Trace-free $$\mathrm{SL}(2,{\pmb {\mathbb {C}}})$$ SL ( 2
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Trace-free SL(2, C )-representations of arborescent links Haimiao Chen1
© Akadémiai Kiadó, Budapest, Hungary 2018
Abstract Given a link L ⊂ S 3 , a representation π1 (S 3 − L) → SL(2, C) is trace-free if it sends each meridian to an element with trace zero. We present a method for completely determining trace-free SL(2, C)-representations for arborescent links. Concrete computations are done for a class of 3-bridge arborescent links. Keywords Trace-free representation · Arborescent link · Arborescent tangle · 3-bridge Mathematics Subject Classification 57M25 · 57M27
1 Introduction Let G be a linear group. Given a link L ⊂ S 3 , a trace-free G-representation is a homomorphism ρ : π1 (S 3 − L) → G which sends each meridian to an element with trace zero. Researchers have paid attention to such representations for long. For a knot K , Lin [6] defined an invariant h(K ) roughly by counting (with signs) conjugacy classes of tracefree SU(2)-representations of K , and showed that h(K ) equals half of the signature of K . Interestingly, it was shown in [5] that if L is an alternating link or a 2-component link, then its Khovanov homology is isomorphic to the singular homology of the space of binary dihedral representations (a special kind of trace-free SU(2)-representations) of L, as graded abelian groups. Also interesting is that, by the result of [7], each trace-free SL(2, C)-representation of L gives rise to a representation π1 (M2 (L)) → SL(2, C), where M2 (L) is the double covering of S 3 branched along L. As is well-known, linear representations of links are useful, but there is no general method to systematically find nontrivial ones. We may first focus on trace-free representations, which turn out to be easier to manipulate. This makes up another motivation. In this paper we aim to determine all trace-free SL(2, C)-representations for each arboresent link, which is a continuation of the previous work [1]. From now on, we abbreviate “trace-free SL(2, C)representation” to “representation”. The main results are Theorems 4.2 and 4.3. Based on them, we are able to explicitly determine all representations for any given arborescent link.
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Haimiao Chen [email protected] Beijing Technology and Business University, Beijing, China
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H. Chen
Fig. 1 The simplest four tangles: a [0], b [∞], c [1], d [− 1] Fig. 2 a T1 + T2 ; b T1 ∗ T2
We introduce some basic notions in Sect. 2, and define representation of a tangle in Sect. 3. In the main body, Sect. 4, we clarify properties of representations of arborescent tangles, uncovering an exquisite structure in the representation space, and then explain how to determine all representations for arborescent tangles. These lead to a practical method to find all representations for arborescent links, as presented in Sect. 5. As an illustration, explicit formulas are given for a class of 3-bridge arborescent links. It should be pointed out that most of the results in this paper will remain valid when C is replaced by a general field, or even a ring.
2 Preliminary 2.1 Tangl
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