Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 TwoBridgeKnotDiagram

Source Notebook

Draw a two-bridge knot

Contributed by: Ed Pegg Jr

ResourceFunction["TwoBridgeKnotDiagram"][p,q]

draws a 2-bridge knot where bridges have length p and offset q, with p>q.

Details

Also called Schubert normal form.
In a knot drawing, a line over a gap indicates an overcrossing and undercrossing. A bridge is a segment in a knot drawing with one or more over-crossings.
All knots with up to 7 crossings can be drawn in a 2-bridge form.
From Wikipedia, 2-bridge knot: Of the 1,701,936 knots with up to sixteen crossings, 5,546 are 2-bridge knots.

Examples

Basic Examples (2) 

Show a Schubert normal form or 2-bridge knot for the overhand knot 31:

In[1]:=
ResourceFunction["TwoBridgeKnotDiagram"][3, 2]
Out[1]=

Show a different 2-bridge form for the overhand knot:

In[2]:=
ResourceFunction["TwoBridgeKnotDiagram"][3, 1]
Out[2]=

Scope (2) 

Show the figure-8 knot (41) in 2-bridge form:

In[3]:=
ResourceFunction["TwoBridgeKnotDiagram"][5, 2]
Out[3]=

Draw the knots 61, 62 and 63 in 2-bridge form:

In[4]:=
Row[{ResourceFunction["TwoBridgeKnotDiagram"][9, 4], ResourceFunction["TwoBridgeKnotDiagram"][11, 4], ResourceFunction["TwoBridgeKnotDiagram"][13, 5]}]
Out[4]=

Possible Issues (1) 

As knots get larger, gaps can obscure the small segments between the bridges:

In[5]:=
ResourceFunction["TwoBridgeKnotDiagram"][29, 11]
Out[5]=

Version History

  • 1.0.0 – 12 September 2025

Related Resources

License Information