Some link homologies in $\mathbb{RP}^3$

This paper introduces new extensions of Khovanov, Lee, and Bar-Natan homologies for links in $\mathbb{RP}^3$ that are distinct from previous constructions and yield novel Rasmussen invariants.

The Gist

Imagine you are trying to understand the shape of a knot, but not just any knot. You are looking at knots tied in a very strange, twisted universe called $\mathbb{R}\mathbb{P}^3$ (Real Projective 3-space).

To visualize this universe, imagine a video game world where if you walk off the right edge of the screen, you instantly reappear on the left edge, but flipped upside down. It's a world with a "twist" built into its very fabric.

For decades, mathematicians have had a powerful tool called Khovanov Homology to study knots in our normal, flat universe (like a standard piece of string). This tool turns a picture of a knot into a complex algebraic "fingerprint" that tells you everything about its shape and how it might be untangled.

The Problem:
When mathematicians tried to use this tool in the "twisted universe" ($\mathbb{R}\mathbb{P}^3$), they hit a wall. Previous attempts to adapt the tool worked, but they were like using a map of the Earth to navigate a Möbius strip—they didn't quite fit the geometry. Some existing methods only worked for specific types of knots, and others missed subtle details.

The Solution (This Paper):
William Rushworth, the author of this paper, has built brand new, custom-made tools specifically designed for this twisted universe. He didn't just tweak the old maps; he invented three new types of "knot scanners":

1. The Doubled Khovanov Scanner: A new way to create the initial fingerprint of the knot.
2. The Doubled Lee Scanner: A more advanced version that filters the fingerprint to find deep geometric secrets (like how "knotted" the knot really is).
3. The Doubled Bar-Natan Scanner: Another advanced version, similar to the Lee one but using a different mathematical "lens" (specifically for a system where numbers behave like on/off switches).

The Creative Analogy: The "Double-Deck" Knot

Think of a knot in this twisted universe as a double-decker bus driving on a road that loops back on itself upside down.

• Old Methods: Previous mathematicians tried to study this bus by looking at just the top deck or just the bottom deck, or by using a map that assumed the road was flat. They missed how the two decks interact with the "twist" of the road.
• Rushworth's New Method: Rushworth's tools look at both decks simultaneously and account for the fact that the road flips the bus upside down. He calls this the "Doubled" theory because it effectively doubles the information, keeping track of two layers of reality at once.

What Did He Discover?

1. New Fingerprints: Rushworth proved that his new tools see things the old tools missed. For example, he found a specific knot (labeled "21" in a math catalog) where the old tools said the knot was "simple" in one way, but his new tool revealed it was actually "complex" in a different way.
2. The "Rasmussen" Score: In knot theory, there is a famous number called the Rasmussen invariant. Think of this as a "difficulty score" for untangling a knot.
   • Rushworth created new difficulty scores for knots in this twisted universe.
   • He showed that his new scores are different from the scores calculated by other recent researchers (Manolescu-Willis and Chen).
   • Why it matters: If you want to know if a knot can be untangled into a perfect circle (a property called "slice-ness"), these scores act as a lie detector. If the score is non-zero, the knot cannot be untangled. Rushworth's new scores might catch a "liar" knot that the old scores let slip by.

The "Twist" in the Story

The paper also highlights a funny quirk of this twisted universe:

• In our normal world, every knot has a clear "front" and "back" (orientation).
• In the twisted universe, some knots are so tangled that they don't have a consistent front or back.
• Rushworth's new tools use a concept called "2-colouring" (painting the knot orange and pink) to handle these weird knots. If you can paint the knot with two colors without the colors clashing at the crossings, the knot is "2-colourable."
• He found that his new tools work on a wider variety of knots than the previous "twisted orientation" tools, making them more versatile.

The Big Question Left Behind

While Rushworth built a better toolbox, he admits there are still mysteries.

• The "E-Page" Mystery: Imagine the tools produce a series of snapshots of the knot as they analyze it. The final snapshot (the "E-infinity page") is the answer.
• Rushworth found that for the knots he tested, his new snapshots and the old snapshots (by Chen) seemed to show the same final picture, even though the journey to get there was totally different.
• The Open Question: He suspects that for some very complex, hidden knots, the final pictures might actually be different. He is asking the math community: "Is there a knot out there where our new tool sees a secret that the old tool completely misses?"

Summary

William Rushworth has updated the mathematical toolkit for studying knots in a twisted, upside-down universe. He built new "Doubled" scanners that are more sensitive and versatile than previous versions. These new tools provide better "difficulty scores" for knots, helping mathematicians determine which knots are truly unbreakable and which can be untangled, while also opening up new questions about the hidden geometry of these strange spaces.

Technical Summary

Here is a detailed technical summary of the paper "Some Link Homologies in $\mathbb{R}\mathbb{P}^3$" by William Rushworth.

1. Problem Statement

The paper addresses the extension of Khovanov homology and its associated spectral sequences (Lee and Bar-Natan) to links in the real projective 3-space ($\mathbb{R}\mathbb{P}^3$). While previous extensions exist—specifically those by Asaeda-Przytycki-Sikora (APS), Gabrovšek, Chen, and Manolescu-Willis—these theories have limitations regarding their construction, coefficient rings, and the invariants they produce.

• The Gap: Existing theories often rely on specific auxiliary data (like 2-colorings or twisted orientations) or are restricted to specific coefficient rings (e.g., $\mathbb{Z}_2$ or rings where 2 is invertible).
• The Goal: To define new "doubled" extensions of Khovanov, Lee, and Bar-Natan homologies that are structurally distinct from previous definitions, work over general commutative rings, and yield new Rasmussen invariants capable of distinguishing links in $\mathbb{R}\mathbb{P}^3$ that previous invariants cannot.

2. Methodology

The author employs a "doubled" construction approach, modifying the standard Khovanov chain complex construction for $\mathbb{R}\mathbb{P}^3$ diagrams.

• Diagrammatic Setup: Links in $\mathbb{R}\mathbb{P}^3$ are represented by tangle diagrams in a disk $D^2$ with antipodal boundary points identified. The author utilizes Reidemeister moves $R_4$ and $R_5$ (in addition to classical moves) to define equivalence.
• The "Doubled" Complex:
  • Algebraic Structure: The coefficient ring $R$ is used to define an algebra $A = R[X]/\langle X^2 \rangle$. The author introduces a grading shift and defines a "doubled" module $A_k = A^{\otimes k} \oplus A^{\otimes k}\{-1\}$.
  • Superscripts: To distinguish the two copies of the algebra, the author uses superscripts $u$ (unshifted) and $\ell$ (shifted/lower).
  • Differentials: The standard Khovanov maps ($m$ for merging, $\Delta$ for splitting) are retained. However, a new map $\eta$ is defined to handle the topology of $\mathbb{R}\mathbb{P}^3$ (specifically the once-punctured Möbius band).
    • $\eta$ maps between the $u$ and $\ell$ copies, effectively "doubling" the state space.
  • Sign Conventions: Signs are assigned to ensure the differential squares to zero, adhering to standard TQFT-like face relations.
• Extensions:
  • Doubled Khovanov ($DKh$): The base homology theory.
  • Doubled Lee ($DKh'$): A perturbation of $DKh$ where the differential includes terms of higher quantum degree (specifically $j$-degree 4). This requires 2 to be invertible in the ground ring.
  • Doubled Bar-Natan ($DBN$): A version over $\mathbb{Z}_2$ where the differential includes terms of $j$-degree 2.
• Spectral Sequences: The author constructs spectral sequences where the $E_2$ page is the Doubled Khovanov homology, abutting to the Doubled Lee or Bar-Natan homology.

3. Key Contributions

A. New Homology Theories

The paper defines three new homology theories:

1. Doubled Khovanov Homology ($DKh$): Distinct from APS, Gabrovšek, and Chen's theories. It does not require auxiliary decorations on diagrams beyond the standard crossing resolutions.
2. Doubled Lee Homology ($DKh'$): Defined over rings where 2 is invertible.
3. Doubled Bar-Natan Homology ($DBN$): Defined over $\mathbb{Z}_2$.

B. Canonical Generators and 2-Colorings

A central technical innovation is the use of 2-colorings of link diagrams (coloring the regions of the diagram with two colors such that crossings satisfy specific parity rules) to define canonical generators.

• Unlike classical Lee homology where generators correspond to link orientations, generators in $\mathbb{R}\mathbb{P}^3$ correspond to 2-colorings.
• The paper proves that a link with $n$ components has $2^{n+1}$ canonical generators if it is 2-colorable, and 0 otherwise.
• This framework allows the author to determine the rank and homological degree support of the homologies without relying on the "twisted orientability" condition required by Chen's theory.

C. Rasmussen Invariants

The author extracts Rasmussen invariants ($d_s$) from the spectral sequences:

• Defined as the maximum $s$-grading (induced by the quantum filtration) of non-zero elements in the homology, minus 1.
• These invariants are concordance invariants and obstruct sliceness.

4. Key Results

• Distinctness of Theories:

  • The new Doubled Khovanov homology is distinct from APS, Gabrovšek, and Chen's theories (demonstrated via rank calculations on specific knots like $3_1$and$2_1$ in Drobotukhina's table).
  • The new Doubled Lee homology is distinct from Manolescu-Willis's extension. For example, on knot $2_1$, Manolescu-Willis's homology is supported in homological degree 0, while the new theory is supported in degree$-2$.
  • The new Doubled Bar-Natan spectral sequence is distinct from Chen's. While they share rank and homological support for null-homologous links, they differ in the number of non-trivial pages and quantum grading support (e.g., on knot $4_1$).

• Rasmussen Invariant Comparisons:

  • The new invariant $d_s^{\mathbb{Q}}$ differs from Manolescu-Willis's $s^{\mathbb{Q}}$.
  • For Chen's Bar-Natan theory ($s^{\mathbb{Z}_2}$) and the new Doubled Bar-Natan ($d_s^{\mathbb{Z}_2}$), the author finds no known counterexamples where they differ (up to a grading shift). However, the author suspects they are distinct in general due to differences in differential degrees.

• Genus Bounds:

  • The new Rasmussen invariants provide genus bounds for 2-colourable cobordisms.
  • The paper establishes that the set of 2-colourable cobordisms is a proper subset of all cobordisms.
  • Crucially, the minimal genus of 2-colourable cobordisms can be arbitrarily larger than the standard slice genus (similar to Chen's result for twisted orientable cobordisms).

5. Significance

• Structural Independence: The new theories are constructed without auxiliary data (like specific orientations or decorations) that previous theories required, making them more "natural" extensions of the classical Khovanov framework to $\mathbb{R}\mathbb{P}^3$.
• New Invariants: The distinct Rasmussen invariants provide new tools for distinguishing links in $\mathbb{R}\mathbb{P}^3$ and studying their concordance properties. The fact that they differ from Manolescu-Willis's invariants suggests a richer structure of link concordance in $\mathbb{R}\mathbb{P}^3$ than previously understood.
• Generalization of Virtual Link Theory: The construction is closely related to homologies for virtual links, suggesting a unified framework for non-orientable surface link theories.
• Open Questions: The paper raises significant questions regarding the relationship between the new spectral sequences and Chen's sequences (Questions A, B, and C), specifically whether the $E_\infty$ pages carry equivalent information and the comparative minimality of genus bounds for 2-colourable vs. twisted orientable cobordisms.

In summary, Rushworth provides a robust, algebraically distinct framework for link homology in $\mathbb{R}\mathbb{P}^3$, expanding the toolkit for low-dimensional topology in non-orientable 3-manifolds and offering new invariants that challenge and refine existing classifications.