I don't know of such a table. One serious problem in making such a table one is choosing generating sets for the mapping class groups of the surfaces $S_{g,1}$. There is no canonical choice and I suspect no good choice.

Following Lickorish, one can cook up an algorithm that turns any monodromy into a product of Dehn twists, then into a product of twists in the Lickorish generating set, and then a product of twists the Humphries generating set. But remember that the monodromy only matters up to conjugacy -- thus finding the "best" representative, if such a thing, is somehow related to the conjugacy problem. It all sounds a bit dicey.

In practice, I would go to KnotInfo to find out if a knot in the tables was fibered. If there was a particular monodromy I wanted to compute I would look at Stallings theorem, compute the action of the monodromy on the fiber subgroup (which is a free group), and then transform that into a product of Dehn twists. If that didn't work, I would consult the experts, probably starting with Nathan Dunfield or Dylan Thurston. See also their paper "A random tunnel number one 3â€“manifold does not fiber over the circle" and its many references.

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One serious problem in making such a table is choosing generating sets for the mapping class groups of the surfaces $S_{g,1}$. There is no canonical choice and I suspect no good choice.

Following Lickorish, one can cook up an algorithm that turns any monodromy into a product of Dehn twists, then into a product of twists in the Lickorish generating set, and then a product of twists the Humphries generating set. But remember that the monodromy only matters up to conjugacy -- thus finding the "best" representative, if such a thing, is somehow related to the conjugacy problem. It all sounds a bit dicey.

In practice, I would go to KnotInfo to find out if a knot in the tables was fibered. If there was a particular monodromy I wanted to compute I would look at Stallings theorem, compute the action of the monodromy on the fiber subgroup (which is a free group), and then transform that into a product of Dehn twists. If that didn't work, I would consult the experts, probably starting with Nathan Dunfield or Dylan Thurston. See also their paper "A random tunnel number one 3â€“manifold does not fiber over the circle" and its many references.