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edited Apr 13, 2017 at 12:57

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Here is an expansion of Ian's answer.

Dehn surgery on a hyperbolic knot or link generically gives another hyperbolic manifold. This follows from the Dehn surgery theorem; see Theorem 5.8.2 in chapter five of Thurston's book. Furthermore, the new manifold has smaller hyperbolic volume than the original. See Theorem 6.5.6 in chapter six. Some further discussion and references can be found in Ian's answer here here.

So: fix a hyperbolic link $L$ in the three-sphere with an unknotted component $C$. Surgering $L$ along $C$ gives an infinite family of hyperbolic links, all with volume less than that of $L$. For explicit pictures of this, see the last few pages of chapter five of Thurston's book. (Actually, the pictures go in the opposite direction -- he explains how the Whitehead link is the "geometric limit" of a sequence of twist knots.)

Here is an expansion of Ian's answer.

Dehn surgery on a hyperbolic knot or link generically gives another hyperbolic manifold. This follows from the Dehn surgery theorem; see Theorem 5.8.2 in chapter five of Thurston's book. Furthermore, the new manifold has smaller hyperbolic volume than the original. See Theorem 6.5.6 in chapter six. Some further discussion and references can be found in Ian's answer here.

So: fix a hyperbolic link $L$ in the three-sphere with an unknotted component $C$. Surgering $L$ along $C$ gives an infinite family of hyperbolic links, all with volume less than that of $L$. For explicit pictures of this, see the last few pages of chapter five of Thurston's book. (Actually, the pictures go in the opposite direction -- he explains how the Whitehead link is the "geometric limit" of a sequence of twist knots.)

Here is an expansion of Ian's answer.

Dehn surgery on a hyperbolic knot or link generically gives another hyperbolic manifold. This follows from the Dehn surgery theorem; see Theorem 5.8.2 in chapter five of Thurston's book. Furthermore, the new manifold has smaller hyperbolic volume than the original. See Theorem 6.5.6 in chapter six. Some further discussion and references can be found in Ian's answer here.

So: fix a hyperbolic link $L$ in the three-sphere with an unknotted component $C$. Surgering $L$ along $C$ gives an infinite family of hyperbolic links, all with volume less than that of $L$. For explicit pictures of this, see the last few pages of chapter five of Thurston's book. (Actually, the pictures go in the opposite direction -- he explains how the Whitehead link is the "geometric limit" of a sequence of twist knots.)

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created Jan 18, 2015 at 17:28

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Here is an expansion of Ian's answer.

Dehn surgery on a hyperbolic knot or link generically gives another hyperbolic manifold. This follows from the Dehn surgery theorem; see Theorem 5.8.2 in chapter five of Thurston's book. Furthermore, the new manifold has smaller hyperbolic volume than the original. See Theorem 6.5.6 in chapter six. Some further discussion and references can be found in Ian's answer here.

So: fix a hyperbolic link $L$ in the three-sphere with an unknotted component $C$. Surgering $L$ along $C$ gives an infinite family of hyperbolic links, all with volume less than that of $L$. For explicit pictures of this, see the last few pages of chapter five of Thurston's book. (Actually, the pictures go in the opposite direction -- he explains how the Whitehead link is the "geometric limit" of a sequence of twist knots.)