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bio website math.princeton.edu/~ssivek
location Princeton, NJ
age 29
visits member for 4 years, 6 months
seen Apr 11 at 20:36

I'm a postdoc in low-dimensional topology at Princeton.


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Nov
21
comment Link surgery on $S^2\times S^1$
For n=2 you do have a product link. You can see it by writing the pages $(S^1\times I)$ of the open book collectively as $(S^1 \times I) \times S^1$, and then gluing in two solid tori $S^1\times D^2$ by using the disks $\{*\} \times D^2$ to cap off the annuli $(\{*\} \times I) \times S^1$ (which consist of one arc from each page). Thus in the surgered manifold the annuli $(\{*\}\times I)\times S^1$ can be closed up to form spheres $\{*\}\times S^2$, one for each point of $S^1$, and the link $L_2$ contributes a pair of points to each of those spheres.
Nov
21
comment Link surgery on $S^2\times S^1$
The argument only works for $n>0$ because when $n=0$ it isn't really an open book decomposition anymore, and then you really do have $S^2\times S^1$. You can check the n=2 case directly by Kirby calculus, though: it's 0-surgery on an unknot and on two of its meridians, and if you blow up one of those meridians you get a chain of unknots with surgery coefficients -1,-1,0,0. Blow down the second to get a chain with coefficients 0,1,0, and then the middle one, and then either of the remaining ones and you're left with a single 0-framed unknot, which gives $S^1\times S^2$.
Nov
21
answered Link surgery on $S^2\times S^1$
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awarded  Citizen Patrol
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answered {0,1} Maslov potentials on Legendrian knots
Jun
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answered Is every quasipositive knot strongly quasipositive?
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Mar
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answered Are there non-compact, non-smoothable manifolds?
Feb
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comment Vassilliev invariants of knots and their cables
@Vivek: it's in section 9.2.2 of "Introduction to Vassiliev Knot Invariants" by Chmutov, Duzhin, and Mostovoy, available at pdmi.ras.ru/~duzhin/papers/cdbook/cdbook.pdf.
Feb
21
comment Computer package to compute HOMFLY polynomial?
@minimax: see katlas.org/wiki/Cabling for an example. You have to import the program CableComponent.m, and then I believe you want to use CableComponent[BR[TorusKnot[19,3]], Knot[3,1]]. (I haven't used CableComponent before, but this knot seems to at least have the right Alexander polynomial according to the formula $\Delta(t) = \Delta_{T_{3,19}}(t) * \Delta_{T_{2,3}}(t^3)$.)