Publications of Keiichi Sakai

Published / Accepted papers (linked to MathSciNet) They can also be found in arXiv.

  1. (with Saki Kanou) The Fox-Hatcher cycle and a Vassiliev invariant of order three, Pacific Journal of Mathematics, Volume 323 (2023), No. 2, pages 281-306
  2. (with Minori Okamura) Span of the Jones polynomials of certain v-adequate virtual links, Journal of Knot Theory and Its Ramifications, volume 30, No. 01 (2021), 2150001
  3. (with Ryutaro Sugiyama) Generalized connected sum formula for the Arnold invariants of generic plane curves, Topology and its Applications, volume 255 (2019), pages 86-108
  4. (with Syunji Moriya) The space of short ropes and the classifying space of the space of long knots, Algebraic and Geometric Topology 18-5 (2018), pages 2859-2873
  5. Lin-Wang type formula for the Haefliger invariant, Homology, Homotopy and Applications, Volume 17 (2015), Number 2, pages 317-341
  6. BV-structures on the homology of the framed long knot space, Journal of Homotopy and Related Structures, Volume 11 (2016), issue 3, pages 425-441
  7. Deloopings of the spaces of long embeddings, Fundamenta Mathematicae, Volume 227 (2014), Number 1, pages 27-34
  8. (with Kenji Daikoku and Masamichi Takase) On a move reducing the genus of a knot diagram, Indiana University Mathematics Journal, Volume 61 (2012), issue 3, pages 1111-1127
  9. (with Tadayuki Watanabe) 1-loop graphs and configuration space integral for embedding spaces, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 152, Issue 03 (May 2012), pages 497-533 Shinshu U Institutional Repository
  10. An integral expression of the first non-trivial one-cocycle of the space of long knots in R3, Pacific Journal of Mathematics, Volume 250 (2011), No. 2, pages 407-419 Shinshu U Institutional Repository
  11. Configuration space integrals for embedding spaces and the Haefliger invariant, Journal of Knot Theory and Its Ramifications, Volume 19, (2010) No. 12, pages 1597-1644
  12. Nontrivalent graph cocycle and the cohomology of long knot space, Algebraic and Geometric Topology 8:3 (2008), pages 1499-1522
  13. Poisson structures on the homology of the space of knots, Geometry and Topology Monographs, Volume 13 (2008), Groups, homotopy and configuration spaces (Tokyo 2005), pages 463-482


  1. Embedding spaces, configuration spaces and Vassiliev invariants (in Japanese), uwv(Sugaku) volume 72, pages 285-309, The Mathematical Society of Japan, English translation in Sugaku Exposition, volume 37 (2024), 53-77, published electronically on May 1 2024

Preprints / In preparation

  1. Bott-Taubes construction for the space of framed long knots, preprint


  • Page 414 in Pacific J Math paper (2011): "the zero-cycle e is given by (ι,1)" should be corrected as "the zero-cycle e is given by (ι,2)". Theorem 3.1 of this paper still holds, and an alternative proof can be found as Corollary 4.9 in Pacific J Math paper (2023). The original proof relies on Lemma 4.5 in AGT paper (2008) that seems to miss factor of 2.
  • Sorry for misspelling the name of Thistleswaite in JKTR paper (2021).
  • AGT paper (2008) and Pacific J Math paper (2011) contain sign errors; the signs of 7th, 8th and 9th graphs in the non-trivalent graph cocycle should be replaced by respectively +, - and +.
    These corrections do not cause any problems on the results of these papers since these graphs do not essentially contribute to the integrals. (3 June 2013)
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