Knot Theory

Pathway

Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.

1. Level 0

   4 pages
   • level 0Crossing signs, writhe, and linking number10 results
     2 definitions, 1 algorithm, 3 lemmas, 4 theoremsExamples & counterexamples →
   • level 0Knot exteriors and peripheral systems6 results
     2 definitions, 3 lemmas, 1 theoremExamples & counterexamples →
   • level 0Satellite knots and JSJ tori16 results
     2 definitions, 12 lemmas, 1 theorem, 1 corollaryExamples & counterexamples →
   • level 0Wirtinger presentations and knot groups8 results
     1 definition, 1 algorithm, 5 lemmas, 1 theoremExamples & counterexamples →

2. Level 1

   8 pages
   • level 1Dehn surgery on knots and homology5 results
     2 definitions, 2 lemmas, 1 theoremExamples & counterexamples →
   • level 1Fox colorings and quandle invariants11 results
     2 definitions, 2 algorithms, 2 lemmas, 5 theoremsExamples & counterexamples →
   • level 1Hyperbolic knot complements and volume25 results
     4 definitions, 17 lemmas, 4 theoremsExamples & counterexamples →
   • level 1Kauffman bracket state sum and the Jones polynomial8 results
     1 definition, 1 algorithm, 3 lemmas, 3 theoremsExamples & counterexamples →
   • level 1Multivariable Alexander modules and chain models14 results
     3 definitions, 1 algorithm, 8 lemmas, 2 theoremsExamples & counterexamples →
   • level 1Satellite operators and concordance filtrations51 results
     2 definitions, 33 lemmas, 15 theoremsExamples & counterexamples →
   • level 1Seifert surfaces and knot genus10 results
     2 definitions, 1 algorithm, 2 lemmas, 5 theoremsExamples & counterexamples →
   • level 1Vassiliev invariants and chord diagrams6 results
     2 definitions, 2 lemmas, 1 theorem, 1 remarkExamples & counterexamples →

3. Level 2

   12 pages
   • level 2Alternating diagrams and crossing-number minimality18 results
     5 definitions, 10 lemmas, 2 theorems, 1 corollaryExamples & counterexamples →
   • level 2Hyperbolic Dehn filling and exceptional slopes17 results
     2 definitions, 12 lemmas, 3 theoremsExamples & counterexamples →
   • level 2Kauffman and Dubrovnik polynomials from BMW algebras15 results
     3 definitions, 1 algorithm, 8 lemmas, 3 theoremsExamples & counterexamples →
   • level 2Khovanov homology and Jones categorification12 results
     2 definitions, 1 algorithm, 5 lemmas, 4 theoremsExamples & counterexamples →
   • level 2Milnor invariants and link peripheral nilpotent quotients20 results
     2 definitions, 1 algorithm, 10 lemmas, 7 theoremsExamples & counterexamples →
   • level 2Planar-diagram codes and knot-table enumeration8 results
     1 definition, 2 algorithms, 2 lemmas, 3 theoremsExamples & counterexamples →
   • level 2Ribbon categories and Reshetikhin–Turaev knot invariants7 results
     2 definitions, 1 algorithm, 1 lemma, 3 theoremsExamples & counterexamples →
   • level 2Seifert matrices and the Alexander module8 results
     2 definitions, 5 lemmas, 1 theoremExamples & counterexamples →
   • level 2Skein modules for links in 3-manifolds9 results
     1 definition, 6 lemmas, 2 theoremsExamples & counterexamples →
   • level 2The Kontsevich integral and universality15 results
     2 definitions, 10 lemmas, 3 theoremsExamples & counterexamples →
   • level 2Three-manifold surgery presentations and Kirby moves10 results
     4 definitions, 3 lemmas, 3 theoremsExamples & counterexamples →
   • level 2Twisted Alexander polynomials and Reidemeister torsion11 results
     2 definitions, 1 algorithm, 4 lemmas, 4 theoremsExamples & counterexamples →

4. Level 3

   7 pages
   • level 3Alexander polynomials from Seifert matrices5 results
     1 lemma, 1 theorem, 2 corollaries, 1 exampleExamples & counterexamples →
   • level 3Colored Jones and the volume conjecture6 results
     1 definition, 1 algorithm, 2 lemmas, 1 theoremExamples & counterexamples →
   • level 3Normal-surface unknot recognition63 results
     2 definitions, 1 algorithm, 50 lemmas, 10 theoremsExamples & counterexamples →
   • level 3Root-of-unity skein TQFT for 3-manifolds11 results
     4 definitions, 6 lemmas, 1 theoremExamples & counterexamples →
   • level 3Smooth and topological knot concordance19 results
     8 definitions, 9 lemmas, 2 theoremsExamples & counterexamples →
   • level 3Terminating computations for classical knot invariants8 results
     1 definition, 2 algorithms, 1 lemma, 4 theoremsExamples & counterexamples →
   • level 3The knot-complement theorem6 results
     1 definition, 3 lemmas, 2 theoremsExamples & counterexamples →

5. Level 4

   6 pages
   • level 4Connected sums and prime decomposition37 results
     2 definitions, 28 lemmas, 7 theoremsExamples & counterexamples →
   • level 4Cyclic branched covers and knot determinants10 results
     1 definition, 7 lemmas, 2 theoremsExamples & counterexamples →
   • level 4Freedman’s theorem and topological sliceness13 results
     4 definitions, 5 lemmas, 4 theoremsExamples & counterexamples →
   • level 4Levine–Tristram signatures and the Arf invariant18 results
     5 definitions, 11 lemmas, 2 theoremsExamples & counterexamples →
   • level 4Ribbon knots and the slice-ribbon problem6 results
     1 definition, 3 lemmas, 1 theoremExamples & counterexamples →
   • level 4Torus knots and Seifert-fibered exteriors7 results
     1 definition, 4 lemmas, 2 theoremsExamples & counterexamples →

6. Level 5

   4 pages
   • level 5Algebraic concordance and metabolic forms8 results
     1 definition, 5 lemmas, 2 theoremsExamples & counterexamples →
   • level 5Casson–Gordon invariants and sliceness obstructions64 results
     14 definitions, 35 lemmas, 15 theoremsExamples & counterexamples →
   • level 5Character varieties and the A-polynomial47 results
     7 definitions, 27 lemmas, 8 theorems, 1 corollaryExamples & counterexamples →
   • level 5Fox–Milnor factorization and slice obstructions4 results
     1 lemma, 2 theorems, 1 corollaryExamples & counterexamples →

7. Level 6

   3 pages
   • level 6Grid homology and combinatorial knot Floer theory24 results
     4 definitions, 1 algorithm, 13 lemmas, 6 theoremsExamples & counterexamples →
   • level 6Open questions in knot polynomials and surgery3 results
     1 definition, 1 lemma, 1 remarkExamples & counterexamples →
   • level 6Smooth sliceness and four-ball genus obstructions17 results
     4 definitions, 8 lemmas, 4 theorems, 1 corollaryExamples & counterexamples →

8. Level 7

   5 pages
   • level 7Branched-cover correction terms and sliceness12 results
     2 definitions, 8 lemmas, 1 theorem, 1 corollaryExamples & counterexamples →
   • level 7Khovanov homology detects the unknot16 results
     3 definitions, 10 lemmas, 3 theoremsExamples & counterexamples →
   • level 7Knot Floer genus and fibering detection10 results
     4 definitions, 3 lemmas, 3 theoremsExamples & counterexamples →
   • level 7Lee deformation and Rasmussen’s s-invariant26 results
     4 definitions, 16 lemmas, 4 theorems, 1 corollary, 1 remarkExamples & counterexamples →
   • level 7The Khovanov to Floer spectral sequence18 results
     3 definitions, 12 lemmas, 3 theoremsExamples & counterexamples →

9. Level 8

   2 pages
   • level 8Knot cobordism maps and functoriality11 results
     4 definitions, 4 lemmas, 3 theoremsExamples & counterexamples →
   • level 8Zero traces and the Conway nonsliceness theorem14 results
     3 definitions, 7 lemmas, 4 theoremsExamples & counterexamples →

10. Level 9

    2 pages
    • level 9Bridge presentations and unknotting number16 results
      1 definition, 1 algorithm, 6 lemmas, 8 theoremsExamples & counterexamples →
    • level 9The tau invariant and four-ball genus6 results
      5 lemmas, 1 theoremExamples & counterexamples →

11. Level 10

    1 page
    • level 10Epsilon and Upsilon concordance invariants11 results
      3 definitions, 4 lemmas, 4 theoremsExamples & counterexamples →