On Triangulations of the 3 Ball And the Solid Torus
On Triangulations of the 3 Ball And the Solid Torus
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. . , u 9, then all the boundary triangles of M are incident on a vertex, i. E. , M is the star of a vertex, which is a contradiction.
Case 3: Assume v falls neither into Case 1 nor into Case 2. Then lk&(v) has k > 2 interior vertices. So p(v) > 2.
Combining the three cases, we see that Y^vP( v ) ^ 2n. D 3 Triangulation of Solid Torus In this section we prove an analogue of Theorem 1 for the solid torus. Both the main proof and the following lemma depend on Proof 1 above.
Lemma 1 There is no tr...iangulation of B 3 in which two tetrahedra that share a vertex v are of type 2, and the remaining tetrahedra are of type 1.
Proof: (We borrow notation from Proof 1 above. ) Assume A is a triangulation that contradicts the claim. One can easily show that lk^(v) contains a 2-ball C with relint(C) C int(B 3 ) and relbd(C) C d(B 3 ). C cuts A into two pieces, say A! and A2. Lk^(v) is contained in one of the pieces, say in A 2 . C must have an almost cutting triangle and arguing as in Proof 1, we find a cutting triangle T in Aj.
What to read after On Triangulations of the 3 Ball And the Solid Torus?
You can find similar books in the "Read Also" column, or choose other free books by Geza Bohus to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
. . , u 9, then all the boundary triangles of M are incident on a vertex, i. E. , M is the star of a vertex, which is a contradiction.
Case 3: Assume v falls neither into Case 1 nor into Case 2. Then lk&(v) has k > 2 interior vertices. So p(v) > 2.
Combining the three cases, we see that Y^vP( v ) ^ 2n. D 3 Triangulation of Solid Torus In this section we prove an analogue of Theorem 1 for the solid torus. Both the main proof and the following lemma depend on Proof 1 above.
Lemma 1 There is no tr...iangulation of B 3 in which two tetrahedra that share a vertex v are of type 2, and the remaining tetrahedra are of type 1.
Proof: (We borrow notation from Proof 1 above. ) Assume A is a triangulation that contradicts the claim. One can easily show that lk^(v) contains a 2-ball C with relint(C) C int(B 3 ) and relbd(C) C d(B 3 ). C cuts A into two pieces, say A! and A2. Lk^(v) is contained in one of the pieces, say in A 2 . C must have an almost cutting triangle and arguing as in Proof 1, we find a cutting triangle T in Aj.
What to read after On Triangulations of the 3 Ball And the Solid Torus?
You can find similar books in the "Read Also" column, or choose other free books by Geza Bohus to read onlineMoreLess
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