) = t 1 + t 3 t 4 Example The Jones polynomial of the figure-eight knot is V (q) = q 2 q q + q 2. This knot is amphicheiral. Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 16 / 18

29 Some problems in knot theory Chirality Identifying if a knot is chiral Example Recall a knot is chiral if its mirror image is a different knot Otherwise it is amphicheiral If the Jones polynomial of a knot is V (t), its mirror image s polynomial is V (t 1 ). If a knot is amphicheiral, then V (t) = V (t 1 ) (palindromic) The right-handed trefoil knot s polynomial is V (t) = t 1 + t 3 t 4, whereas the left-handed version is V (t) = t 1 + t 3 t 4 Example The Jones polynomial of the figure-eight knot is V (q) = q 2 q q + q 2. This knot is amphicheiral. Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 16 / 18

30 Some problems in knot theory Chirality Chirality: a final (and real-world) example In chemistry, a molecular knot is a molecular architecture that contains a knot DNA and certain proteins are naturally-occuring examples Figure: Molecular trefoil knot containing two copper ions Mathematical knots are flexible, molecular bonds are (more) rigid A molecular knot with a chiral (mathematical) knot architecture is chiral (in the chemistry sense) Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 17 / 18

31 Questions? Some problems in knot theory Chirality Nick Brettell (UC) An Introduction to Mathematical Knots Postgrad talk 18 / 18