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1

3-coloring and other elementary invariants of knots

100%

Przytycki J. H.

Banach Center Publications

|

1998

|

tom 42

|

nr 1

275-295

2

Search for different links with the same Jones' type polynomials: Ideas from graph theory and statistical mechanics

100%

Przytycki J. H.

Banach Center Publications

|

1995

|

tom 34

|

nr 1

121-148

EN

We describe in this talk three methods of constructing different links with the same Jones type invariant. All three can be thought as generalizations of mutation. The first combines the satellite construction with mutation. The second uses the notion of rotant, taken from the graph theory, the third, invented by Jones, transplants into knot theory the idea of the Yang-Baxter equation with the spectral parameter (idea employed by Baxter in the theory of solvable models in statistical mechanics). We extend the Jones result and relate it to Traczyk's work on rotors of links. We also show further applications of the Jones idea, e.g. to 3-string links in the solid torus. We stress the fact that ideas coming from various areas of mathematics (and theoretical physics) has been fruitfully used in knot theory, and vice versa.

3

Skein algebra of a group

64%

Przytycki J. H., Sikora A. S.

Banach Center Publications

|

1998

|

tom 42

|

nr 1

297-306

EN

We define for each group G the skein algebra of G. We show how it is related to the Kauffman bracket skein modules. We prove that skein algebras of abelian groups are isomorphic to symmetric subalgebras of corresponding group rings. Moreover, we show that, for any abelian group G, homomorphisms from the skein algebra of G to C correspond exactly to traces of SL(2,C)-representations of G. We also solve, for abelian groups, the conjecture of Bullock on SL(2,C) character varieties of groups - we show that skein algebras are isomorphic to the coordinate rings of the corresponding character varieties.

4

Lissajous knots and billiard knots

64%

Jones V. F. R., Przytycki J. H.

Banach Center Publications

|

1998

|

tom 42

|

nr 1

145-163

EN

We show that Lissajous knots are equivalent to billiard knots in a cube. We consider also knots in general 3-dimensional billiard tables. We analyse symmetry of knots in billiard tables and show in particular that the Alexander polynomial of a Lissajous knot is a square modulo 2.

5

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Actions of $Z_n$ on some surface-bundles over $S^1$

44%

Przytycki J. H.

Colloquium Mathematicum

|

1982

|

tom 47

|

nr 2

221-239

6

Cyclic actions on $S^2$- and $P^2$-bundles over $S^1$

44%

Przytycki J. H.

Colloquium Mathematicum

|

1982

|

tom 47

|

nr 2

241-254

7

$Z_n$-actions on 3-manifolds

32%

Przytycki J. H.

Colloquium Mathematicum

|

1982

|

tom 47

|

nr 2

199-219

Ograniczanie wyników

4 Banach Center Publications

3 Colloquium Mathematicum

7 Przytycki J. H.

1 Jones V. F. R.

1 Sikora A. S.

3 1998

1 1995

3 1982

3-coloring and other elementary invariants of knots

Search for different links with the same Jones' type polynomials: Ideas from graph theory and statistical mechanics

Skein algebra of a group

Lissajous knots and billiard knots

Actions of $Z_n$ on some surface-bundles over $S^1$

Cyclic actions on $S^2$- and $P^2$-bundles over $S^1$

$Z_n$-actions on 3-manifolds