An algorithm for Seifert surfaces in 3-manifolds via surgery presentations

Abstract: The classical Seifert algorithm provides an explicit construction of a Seifert surface for any link in $S<sup>3$. Alegria and Menasco extended this construction to integral homology $3$-spheres using Heegaard splittings. In this paper, we extend the Seifert algorithm to null-homologous links in arbitrary $3$-manifolds via surgery on framed links in $S<sup>3$.

• The paper presents an explicit Seifert-surface construction algorithm for null-homologous links in oriented 3-manifolds via surgery on framed links in $S^3$.
• The method uses integral linear algebra with the surgery linking matrix to convert the problem to the classical Seifert algorithm, providing explicit formulas for linking numbers, framing coefficients, and Seifert matrices.
• This method is applicable to integral homology spheres and helps demonstrate topological properties, such as Alexander polynomials, signatures, and genus.
• This method is applicable to integral homology spheres and helps demonstrate topological properties, such as Alexander polynomials and signatures, providing concrete examples and topological affirmations.

The paper develops an explicit Seifert-surface construction for null-homologous links in closed, orientable 3-manifolds presented by surgery on framed links in $S^3$. Its central contribution is an algorithm that reduces the problem in the surgered manifold to the classical Seifert algorithm in $S^3$. The reduction is controlled by integral linear algebra associated with the surgery linking matrix. In addition to constructing surfaces, the method yields formulas for linking numbers, converts framing coefficients between the surgered manifold and the original surgery diagram, and supports computations of Seifert matrices, signatures, Alexander polynomials, and non-locality phenomena.

Surgery presentations and the algebraic encoding

Let $Y=S^3(L,\phi)$ be obtained by surgery on an oriented framed link $L=L_1\cup\cdots\cup L_n\subset S^3$. The surgery data are encoded by the symmetric linking matrix

$$M_L=(m_{ij}),\qquad m_{ij}=\operatorname{lk}_{S^3}(L_i,\widetilde L_j),$$

whose diagonal entries are the framing coefficients and whose off-diagonal entries are pairwise linking numbers. The first homology group of the surgered manifold is presented by

$H_1(Y)\cong \mathbb Z^n/M_L\mathbb Z^n.$

For an oriented link $K\subset S^3\setminus\operatorname{int}\nu(L)$, the paper introduces its linking vector

$$V_K= \begin{pmatrix} \operatorname{lk}_{S^3}(K,L_1)\ \vdots\ \operatorname{lk}_{S^3}(K,L_n) \end{pmatrix}.$$

Under the standard identification of the homology of the surgery complement with $\mathbb Z^n$, the class of $K$ is represented by $S^3$0. Consequently,

$S^3$1

for some $S^3$2. Such an $S^3$3 is called a solution vector. This reformulates null-homology as an explicitly solvable integral system. In the integral homology-sphere case, $S^3$4 is unimodular, so the solution vector is unique and equals $S^3$5.

This algebraic formulation is the essential input to the geometric construction. It also clarifies the distinction between the general case and the homology-sphere case: for arbitrary $S^3$6, $S^3$7 need not be invertible over $S^3$8, but null-homology guarantees that $S^3$9 lies in its image. The algorithm therefore requires solving an integral system rather than formally applying an inverse.

The Seifert-surface algorithm

The main theorem states that if $Y=S^3(L,\phi)$0 is null-homologous in $Y=S^3(L,\phi)$1, then it can be isotoped in $Y=S^3(L,\phi)$2 to a link $Y=S^3(L,\phi)$3 that bounds a Seifert surface in the surgery complement: $Y=S^3(L,\phi)$4 The construction has two stages.

First, the link is modified by slides over components of the surgery link. A positive slide over $Y=S^3(L,\phi)$5 adds the $Y=S^3(L,\phi)$6th column of $Y=S^3(L,\phi)$7 to the linking vector, while a negative slide subtracts it. If

$Y=S^3(L,\phi)$8

satisfies $Y=S^3(L,\phi)$9, then one performs $L=L_1\cup\cdots\cup L_n\subset S^3$0 slides over $L=L_1\cup\cdots\cup L_n\subset S^3$1, with the sign chosen to subtract $L=L_1\cup\cdots\cup L_n\subset S^3$2 times the corresponding column. The resulting link $L=L_1\cup\cdots\cup L_n\subset S^3$3 satisfies

$L=L_1\cup\cdots\cup L_n\subset S^3$4

These slides are not generally isotopies in the original $L=L_1\cup\cdots\cup L_n\subset S^3$5; their bands can produce different links there. They are, however, isotopies in the surgered manifold, because a slide over a surgery component becomes a band move across the meridional disk of the attached solid torus. This distinction is important: the algorithm preserves the isotopy class in $L=L_1\cup\cdots\cup L_n\subset S^3$6, not necessarily in the surgery presentation of $L=L_1\cup\cdots\cup L_n\subset S^3$7.

Second, one applies the ordinary Seifert algorithm to $L=L_1\cup\cdots\cup L_n\subset S^3$8 as a link in $L=L_1\cup\cdots\cup L_n\subset S^3$9. Let $$M_L=(m_{ij}),\qquad m_{ij}=\operatorname{lk}_{S^3}(L_i,\widetilde L_j),$$0 be a Seifert surface for $$M_L=(m_{ij}),\qquad m_{ij}=\operatorname{lk}_{S^3}(L_i,\widetilde L_j),$$1. Since every component of $$M_L=(m_{ij}),\qquad m_{ij}=\operatorname{lk}_{S^3}(L_i,\widetilde L_j),$$2 has algebraic intersection zero with $$M_L=(m_{ij}),\qquad m_{ij}=\operatorname{lk}_{S^3}(L_i,\widetilde L_j),$$3, the intersections $$M_L=(m_{ij}),\qquad m_{ij}=\operatorname{lk}_{S^3}(L_i,\widetilde L_j),$$4 can be paired into oppositely signed points. For each pair, one removes small disks from $$M_L=(m_{ij}),\qquad m_{ij}=\operatorname{lk}_{S^3}(L_i,\widetilde L_j),$$5 and attaches a tube along an arc of $$M_L=(m_{ij}),\qquad m_{ij}=\operatorname{lk}_{S^3}(L_i,\widetilde L_j),$$6. Repeating this operation removes all intersections with the surgery link and produces a surface

$$M_L=(m_{ij}),\qquad m_{ij}=\operatorname{lk}_{S^3}(L_i,\widetilde L_j),$$7

with boundary $$M_L=(m_{ij}),\qquad m_{ij}=\operatorname{lk}_{S^3}(L_i,\widetilde L_j),$$8.

Thus the procedure is explicit:

1. compute $$M_L=(m_{ij}),\qquad m_{ij}=\operatorname{lk}_{S^3}(L_i,\widetilde L_j),$$9 and $H_1(Y)\cong \mathbb Z^n/M_L\mathbb Z^n.$0;
2. solve $H_1(Y)\cong \mathbb Z^n/M_L\mathbb Z^n.$1 over $H_1(Y)\cong \mathbb Z^n/M_L\mathbb Z^n.$2;
3. perform the prescribed slides over the surgery components;
4. apply the classical Seifert algorithm to the resulting link;
5. tube the surface away from the surgery link.

The tubing step gives a direct complexity estimate. Each tube decreases Euler characteristic by two. If the initial surface is connected, each tube increases its genus by one. The theorem therefore provides existence and a concrete construction, but it does not assert that the resulting surface has minimal genus. Indeed, the number of slides and intersections introduced by a chosen surgery presentation can substantially affect the genus of the constructed surface.

The result extends the Heegaard-splitting construction for integral homology spheres due to Alegria and Menasco (Alegria et al., 2024). Its distinguishing feature is that it applies to arbitrary surgery presentations and uses the surgery matrix as the organizing structure.

Linking numbers in a surgered manifold

The same slide construction yields a surgery formula for linking numbers. Suppose $H_1(Y)\cong \mathbb Z^n/M_L\mathbb Z^n.$3 and $H_1(Y)\cong \mathbb Z^n/M_L\mathbb Z^n.$4 are null-homologous knots in $H_1(Y)\cong \mathbb Z^n/M_L\mathbb Z^n.$5. If $H_1(Y)\cong \mathbb Z^n/M_L\mathbb Z^n.$6 solves

$H_1(Y)\cong \mathbb Z^n/M_L\mathbb Z^n.$7

then

$H_1(Y)\cong \mathbb Z^n/M_L\mathbb Z^n.$8

Geometrically, one replaces $H_1(Y)\cong \mathbb Z^n/M_L\mathbb Z^n.$9 by the link obtained after the slides determined by $K\subset S^3\setminus\operatorname{int}\nu(L)$0. The modified knot has zero linking with every component of $K\subset S^3\setminus\operatorname{int}\nu(L)$1, so a Seifert surface for it can be chosen entirely in the surgery complement. Its intersection with $K\subset S^3\setminus\operatorname{int}\nu(L)$2 is therefore computed in $K\subset S^3\setminus\operatorname{int}\nu(L)$3. Each slide contributes the corresponding linking number with $K\subset S^3\setminus\operatorname{int}\nu(L)$4, and the total correction is $K\subset S^3\setminus\operatorname{int}\nu(L)$5.

Although the solution vector need not be unique when $K\subset S^3\setminus\operatorname{int}\nu(L)$6 is singular, the formula is still well defined. If $K\subset S^3\setminus\operatorname{int}\nu(L)$7 and $K\subset S^3\setminus\operatorname{int}\nu(L)$8 are two solutions, then $K\subset S^3\setminus\operatorname{int}\nu(L)$9. Since $$V_K= \begin{pmatrix} \operatorname{lk}_{S^3}(K,L_1)\ \vdots\ \operatorname{lk}_{S^3}(K,L_n) \end{pmatrix}.$$0 lies in the image of $$V_K= \begin{pmatrix} \operatorname{lk}_{S^3}(K,L_1)\ \vdots\ \operatorname{lk}_{S^3}(K,L_n) \end{pmatrix}.$$1 and $$V_K= \begin{pmatrix} \operatorname{lk}_{S^3}(K,L_1)\ \vdots\ \operatorname{lk}_{S^3}(K,L_n) \end{pmatrix}.$$2 is symmetric,

$$V_K= \begin{pmatrix} \operatorname{lk}_{S^3}(K,L_1)\ \vdots\ \operatorname{lk}_{S^3}(K,L_n) \end{pmatrix}.$$3

Therefore the correction term is independent of the chosen solution. In the homology-sphere case, the formula becomes the familiar matrix expression

$$V_K= \begin{pmatrix} \operatorname{lk}_{S^3}(K,L_1)\ \vdots\ \operatorname{lk}_{S^3}(K,L_n) \end{pmatrix}.$$4

This formula is not merely auxiliary. It converts all pairwise linking computations on a Seifert surface in $$V_K= \begin{pmatrix} \operatorname{lk}_{S^3}(K,L_1)\ \vdots\ \operatorname{lk}_{S^3}(K,L_n) \end{pmatrix}.$$5 into computations in $$V_K= \begin{pmatrix} \operatorname{lk}_{S^3}(K,L_1)\ \vdots\ \operatorname{lk}_{S^3}(K,L_n) \end{pmatrix}.$$6 plus a finite-dimensional correction determined by the surgery data. It consequently gives an effective method for constructing Seifert matrices: if $$V_K= \begin{pmatrix} \operatorname{lk}_{S^3}(K,L_1)\ \vdots\ \operatorname{lk}_{S^3}(K,L_n) \end{pmatrix}.$$7 is a basis for $$V_K= \begin{pmatrix} \operatorname{lk}_{S^3}(K,L_1)\ \vdots\ \operatorname{lk}_{S^3}(K,L_n) \end{pmatrix}.$$8, the entries

$$V_K= \begin{pmatrix} \operatorname{lk}_{S^3}(K,L_1)\ \vdots\ \operatorname{lk}_{S^3}(K,L_n) \end{pmatrix}.$$9

can be evaluated using the corresponding $\mathbb Z^n$0 linking numbers and solution vectors.

Conversion of framing coefficients

The paper also derives a surgery-diagram conversion formula. Let $\mathbb Z^n$1 be a null-homologous knot in $\mathbb Z^n$2 with intrinsic framing coefficient

$\mathbb Z^n$3

The framing coefficient of the same push-off in the original $\mathbb Z^n$4 surgery diagram is not generally $\mathbb Z^n$5. Applying the linking formula to $\mathbb Z^n$6 and $\mathbb Z^n$7, and using $\mathbb Z^n$8, gives

$\mathbb Z^n$9

Therefore, if $K$0, then $K$1-surgery on $K$2 in $K$3 is represented by

$K$4

The correction term is essential: the coefficient visible in the $K$5 diagram records the ambient $K$6 linking of $K$7 with its push-off, whereas $K$8 records the linking in the surgered manifold. The difference is precisely the algebraic effect of the surgery components. This result makes surgery on knots in a presented 3-manifold accessible to ordinary Kirby calculus.

Computations in homology spheres

The first example uses a homology sphere presented by three $K$9-framed components with identity linking matrix. The knot has linking vector

$S^3$00

so the unique solution vector is the same vector. A single slide over the first surgery component produces a knot $S^3$01 with zero linking vector. The resulting disk in $S^3$02 intersects the surgery link, and tubing removes those intersections to produce a genus-one surface in the surgered manifold.

For a basis $S^3$03 of the resulting surface, the paper computes the Seifert matrix

$S^3$04

It follows that

$S^3$05

and

$S^3$06

The Alexander polynomial is nontrivial, and the constructed genus-one surface establishes that $S^3$07 does not bound a disk. In the argument presented, the Seifert-matrix calculation yields $S^3$08.

The framing-conversion formula assigns coefficient zero to $S^3$09 in the original $S^3$10 diagram when performing $S^3$11-surgery on $S^3$12 in $S^3$13. Kirby calculus then identifies the resulting manifold with $S^3$14. If $S^3$15 were isotopic to a knot $S^3$16 contained in a 3-ball in $S^3$17, one would have

$S^3$18

Since the left-hand side is $S^3$19, this would force $S^3$20, contradicting the nontriviality of $S^3$21. Hence $S^3$22 is not a local knot. The paper also gives an alternative contradiction using Property P [math/0407152].

The second example concerns the boundary of a contractible 4-manifold, represented by a two-component surgery diagram with linking matrix

$S^3$23

Although the diagonal framings are zero, the matrix is unimodular, so the boundary is an integral homology sphere. For the knot under consideration,

$S^3$24

The algorithm again produces a genus-one surface with the same Seifert matrix

$S^3$25

and hence the same numerical invariants: $S^3$26

This example separates the topology of the boundary from that of the bounding 4-manifold. The knot has genus one in $S^3$27, while it bounds an embedded disk in the contractible 4-manifold. The paper therefore exhibits a knot whose boundary genus is positive even though its 4-dimensional genus is zero. Applying the framing conversion to zero-surgery gives

$S^3$28

If $S^3$29 were local, then

$S^3$30

which would imply $S^3$31. Since the boundary is a non-simply connected homology sphere, this is impossible. The alternative argument invokes Property R [math/9709211] to exclude zero-surgery on a nontrivial knot in $S^3$32 producing $S^3$33.

Limitations and open questions

The construction depends on a chosen surgery presentation and on explicit band choices for the slides. Different choices can produce links that are not isotopic in $S^3$34, and different surgery presentations can produce surfaces with different genera and combinatorial complexity. The theorem guarantees a Seifert surface, not a genus-minimizing one. The tubing procedure can also increase genus substantially, with one genus increment for each tube when the initial surface is connected.

The computational content is clearest for homology spheres, where $S^3$35 exists over $S^3$36. In a general 3-manifold, the algorithm requires finding an integral solution of $S^3$37 and handling possible nonuniqueness. The paper proves that the linking formula remains well defined, but it does not provide complexity bounds for solving the integral system, minimizing the number of slides, minimizing the number of tubes, or optimizing the genus of the resulting surface.

The examples establish the method through explicit diagrams and Kirby calculus, but the supplied text does not develop a general implementation for diagrammatic input, nor does it analyze the algorithmic complexity of computing Seifert matrices or Alexander polynomials for arbitrary surgery presentations. These remain concrete questions about the relationship between presentation complexity, solution-vector size, tubing complexity, and the topology of the resulting surface.

Conclusion

The paper gives a surgery-theoretic extension of the classical Seifert algorithm. Null-homology is converted into an integral equation involving the surgery linking matrix; its solution determines slides that eliminate the link’s algebraic interaction with the surgery components; and a tubing construction then transfers a classical Seifert surface from $S^3$38 into the surgery complement. The same framework produces explicit linking-number and framing-conversion formulas. The examples demonstrate that the method supports nontrivial calculations of Seifert matrices, signatures, Alexander polynomials, genus, and locality in homology 3-spheres.