Let us consider the two trefoils with Gauss codes $(1U+,2O+,3U+,1O+,2U+,3O+)$ and $(1O-,2U-,3O-,1U-,2O-,3U-)$ respectively ( $O/U$ - over/under, $-/+$ - negative/positive crossing type). These knots are not isotopic. They are, however, the mirror image of each other. Is it true, that if we neglect the information about the crossing types (positive/negative) in Gauss codes, they will determine knots up to mirror images? What would be the effect of such negletion on knot invariants (such as Jones Polynomial, Khovanov homology, etc. )?

Remark. We consider also these Gauss codes which define proper virtual knots, i.e. the knot diagrams of these knots cannot have only classical crossings.


1 Answer 1


If you neglect the $+/-$, then the virtual knot is well-defined up to an operation Kauffman calls virtualization, which is "flanking by virtuals" then flipping the crossing:

example of flanking by virtuals

Kauffman, Louis H., Virtual knot theory, Eur. J. Comb. 20, No. 7, 663-690 (1999). ZBL0938.57006.

If you take the virtual trefoil, then virtualize one of the two crossings, you get the unknot. Since the virtual trefoil is not virtually equivalent to the unknot, this implies that for proper virtual knots, Gauss codes without the $+/-$ information do not determine the virtual knot, even up to mirror image and orientation reversal.

That paper shows how the Kauffman bracket is invariant under flanking by virtuals. If the Jones polynomial were well defined for Gauss codes without $+/-$, then it'd have to be well-defined up to flipping crossings, and yet it is sensitive to flipping crossings. For example, the trefoil knot and the unknot have different Jones polynomials, yet they are related by a single crossing flip.

For classical knots, you can virtualize all the crossings in a connect summand to mirror it. In general, $K_1\mathbin{\#}K_2$ is not equivalent to $K_1\mathbin{\#}mK_2$. While a prime unoriented knot can be recovered up to mirror image from its unsigned Gauss code (mentioned in Lickorish, An introduction to knot theory), that doesn't imply that there aren't pairs of distinct classical knots that are related by virtualization moves.

I am not sure of examples of virtualization moves that connect two distinct classical knots (beyond mirror images), but in principal I do not see why there shouldn't be. As a start for where not to look, Lemma 5 says that the involutory knot quandle is invariant under virtualization, so since for classical knotted knots this involutory quandle is nontrivial, virtualization of the unknot only yields proper virtual knots or the unknot.


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