{\it Skein modules}, which are invariants of 3-manifolds as well as of links in these manifolds, were introduced by Przytycki ([Bull. Pol. Acad. Sci., Math. 39, No.1-2, 91-100 (1991; Zbl 0762.57013)]) and Turaev ([J. Sov. Math. 52, No.1, 2799-2805 (1990; Zbl 0706.57004)]). In the present paper, diagrams and Reidemeister moves for links in a twisted $\mathbb S^1$-bundles over a non-orientable surface are introduced, and the Kauffman bracket skein module (KBSM) of $\mathbb R P^3 \times \mathbb R P^3$ is computed. Note that the notion of diagrams of links in $F \times \mathbb S^1$ ($F$ being an orientable surface)), was introduced in [Topology Appl. 156, No. 10, 1831-1849 (2009; Zbl 1168.57010)], together with Reidemeister moves for such diagrams. After having extended the above notions to the case $N \hat \times \mathbb S^1$ ($N$ being a non-orientable surface), the author takes into account the particular case $N= \mathbb RP^2$, so that $N \hat \times \mathbb S^1 = \mathbb R P^3 \times \mathbb R P^3.$ The full computation of KBSM for $\mathbb R P^3 \times \mathbb R P^3$ shows that it has torsion (as it happens for $\mathbb S^1 \times \mathbb S^2$: see [Math. Z. 220, No.1, 65-73 (1995; Zbl 0826.57007)]), but - unlike the KBSM of $\mathbb S^1 \times \mathbb S^2$ - it does not split as a sum of cyclic modules. A new computation of KBSM of both $\mathbb S^1 \times \mathbb S^2$ and the lens space $L(p,1)$ completes the paper.

REVIEW OF: "Mroczkowski Maciej, Kauffman Bracket Skein module of the connected sum of two projective spaces, J. Knot Theory Ramifications 20, No. 5, 651-675 (2011)".[DE059180169] / Casali, Maria Rita. - In: EXCERPTS FROM ZENTRALBLATT MATH. - ISSN 2190-3484. - STAMPA. - Zbl 1227.57019:(2011).

REVIEW OF: "Mroczkowski Maciej, Kauffman Bracket Skein module of the connected sum of two projective spaces, J. Knot Theory Ramifications 20, No. 5, 651-675 (2011)".[DE059180169]

CASALI, Maria Rita
2011

Abstract

{\it Skein modules}, which are invariants of 3-manifolds as well as of links in these manifolds, were introduced by Przytycki ([Bull. Pol. Acad. Sci., Math. 39, No.1-2, 91-100 (1991; Zbl 0762.57013)]) and Turaev ([J. Sov. Math. 52, No.1, 2799-2805 (1990; Zbl 0706.57004)]). In the present paper, diagrams and Reidemeister moves for links in a twisted $\mathbb S^1$-bundles over a non-orientable surface are introduced, and the Kauffman bracket skein module (KBSM) of $\mathbb R P^3 \times \mathbb R P^3$ is computed. Note that the notion of diagrams of links in $F \times \mathbb S^1$ ($F$ being an orientable surface)), was introduced in [Topology Appl. 156, No. 10, 1831-1849 (2009; Zbl 1168.57010)], together with Reidemeister moves for such diagrams. After having extended the above notions to the case $N \hat \times \mathbb S^1$ ($N$ being a non-orientable surface), the author takes into account the particular case $N= \mathbb RP^2$, so that $N \hat \times \mathbb S^1 = \mathbb R P^3 \times \mathbb R P^3.$ The full computation of KBSM for $\mathbb R P^3 \times \mathbb R P^3$ shows that it has torsion (as it happens for $\mathbb S^1 \times \mathbb S^2$: see [Math. Z. 220, No.1, 65-73 (1995; Zbl 0826.57007)]), but - unlike the KBSM of $\mathbb S^1 \times \mathbb S^2$ - it does not split as a sum of cyclic modules. A new computation of KBSM of both $\mathbb S^1 \times \mathbb S^2$ and the lens space $L(p,1)$ completes the paper.
Casali, Maria Rita
REVIEW OF: "Mroczkowski Maciej, Kauffman Bracket Skein module of the connected sum of two projective spaces, J. Knot Theory Ramifications 20, No. 5, 651-675 (2011)".[DE059180169] / Casali, Maria Rita. - In: EXCERPTS FROM ZENTRALBLATT MATH. - ISSN 2190-3484. - STAMPA. - Zbl 1227.57019:(2011).
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