Fraction:

Calculate a continued fraction expansion: Enter a fraction of the form $m/n$.

S ( , )

Calculate a Dedekind sum: Enter two relatively prime integers.

( , , )

p =

    • Calculate the Seifert invariants of a manifold of finite fundamental group (other than a lens space), i.e., a Seifert fibered 3-manifold with 3 singular fibers: Enter the multiplicities of three singular fibers, the order of the fundamental group "p", and whether the manifold should bound a negative or positive definite 4-manifold (equivalently, if "e" is the orbifold euler number, enter p=|e| and sgn(e)).

( , , )

p =

Format:

    • Calculate the intersection form of a definite 4-manifold bounded by a manifold of finite fundamental group (equivalently, a recipe to build the 3-manifold via link surgery): Enter the multiplicities of three singular fibers, the order of the fundamental group "p", and whether the manifold should bound a negative or positive definite 4-manifold (equivalently, if "e" is the orbifold euler number, enter p=|e| and sgn(e)).

L ( , )

Calculate the d-invariants of a lens space: Enter 2 natural numbers.
NB: We follow the HF conventions and write $L(2,1)$ to be -2-surgery on the unknot. Recall that $d(L(p,q),s)=-d(L(p,-q),-s)$.

Surgery coefficient:

Calculate the d-invariants of a manifold which is an L-space and is surgery on a knot in the 3-sphere: Enter the coefficients of the symmetrized Alexander polynomial, i.e., $t^3 - t + 2 -t^{-1} + t^{-3}$ should be entered as 2, -1, 3.

( , , )

p =

Calculate the d-invariants of a manifold of finite fundamental group: Enter the multiplicities of three singular fibers, the order of the fundamental group "p", and whether the manifold should bound a negative or positive definite 4-manifold (equivalently, if "e" is the orbifold euler number, enter p=|e| and sign(e)).


Calculate the d-invariants of the boundary of a plumbed 4-manifold with at most 2 non-nice plumbing vertices: Please enter a matrix in the following (boost) form: [8,8]((-3,1,1,0,1,0,0,0,),(1,-2,0,0,0,0,0,0,),(1,0,-2,1,0,0,0,0,),(0,0,1,-2,0,0,0,0,),(1,0,0,0,-2,1,0,0,),(0,0,0,0,1,-2,1,0,),(0,0,0,0,0,1,-2,1,),(0,0,0,0,0,0,1,-2,))

L ( , )

Calculate the Casson-Walker invariant of a lens space: Enter two natural numbers.
NB: We follow Heegaard Floer conventions and write $L(2,1)$ for -2-surgery on the unknot. Recall $\lambda(L(p,q))=-\lambda(L(p,-q))$.

Surgery coefficient:

Calculate the Casson-Walker invariant of a manifold which is surgery on a knot in the 3-sphere: enter the surgery coefficient as a positive fraction and the coefficients of the symmetrized Alexander polynomial, i.e., $t^3 - t + 2 - t^{-1} + t^{-3}$ should be entered as 2, -1, 3.

Knot in arc presentation format:

Calculate knot genus (via highest Alexander grading of a generator of $\widehat{HFK}$).

Example input: {1, 3}, {2, 4}, {2, 3}, {1, 4}

You may click "Draw" to enter vertices manually or select from some common knots.

Knot in arc presentation format:

Calculate whether a knot is fibered (via rank of $\widehat{HFK}$ in highest Alexander grading).

Example input: {1, 3}, {2, 4}, {2, 3}, {1, 4}

You may click "Draw" to enter vertices manually or select from some common knots.

Math Toolkit

© Margaret Doig 2020