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On the quantum $\mathfrak{sl}_3$ invariant of positive links

Matthew Harper, Efstratia Kalfagianni

Abstract

We use the skein theory of $\mathfrak{sl}_3$-webs to study the properties of the quantum $\mathfrak{sl}_3$-link polynomial of positive links. We give explicit formulae for the three leading terms of the polynomial on positive links in terms of diagrammatic quantities of their positive diagrams. We show that a positive link is fibered if and only the second coefficient of the polynomial is equal to one. We also show that the third coefficient provides obstructions to representing links by positive braids.

On the quantum $\mathfrak{sl}_3$ invariant of positive links

Abstract

We use the skein theory of -webs to study the properties of the quantum -link polynomial of positive links. We give explicit formulae for the three leading terms of the polynomial on positive links in terms of diagrammatic quantities of their positive diagrams. We show that a positive link is fibered if and only the second coefficient of the polynomial is equal to one. We also show that the third coefficient provides obstructions to representing links by positive braids.

Paper Structure

This paper contains 16 sections, 22 theorems, 68 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. The quantum $\sl_3$ invariant of unframed links
  3. Definition of the invariant
  4. Computation of $\langle\!\langle\cdot\rangle\!\rangle$ by states
  5. Relating maximum degrees of weights
  6. The two leading terms of the $\mathfrak{sl}_3$ invariant for positive links
  7. The $O$ and $W$ state graphs
  8. Proof of Theorem \ref{['thm.second']}
  9. A characterization of positive fibered links
  10. The third coefficient of the $\mathfrak{sl}_3$ invariant
  11. Separated and mixed states
  12. Some web identities
  13. The proof of Theorem \ref{['thm.third']}
  14. Connected sum and disjoint union
  15. Obstructing positive braiding
  16. ...and 1 more sections

Key Result

Theorem 1.1

For any connected positive diagram $D=D(L)$, we have: Here $v$ and $e'$ are as defined above, $\mu$ is the number of edges in $\mathbb{G}'_W$ that have multiplicity greater than one in $\mathbb{G}_W$, and $\theta$ is the number of pairs of edges in $\mathbb{G}'_W$ which are mixed at a vertex in $\mathbb{G}'_W$.

Figures (8)

  • Figure 1: A positive crossing and a negative crossing.
  • Figure 2: The oriented ($O$) and web ($W$) -resolutions of a crossing.
  • Figure 3: An $OW$-move between two state webs $\mathbb{W}$(left) and $\mathbb{W}'$(right) in the disc $E$ whose boundary is indicated by the dotted line.
  • Figure 4: The clearing region between the two arcs of an $O$-resolution.
  • Figure 5: The $A$-resolution (left) the $B$-resolution (right) of a crossing and their contribution to state surfaces. In both cases the edges of the state graph are shown in red.
  • ...and 3 more figures

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3: Ohtsuki
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • ...and 41 more