Geometry & Topology

Additive invariants for knots, links and graphs in $3$–manifolds

Scott A Taylor and Maggy Tomova

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We define two new families of invariants for ( 3 –manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and ( 1 2 ) additive under trivalent vertex sum of pairs. The first of these families is closely related to both bridge number and tunnel number. The second of these families is a variation and generalization of Gabai’s width for knots in the 3 –sphere. We give applications to the tunnel number and higher-genus bridge number of connected sums of knots.

Article information

Source
Geom. Topol., Volume 22, Number 6 (2018), 3235-3286.

Dates
Received: 16 July 2016
Revised: 6 October 2017
Accepted: 15 October 2017
First available in Project Euclid: 29 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.gt/1538186737

Digital Object Identifier
doi:10.2140/gt.2018.22.3235

Mathematical Reviews number (MathSciNet)
MR3858764

Zentralblatt MATH identifier
06945126

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds

Keywords
thin position bridge number tunnel number

Citation

Taylor, Scott A; Tomova, Maggy. Additive invariants for knots, links and graphs in $3$–manifolds. Geom. Topol. 22 (2018), no. 6, 3235--3286. doi:10.2140/gt.2018.22.3235. https://projecteuclid.org/euclid.gt/1538186737


Export citation

References

  • R Blair, M Tomova, Companions of the unknot and width additivity, J. Knot Theory Ramifications 20 (2011) 497–511
  • R Blair, M Tomova, Width is not additive, Geom. Topol. 17 (2013) 93–156
  • A J Casson, C M Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987) 275–283
  • B Clark, The Heegaard genus of manifolds obtained by surgery on links and knots, Internat. J. Math. Math. Sci. 3 (1980) 583–589
  • M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987) 237–300
  • H Doll, A generalized bridge number for links in $3$–manifolds, Math. Ann. 294 (1992) 701–717
  • D Gabai, Foliations and the topology of $3$–manifolds, III, J. Differential Geom. 26 (1987) 479–536
  • C M Gordon, J Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989) 371–415
  • C Hayashi, K Shimokawa, Heegaard splittings of the pair of the solid torus and the core loop, Rev. Mat. Complut. 14 (2001) 479–501
  • C Hayashi, K Shimokawa, Thin position of a pair (\hspace*-1pt$3$–manifold, $1$–submanifold\hspace*1pt), Pacific J. Math. 197 (2001) 301–324
  • J Hempel, $3$–manifolds as viewed from the curve complex, Topology 40 (2001) 631–657
  • W Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics 43, Amer. Math. Soc., Providence, RI (1980)
  • T Kobayashi, Y Rieck, Heegaard genus of the connected sum of $m$–small knots, Comm. Anal. Geom. 14 (2006) 1037–1077
  • T Kobayashi, Y Rieck, The growth rate of the tunnel number of $m$–small knots, preprint (2015)
  • M Lackenby, An algorithm to determine the Heegaard genus of simple $3$–manifolds with nonempty boundary, Algebr. Geom. Topol. 8 (2008) 911–934
  • S Matveev, V Turaev, A semigroup of theta-curves in $3$–manifolds, Mosc. Math. J. 11 (2011) 805–814
  • J W Milnor, On the total curvature of knots, Ann. of Math. 52 (1950) 248–257
  • K Miyazaki, Conjugation and the prime decomposition of knots in closed, oriented $3$–manifolds, Trans. Amer. Math. Soc. 313 (1989) 785–804
  • Y Moriah, On boundary primitive manifolds and a theorem of Casson–Gordon, Topology Appl. 125 (2002) 571–579
  • K Morimoto, There are knots whose tunnel numbers go down under connected sum, Proc. Amer. Math. Soc. 123 (1995) 3527–3532
  • K Morimoto, Tunnel number, connected sum and meridional essential surfaces, Topology 39 (2000) 469–485
  • K Morimoto, On composite types of tunnel number two knots, J. Knot Theory Ramifications 24 (2015) art. id. 1550013, 10 pages
  • K Morimoto, M Sakuma, Y Yokota, Examples of tunnel number one knots which have the property “$1+1=3$”, Math. Proc. Cambridge Philos. Soc. 119 (1996) 113–118
  • K Morimoto, J Schultens, Tunnel numbers of small knots do not go down under connected sum, Proc. Amer. Math. Soc. 128 (2000) 269–278
  • F H Norwood, Every two-generator knot is prime, Proc. Amer. Math. Soc. 86 (1982) 143–147
  • Y Rieck, E Sedgwick, Thin position for a connected sum of small knots, Algebr. Geom. Topol. 2 (2002) 297–309
  • M Scharlemann, Heegaard splittings of compact $3$–manifolds, from “Handbook of geometric topology” (R J Daverman, R B Sher, editors), North-Holland, Amsterdam (2002) 921–953
  • M Scharlemann, J Schultens, The tunnel number of the sum of $n$ knots is at least $n$, Topology 38 (1999) 265–270
  • M Scharlemann, J Schultens, $3$–manifolds with planar presentations and the width of satellite knots, Trans. Amer. Math. Soc. 358 (2006) 3781–3805
  • M Scharlemann, A Thompson, Thin position for $3$–manifolds, from “Geometric topology” (C Gordon, Y Moriah, B Wajnryb, editors), Contemp. Math. 164, Amer. Math. Soc., Providence, RI (1994) 231–238
  • M Scharlemann, A Thompson, On the additivity of knot width, from “Proceedings of the Casson Fest” (C Gordon, Y Rieck, editors), Geom. Topol. Monogr. 7, Geom. Topol. Publ., Coventry (2004) 135–144
  • M Scharlemann, M Tomova, Uniqueness of bridge surfaces for $2$–bridge knots, Math. Proc. Cambridge Philos. Soc. 144 (2008) 639–650
  • H Schubert, Über eine numerische Knoteninvariante, Math. Z. 61 (1954) 245–288
  • J Schultens, The classification of Heegaard splittings for (compact orientable surface)$\,\times\, S^1$, Proc. London Math. Soc. 67 (1993) 425–448
  • J Schultens, Additivity of bridge numbers of knots, Math. Proc. Cambridge Philos. Soc. 135 (2003) 539–544
  • S A Taylor, M Tomova, Thin position for knots, links, and graphs in $3$–manifolds, preprint (2016)
  • A Thompson, Thin position and the recognition problem for $S^3$, Math. Res. Lett. 1 (1994) 613–630
  • R Weidmann, On the rank of amalgamated products and product knot groups, Math. Ann. 312 (1998) 761–771
  • Y Yokota, On quantum ${\rm SU}(2)$ invariants and generalized bridge numbers of knots, Math. Proc. Cambridge Philos. Soc. 117 (1995) 545–557