Interactive tools to draw knots, explore properties, and test your knowledge
Welcome to the final module! You've journeyed through basic definitions, knot invariants, polynomial calculations, advanced topics, and real-world applications. Now it's time to consolidate your knowledge through hands-on experimentation and comprehensive testing.
Use the interactive tools to draw knots, explore their properties, and investigate mathematical relationships. Then take the comprehensive quiz to test your understanding across all topics. Are you ready to become a knot theory expert?
Draw a knot diagram freehand! Try sketching a trefoil, figure-eight, or unknot. The canvas will help you visualize knot structures and understand their properties.
A knot diagram is a 2D projection of a 3D knot showing crossings:
Explore knot properties, polynomials, and operations! Select a knot and investigate its mathematical properties, invariants, and how it relates to other knots.
Crossing Number c(K): Minimum crossings in any diagram
Unknotting Number u(K): Min crossing changes to unknot
Genus g(K): Minimal genus of spanning Seifert surface
Fibered: Complement fibers over S¹
Alternating: Crossings alternate over/under
Chiral: Not equivalent to mirror image
Test your knowledge! This comprehensive quiz covers all topics from basic definitions to advanced applications. Choose your difficulty level and see how well you understand knot theory.
Draw a complex-looking knot that is actually the unknot. Use Reidemeister moves to simplify it. How many moves does it take?
Calculate the Jones polynomial for the figure-eight knot using the skein relation. Verify your result matches: V(t) = t² - t + 1 - t⁻¹ + t⁻²
The trefoil knot is chiral. Draw both the right-handed and left-handed trefoils. Calculate their Jones polynomials and verify they are different.
What is the crossing number of trefoil # trefoil (two trefoils connected)? What is its genus? Is it prime?
The figure-eight knot has the smallest hyperbolic volume (~2.02988). Why can't the trefoil have a hyperbolic volume? What geometry does its complement admit?
A categorification of the Jones polynomial discovered by Mikhail Khovanov (2000). Provides stronger invariants and has connections to physics (gauge theory).
Developed by Ozsváth-Szabó (2004), connects knot theory to symplectic geometry and 4-manifold topology. Can detect genus and fibering.
Relates the colored Jones polynomial to hyperbolic volume. One of the most important open conjectures in knot theory (Kashaev, Murakami-Murakami).
Every slice knot is ribbon. Relates 3D and 4D topology. Remains open despite decades of effort - perhaps you'll solve it!
You've completed the Topology & Knot Theory learning module! You now understand fundamental concepts, can calculate knot invariants, recognize applications to science and technology, and have the foundation to explore advanced topics.
✓ Mastered: Knot definitions, Reidemeister moves, crossing number, knot table
✓ Calculated: Alexander, Jones, and HOMFLY polynomials
✓ Explored: Seifert surfaces, genus, fibered knots, hyperbolic geometry
✓ Applied: DNA topology, quantum computing, statistical mechanics, materials science
✓ Practiced: Interactive drawing, property exploration, comprehensive quiz