Compositions of Knots, and Knot Catalogues

Knots can be composed: Take two knots J, K, form a new one, called J#K. Compositions of Knots, and Knot Catalogues

Knots can be composed: Take two knots J, K, form a new one, called J#K.

To compose two knots, break them open and connect the openings in such a way that no new crossings are created Compositions of Knots, and Knot Catalogues

Knots can be composed: Take two knots J, K, form a new one, called J#K.

To compose two knots, break them open and connect the openings in such a way that no new crossings are created Compositions of Knots, and Knot Catalogues

Knots can be composed: Take two knots J, K, form a new one, called J#K.

To compose two knots, break them open and connect the openings in such a way that no new crossings are created Picture taken from Adams’ book Question: Is the composition J#K well-defined, i.e. independent of the choices of diagrams for J, K, and the choice where the knots are opened? Question: Is the composition J#K well-defined, i.e. independent of the choices of diagrams for J, K, and the choice where the knots are opened? Question: Is the composition J#K well-defined, i.e. independent of the choices of diagrams for J, K, and the choice where the knots are opened?

Answer: In general, no. Question: Is the composition J#K well-defined, i.e. independent of the choices of diagrams for J, K, and the choice where the knots are opened?

Answer: In general, no.. But the composition is well-defined if J, K are oriented, and the orientations match up in J#K. Question: Is the composition J#K well-defined, i.e. independent of the choices of diagrams for J, K, and the choice where the knots are opened? Question: Is the composition J#K well-defined, i.e. independent of the choices of diagrams for J, K, and the choice where the knots are opened? Question: Is the composition J#K well-defined, i.e. independent of the choices of diagrams for J, K, and the choice where the knots are opened? Picture taken from Adams’ book Picture taken from Adams’ book 817#817 ≠ 817#m(817)

Picture taken from Adams’ book Theorem 3.10 ■ The composition J #K of oriented topological knots J , K is well-defined. ■ If K is any oriented topological knot, and U the oriented topological unknot, then

K#U = K . Theorem 3.10 ■ The composition J #K of oriented topological knots J , K is well-defined. ■ If K is any oriented topological knot, and U the oriented topological unknot, then

K#U = K .

Proof of second part: Theorem 3.10 ■ The composition J #K of oriented topological knots J , K is well-defined. ■ If K is any oriented topological knot, and U the oriented topological unknot, then

K#U = K .

Proof of second part: Theorem 3.10 ■ The composition J #K of oriented topological knots J , K is well-defined. ■ If K is any oriented topological knot, and U the oriented topological unknot, then

K#U = K .

Proof of second part: Theorem 3.10 ■ The composition J #K of oriented topological knots J , K is well-defined. ■ If K is any oriented topological knot, and U the oriented topological unknot, then

K#U = K .

Proof of second part: Definition 3.11

A topological knot K which can be written as K = K1#K2, with both K1 and K2 non-trivial, is called composite.

Picture taken from Adams’ book Example: Analogy to integers and primes:

integers knots multiplication of integers n · m composition of knots J #K 1 unknot composite integer n = m · k, m, k ≠ 1 composite knot Analogy to integers and primes:

integers knots multiplication of integers n · m composition of knots J #K 1 unknot composite integer n = m · k, m, k ≠ 1 composite knot

The analogy goes even further:

No. Like 1 ≠ m · k for integers m, k ≠ 1, the unknot is not composite. Analogy to integers and primes:

integers knots multiplication of integers n · m composition of knots J #K 1 unknot composite integer n = m · k, m, k ≠ 1 composite knot

The analogy goes even further:

No. Like 1 ≠ m · k for integers m, k ≠ 1, the unknot is not composite. Analogy to integers and primes:

integers knots multiplication of integers n · m composition of knots J #K 1 unknot composite integer n = m · k, m, k ≠ 1 composite knot

The analogy goes even further:

integers knots 1 is not composite The unknot is not composite Define a prime number to be a Define a prime knot to be a knot that number that is not composite, and not 1 is not composite, and not the unknot Each integer n factors into a Each knot K factors into a unique set of prime numbers unique set of prime knots n = p1 ··· pN K = P1# ... #PN Knot Catalogues

■ Knot catalogues list prime knots (and the unknot), like a list of prime numbers (and 1): 1, 2, 3, 5, 7, 11, 13, 17, ...

■ Notation: nm means knot “number m” (in an arbitrary order) with crossing number n. ■ Note: Mirror images are not listed, although they can be ̸ ∗ inequivalent (as in the case of the trefoil 31 = 31, for example) Knot Catalogues

■ Knot catalogues list prime knots (and the unknot), like a list of prime numbers (and 1): 1, 2, 3, 5, 7, 11, 13, 17, ...

■ Number of knots with crossing number up to 10: 249. The power of the Jones polynomial

■ Number of knots with crossing number up to 10: 249. ■ Number of different Jones polynomials for these knots: 248. The power of the Jones polynomial

■ Number of knots with crossing number up to 10: 249. ■ Number of different Jones polynomials for these knots: 248.

■ Only 51 and 10132 have the same Jones polynomial,

− −7 −6 − −5 −4 −2 V51 (t) = V10132 (t) = t + t t + t + t The power of the Jones polynomial

■ Number of knots with crossing number up to 10: 249. ■ Number of different Jones polynomials for these knots: 248.

■ Only 51 and 10132 have the same Jones polynomial,

− −7 −6 − −5 −4 −2 V51 (t) = V10132 (t) = t + t t + t + t

■ Number of knots with crossing number up to 10 that are different from their mirror image (chiral): 229. The power of the Jones polynomial

■ Number of knots with crossing number up to 10: 249. ■ Number of different Jones polynomials for these knots: 248.

■ Only 51 and 10132 have the same Jones polynomial,

− −7 −6 − −5 −4 −2 V51 (t) = V10132 (t) = t + t t + t + t

■ Number of knots with crossing number up to 10 that are different from their mirror image (chiral): 229.

■ Up to two exceptions, 942 and 10125, the Jones polynomial ∗ detects these chiralities: V(K) ≠ V(K ) unless K = 942 or K = 10125. Question: Is the unknot the only top. knot K with VK(t) = 1? Question: Is the unknot the only top. knot K with VK(t) = 1?

Answer: Unknown.