Thursday, January 31, 2019

KLO 0.921a

KLO 0.921a is online at

Small fix so that KLO can read its files with band-slides and band-swims in them (thanks to Brendan Owens for pointing that out), plus changed the web link and gave one line of instructions when entering braids symbolically. 

Tuesday, January 29, 2019

KLO 0.92a

KLO 0.92 alpha is now online at

  • Bands can be drawn with knots (this is not new) to represent surfaces in 4-space; band-slides and band-swims for these diagrams have now been implemented:
    • To perform a band-slide, drag the end of the band to slide to an adjacent end from a different band.
    • To perform a band-swim, drag the end of the band to be swum along to an edge for a different band on the same face (note that this is effectively the opposite of the band-swim being done; starting with the band-end makes the interfacing simpler).
  • When constructing a new knot, a new checkbox now exists to specify a braid to close; selecting this opens up a dialog in which the number of strands and the braid can be entered (braid generators are a, b, c, ... from the top, with the inverses given by A, B, C, ...).  The braid is visually previewed as you enter it. 

Saturday, January 26, 2019

KLO 0.91a

Two new features of KLO have been introduced in KLO 0.91 alpha, along with one improvement:
  1. Relative Kirby diagrams (bottom components <bracketed>) are now possible, and a new button for exists for computer-aided turning upside-down of a Kirby diagram.  The initial steps are performed by KLO, after which the user manipulates the bracketed components into a zero-framed unlink (with the top components following along).  A button finishes the mode and reports the number of 3- and 4-handles.
  2. Kirby diagrams for 2-fold branched covers of knots and links are now a button-click away for any knot.
  • Blow-downs of ±1-framed 2-handles now only requires that there are no crossings inside the loop (crossings inside a loop with no self-crossings can still be cleared in one click by setting focus on the loop and clicking inside it).