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fixed arxiv front-end links and gave doi links
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David Roberts ♦
David Roberts ♦
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  1. Regarding whether or not Vassiliev invariants separate knots, the sticking point is whether they separate a knot from its reverse, where the reverse of a knot is the same knot with the opposite orientation. If this were false, Vassiliev invariants would not separate knots by a result of Kuperberg: result of KuperbergDetecting knot invertibility, J. Knot Theory Ramifications 5 (1996), 173-181 arXiv:q-alg/9712048. This boils down to the question of whether the space of Jacobi diagrams with an odd number of legs vanishes. As a piece of shameless self-promotion I will refer to my own papermy own paper (joint with Tomotoda Ohtsuki), Vanishing of 3-Loop Jacobi Diagrams of Odd Degree (J. Combin. Theory Ser. A. 114 (2007) 919-931 https://doi.org/10.1016/j.jcta.2006.10.005) where we prove this for 3-loop Jacobi diagrams. You could ask the same question about homotopy string links, where we know that Vassiliev invariants do separate homotopy string links (see the paper this paperVassiliev homotopy string link invariants by Bar-Natan (Journal of Knot Theory and its Ramifications 4-1 (1995) 13–32. https://doi.org/10.1142/S021821659500003X).
  2. I might be way off-base here, but can't you just take the formal product of an element of $\mathbb{C}$ with the knot? So a negative knot looks like -1 times K. So no, singular knots are not global points of V. Resolutions (via the Vassiliev skein relation) of singular knots as formal sums of knots over $\mathbb{C}$ are global points of V.
  3. You sort-of say this, but the Fundamental Theorem of Vassiliev Invariants (proved over $\mathbb{C}$ by Kontsevich and others) tells us that weight systems integrate, which is the statement that any weight system (with some natural conditions) gives rise to a Vassiliev invariant (which is nothing short of miraculous when you think about it). A reference is the survey paper this survey paper by Bar-Natan and StoimenowThe Fundamental Theorem of Vassiliev Invariants by Bar-Natan and Stoimenow.
  1. Regarding whether or not Vassiliev invariants separate knots, the sticking point is whether they separate a knot from its reverse, where the reverse of a knot is the same knot with the opposite orientation. If this were false, Vassiliev invariants would not separate knots by a result of Kuperberg. This boils down to the question of whether the space of Jacobi diagrams with an odd number of legs vanishes. As a piece of shameless self-promotion I will refer to my own paper (joint with Tomotoda Ohtsuki), where we prove this for 3-loop Jacobi diagrams. You could ask the same question about homotopy string links, where we know that Vassiliev invariants do separate homotopy string links (see this paper by Bar-Natan).
  2. I might be way off-base here, but can't you just take the formal product of an element of $\mathbb{C}$ with the knot? So a negative knot looks like -1 times K. So no, singular knots are not global points of V. Resolutions (via the Vassiliev skein relation) of singular knots as formal sums of knots over $\mathbb{C}$ are global points of V.
  3. You sort-of say this, but the Fundamental Theorem of Vassiliev Invariants (proved over $\mathbb{C}$ by Kontsevich and others) tells us that weight systems integrate, which is the statement that any weight system (with some natural conditions) gives rise to a Vassiliev invariant (which is nothing short of miraculous when you think about it). A reference is this survey paper by Bar-Natan and Stoimenow.
  1. Regarding whether or not Vassiliev invariants separate knots, the sticking point is whether they separate a knot from its reverse, where the reverse of a knot is the same knot with the opposite orientation. If this were false, Vassiliev invariants would not separate knots by a result of Kuperberg: Detecting knot invertibility, J. Knot Theory Ramifications 5 (1996), 173-181 arXiv:q-alg/9712048. This boils down to the question of whether the space of Jacobi diagrams with an odd number of legs vanishes. As a piece of shameless self-promotion I will refer to my own paper (joint with Tomotoda Ohtsuki), Vanishing of 3-Loop Jacobi Diagrams of Odd Degree (J. Combin. Theory Ser. A. 114 (2007) 919-931 https://doi.org/10.1016/j.jcta.2006.10.005) where we prove this for 3-loop Jacobi diagrams. You could ask the same question about homotopy string links, where we know that Vassiliev invariants do separate homotopy string links (see the paper Vassiliev homotopy string link invariants by Bar-Natan (Journal of Knot Theory and its Ramifications 4-1 (1995) 13–32. https://doi.org/10.1142/S021821659500003X).
  2. I might be way off-base here, but can't you just take the formal product of an element of $\mathbb{C}$ with the knot? So a negative knot looks like -1 times K. So no, singular knots are not global points of V. Resolutions (via the Vassiliev skein relation) of singular knots as formal sums of knots over $\mathbb{C}$ are global points of V.
  3. You sort-of say this, but the Fundamental Theorem of Vassiliev Invariants (proved over $\mathbb{C}$ by Kontsevich and others) tells us that weight systems integrate, which is the statement that any weight system (with some natural conditions) gives rise to a Vassiliev invariant (which is nothing short of miraculous when you think about it). A reference is the survey paper The Fundamental Theorem of Vassiliev Invariants by Bar-Natan and Stoimenow.
combined 2 and 4; added reference to Kuperberg result.
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Daniel Moskovich
Daniel Moskovich
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  1. Regarding whether or not Vassiliev invariants separate knots, the sticking point is whether they separate a knot from its reverse, where the reverse of a knot is the same knot with the opposite orientation. If this were false, Vassiliev invariants would not separate knots by a result of Kuperberg. This boils down to the question of whether the space of Jacobi diagrams with an odd number of legs vanishes. As a piece of shameless self-promotion I will refer to my own paper (joint with Tomotoda Ohtsuki), where we prove this for 3-loop Jacobi diagrams. You could ask the same question about homotopy string links, where we know that Vassiliev invariants do separate homotopy string links (see this paper by Bar-Natan).
  2. I might be way off-base here, but can't you just take the formal product of an element of $\mathbb{C}$ with the knot? So a negative knot looks like -1 times K. So no, singular knots are not global points of V. Resolutions (via the Vassiliev skein relation) of singular knots as formal sums of knots over $\mathbb{C}$ are global points of V.
  3. You sort-of say this, but the Fundamental Theorem of Vassiliev Invariants (proved over $\mathbb{C}$ by Kontsevich and others) tells us that weight systems integrate, which is the statement that any weight system (with some natural conditions) gives rise to a Vassiliev invariant (which is nothing short of miraculous when you think about it). A reference is this survey paper by Bar-Natan and Stoimenow.
  4. You could ask the same question about homotopy string links, where we know that Vassiliev invariants do separate homotopy string links ("Vassiliev Homotopy String Link Invariants" by Bar-Natan).
  1. Regarding whether or not Vassiliev invariants separate knots, the sticking point is whether they separate a knot from its reverse, where the reverse of a knot is the same knot with the opposite orientation. This boils down to the question of whether the space of Jacobi diagrams with an odd number of legs vanishes. As a piece of shameless self-promotion I will refer to my own paper (joint with Tomotoda Ohtsuki), where we prove this for 3-loop Jacobi diagrams.
  2. I might be way off-base here, but can't you just take the formal product of an element of $\mathbb{C}$ with the knot? So a negative knot looks like -1 times K. So no, singular knots are not global points of V. Resolutions (via the Vassiliev skein relation) of singular knots as formal sums of knots over $\mathbb{C}$ are global points of V.
  3. You sort-of say this, but the Fundamental Theorem of Vassiliev Invariants (proved over $\mathbb{C}$ by Kontsevich and others) tells us that weight systems integrate, which is the statement that any weight system (with some natural conditions) gives rise to a Vassiliev invariant (which is nothing short of miraculous when you think about it). A reference is this survey paper by Bar-Natan and Stoimenow.
  4. You could ask the same question about homotopy string links, where we know that Vassiliev invariants do separate homotopy string links ("Vassiliev Homotopy String Link Invariants" by Bar-Natan).
  1. Regarding whether or not Vassiliev invariants separate knots, the sticking point is whether they separate a knot from its reverse, where the reverse of a knot is the same knot with the opposite orientation. If this were false, Vassiliev invariants would not separate knots by a result of Kuperberg. This boils down to the question of whether the space of Jacobi diagrams with an odd number of legs vanishes. As a piece of shameless self-promotion I will refer to my own paper (joint with Tomotoda Ohtsuki), where we prove this for 3-loop Jacobi diagrams. You could ask the same question about homotopy string links, where we know that Vassiliev invariants do separate homotopy string links (see this paper by Bar-Natan).
  2. I might be way off-base here, but can't you just take the formal product of an element of $\mathbb{C}$ with the knot? So a negative knot looks like -1 times K. So no, singular knots are not global points of V. Resolutions (via the Vassiliev skein relation) of singular knots as formal sums of knots over $\mathbb{C}$ are global points of V.
  3. You sort-of say this, but the Fundamental Theorem of Vassiliev Invariants (proved over $\mathbb{C}$ by Kontsevich and others) tells us that weight systems integrate, which is the statement that any weight system (with some natural conditions) gives rise to a Vassiliev invariant (which is nothing short of miraculous when you think about it). A reference is this survey paper by Bar-Natan and Stoimenow.
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Daniel Moskovich
Daniel Moskovich
  • 22.1k
  • 15
  • 139
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  1. Regarding whether or not Vassiliev invariants separate knots, the sticking point is whether they separate a knot from its reverse, where the reverse of a knot is the same knot with the opposite orientation. This boils down to the question of whether the space of Jacobi diagrams with an odd number of legs vanishes. As a piece of shameless self-promotion I will refer to my own paper (joint with Tomotoda Ohtsuki), where we prove this for 3-loop Jacobi diagrams.
  2. I might be way off-base here, but can't you just take the formal product of an element of $\mathbb{C}$ with the knot? So a negative knot looks like -1 times K. So no, singular knots are not global points of V. Resolutions (via the Vassiliev skein relation) of singular knots as formal sums of knots over $\mathbb{C}$ are global points of V.
  3. You sort-of say this, but the Fundamental Theorem of Vassiliev Invariants (proved over $\mathbb{C}$ by Kontsevich and others) tells us that weight systems integrate, which is the statement that any weight system (with some natural conditions) gives rise to a Vassiliev invariant (which is nothing short of miraculous when you think about it). A reference is this survey paper by Bar-Natan and Stoimenow.
  4. You could ask the same question about homotopy string links, where we know that Vassiliev invariants do separate homotopy string links ("Vassiliev Homotopy String Link Invariants" by Bar-Natan).