Algebraic & Geometric Topology

A signature invariant for knotted Klein graphs

Catherine Gille and Louis-Hadrien Robert

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

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


We define some signature invariants for a class of knotted trivalent graphs using branched covers. We relate them to classical signatures of knots and links. Finally, we explain how to compute these invariants through the example of Kinoshita’s knotted theta graph.

Article information

Algebr. Geom. Topol., Volume 18, Number 6 (2018), 3719-3747.

Received: 9 May 2018
Revised: 19 June 2018
Accepted: 30 June 2018
First available in Project Euclid: 27 October 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 57M12: Special coverings, e.g. branched 57M15: Relations with graph theory [See also 05Cxx]
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds

knotted trivalent graphs branched covering signature invariants


Gille, Catherine; Robert, Louis-Hadrien. A signature invariant for knotted Klein graphs. Algebr. Geom. Topol. 18 (2018), no. 6, 3719--3747. doi:10.2140/agt.2018.18.3719.

Export citation


  • M F Atiyah, I M Singer, The index of elliptic operators, III, Ann. of Math. 87 (1968) 546–604
  • J S Calcut, J R Metcalf-Burton, Double branched covers of theta-curves, J. Knot Theory Ramifications 25 (2016) art. id. 1650046
  • E Flapan, When topology meets chemistry: a topological look at molecular chirality, Cambridge Univ. Press (2000)
  • P Gilmer, Real algebraic curves and link cobordism, Pacific J. Math. 153 (1992) 31–69
  • P M Gilmer, Link cobordism in rational homology $3$–spheres, J. Knot Theory Ramifications 2 (1993) 285–320
  • C M Gordon, On the $G$–signature theorem in dimension four, from “À la recherche de la topologie perdue” (L Guillou, A Marin, editors), Progr. Math. 62, Birkhäuser, Boston, MA (1986) 159–180
  • C M Gordon, R A Litherland, On the signature of a link, Invent. Math. 47 (1978) 53–69
  • C M Gordon, R A Litherland, On a theorem of Murasugi, Pacific J. Math. 82 (1979) 69–74
  • F Hirzebruch, The signature of ramified coverings, from “Global analysis” (D C Spencer, S Iyanaga, editors), Univ. Tokyo Press (1969) 253–265
  • B Jang, A Kronaeur, P Luitel, D Medici, S A Taylor, A Zupan, New examples of Brunnian theta graphs, Involve 9 (2016) 857–875
  • L H Kauffman, L R Taylor, Signature of links, Trans. Amer. Math. Soc. 216 (1976) 351–365
  • S Kinoshita, Alexander polynomials as isotopy invariants, I, Osaka Math. J. 10 (1958) 263–271
  • S Kinoshita, On elementary ideals of polyhedra in the $3$–sphere, Pacific J. Math. 42 (1972) 89–98
  • R C Kirby, The topology of $4$–manifolds, Lecture Notes in Mathematics 1374, Springer (1989)
  • P B Kronheimer, T S Mrowka, Tait colorings, and an instanton homology for webs and foams, preprint (2015)
  • C Lescop, An introduction to finite type invariants of knots and $3$–manifolds defined by counting graph configurations, Vestn. Chelyab. Gos. Univ. Mat. Mekh. Inform. (2015) 67–117
  • L Lewark, Homologies de Khovanov–Rozansky, toiles nouées pondérées et genre lisse, PhD thesis, Université Paris 7 - Denis Diderot (2013) Available at \setbox0\makeatletter\@url {\unhbox0
  • J M Montesinos, Surgery on links and double branched covers of $S^{3}$, from “Knots, groups, and $3$–manifolds (Papers dedicated to the memory of R H Fox)” (L P Neuwirth, editor), Ann. of Math. Studies 84, Princeton Univ. Press (1975) 227–259
  • K Murasugi, On the signature of links, Topology 9 (1970) 283–298
  • J H Przytycki, A Yasuhara, Linking numbers in rational homology $3$–spheres, cyclic branched covers and infinite cyclic covers, Trans. Amer. Math. Soc. 356 (2004) 3669–3685
  • D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish, Houston, TX (1990)
  • J Simon, Molecular graphs as topological objects in space, J. Comput. Chem. 8 (1987) 718–726
  • D P Thurston, The algebra of knotted trivalent graphs and Turaev's shadow world, from “Invariants of knots and $3$–manifolds” (T Ohtsuki, T Kohno, T Le, J Murakami, J Roberts, V Turaev, editors), Geom. Topol. Monogr. 4, Geom. Topol. Publ., Coventry (2002) 337–362