The word problem for braided monoidal categories is unknot-hard
Antonin Delpeuch Jamie Vicary
July 15, 2021 Applied Category Theory 2021
A. Delpeuch Unknot-hardness ACT 2021 1 / 16 Periodic table of n-categories
k \n 0 1 2 3 0 set category 2-category 3-category 1 {?} monoid monoidal cat. monoidal 2-cat. . 2 . {?} comm. monoid braided monoidal cat. . 3 . {?} comm. monoid . 4 . {?}
A. Delpeuch Unknot-hardness ACT 2021 2 / 16 Braided monoidal categories
Definition A braided monoidal category C is a monoidal category equipped with a natural isomorphism σA,B : A ⊗ B → B ⊗ A satisfying the hexagon equations.
A B B A
B A A B
−1 (a) String diagram for σA,B (b) String diagram for σA,B
A. Delpeuch Unknot-hardness ACT 2021 3 / 16 Axioms of braided monoidal categories
A B C A B C A B ⊗ C A ⊗ B C
= =
Figure: Hexagon equations
A. Delpeuch Unknot-hardness ACT 2021 4 / 16 Axioms of braided monoidal categories
= =
(a) Reidemeister 2 move (σ is an iso)
......
... = ...
......
(b) Pull-through move (naturality of σ)
A. Delpeuch Unknot-hardness ACT 2021 4 / 16 Word problem for braided monoidal categories
Decision problem Given two expressions of morphisms in a free braided monoidal category, determine if the morphisms they represent are equal.
= ?
A. Delpeuch Unknot-hardness ACT 2021 5 / 16 A few examples
= ?
A. Delpeuch Unknot-hardness ACT 2021 6 / 16 A few examples
6= = ?
A. Delpeuch Unknot-hardness ACT 2021 6 / 16 A few examples
6= 6=
= ?
A. Delpeuch Unknot-hardness ACT 2021 6 / 16 A few examples
6= 6=
6=
A. Delpeuch Unknot-hardness ACT 2021 6 / 16 This decision problem is known to be in NP and coNP, but no polynomial time algorithm is known for it.
Unknotting knots: a well-studied problem
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A. Delpeuch Unknot-hardness ACT 2021 7 / 16 Unknotting knots: a well-studied problem
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This decision problem is known to be in NP and coNP, but no polynomial time algorithm is known for it.
A. Delpeuch Unknot-hardness ACT 2021 7 / 16 Extra equations for knots
= =
(a) Yanking moves
= =
= =
(b) Reidemeister type 1 moves
A. Delpeuch Unknot-hardness ACT 2021 8 / 16 = =
= = =
Braided
= =
= = Knot
A. Delpeuch Unknot-hardness ACT 2021 9 / 16 Our result
Theorem The unknotting problem can be polynomially reduced to the word problem for braided monoidal categories.
A. Delpeuch Unknot-hardness ACT 2021 10 / 16 First attempt
A. Delpeuch Unknot-hardness ACT 2021 11 / 16 First attempt
=?
A. Delpeuch Unknot-hardness ACT 2021 11 / 16 First attempt
=?
Cap-cup cycle: ( , , , , , )
A. Delpeuch Unknot-hardness ACT 2021 11 / 16 Writhe
Definition Given an oriented knot diagram, its writhe is obtained by summing the local writhe at each crossing:
w( ) = +1 w( ) = −1
Lemma The axioms of braided monoidal categories preserve the writhe.
A. Delpeuch Unknot-hardness ACT 2021 12 / 16 Idea of the proof
A. Delpeuch Unknot-hardness ACT 2021 13 / 16 Idea of the proof
A. Delpeuch Unknot-hardness ACT 2021 13 / 16 Idea of the proof
A. Delpeuch Unknot-hardness ACT 2021 13 / 16 Idea of the proof
+ −
+ +
+ −
A. Delpeuch Unknot-hardness ACT 2021 13 / 16 Idea of the proof
+ + −
+ + =BMC?
+ − +
A. Delpeuch Unknot-hardness ACT 2021 13 / 16 Is the word problem for braided monoidal categories even decidable?
Conclusion
The word problem for braided monoidal categories is at least as hard as the unknotting problem.
A. Delpeuch Unknot-hardness ACT 2021 14 / 16 Conclusion
The word problem for braided monoidal categories is at least as hard as the unknotting problem.
Is the word problem for braided monoidal categories even decidable?
A. Delpeuch Unknot-hardness ACT 2021 14 / 16 Braids
The free braided monoidal category on a single object is the category of braids.
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A. Delpeuch Unknot-hardness ACT 2021 15 / 16 But braided monoidal categories can have non-braid morphisms!
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Word problem for braids
Theorem The word problem for braids can be solved in quadratic time in the length of the braids.
A. Delpeuch Unknot-hardness ACT 2021 16 / 16 Word problem for braids
Theorem The word problem for braids can be solved in quadratic time in the length of the braids.
But braided monoidal categories can have non-braid morphisms!
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A. Delpeuch Unknot-hardness ACT 2021 16 / 16