1 Introduction↩︎

Integer-valued knot invariants arising from diagrammatic topology, knot homologies, four-dimensional topology, and skein theory are connected by a large network of inequalities. Examples include relations between the crossing number, braid index, bridge number, genus, unknotting number, and polynomial invariants. More recently, analogous inequalities involving four-dimensional and Floer-theoretic invariants have appeared in the literature. Despite the abundance of individual results, there is no unified picture of these relations and of their mutual interactions.

The purpose of this paper is threefold. First, we present a directed graph of \(46\) inequalities between \(33\) integer knot invariants. The graph includes invariants from several different settings, including crossing-type invariants, surface genera, concordance invariants, Floer-theoretic quantities, and polynomial degrees and spreads. Second, using these inequalities together with parity constraints, we propagate known values from KnotInfo to construct the database NewDB for knots up to \(13\) crossings. This computation yields numerous improved bounds and determines \(139\) new exact values for the unknotting number and doubly slice genus. Finally, we formulate a collection of conjectural inequalities selected by a transitivity-minimal criterion.

Here, we present the graph of selected \(46\) invariant inequalities for any knot \(K\hookrightarrow \mathbb{S}^3\). In the diagram, shown in Figure 1, source/target convention: \(X\to Y\) means \(X(K)\ge Y(K)\) for any (non-trivial) knot \(K\). The knot invariants we use are abbreviated as follows. Definitions of the Invariants are in the Appendix ¿sec:app?.

• \(\color{blue} c\) - the crossing number,

• \(\color{blue} c_3\) - triple crossing index,

• \(\color{blue} c_{\Delta}\) - the delta-crossing number,

• \(\color{blue} g_f\) - the free genus,

• \(\color{blue} g_c\) - the canonical genus,

• \(\color{blue} u\) - the unknotting number,

• \(\color{blue} u_b\) - the band-unknotting number,

• \(\color{blue} ul_b\) - the band-unlinking number,

• \(\color{blue} cl\) - the clasp number,

• \(\color{blue} td\) - the skein tree depth,

• \(\color{blue} tr\) - the trivializing number,

• \(\color{blue} br\) - the bridge number,

• \(\color{blue} b\) - the braid index,

• \(\color{blue} \alpha\) - the arc index,

• \(\color{blue} m\) - the mosaic number,

• \(\color{blue} a\) - the ascending number.

• \(\color{blue} cl_4\) - the 4D clasp number,

• \(\color{blue} g_4\) - the slice genus,

• \(\color{blue} \sigma\) - the knot signature,

• \(\color{blue} g_{ds}\) - the doubly slice genus,

• \(\color{blue} g_r\) - the ribbon slice genus,

• \(\color{blue} u^*_r\) - the weak ribbon unknotting number,

• \(\color{blue} u_s\) - the slicing number,

• \(\color{blue} u_c\) - the concordance unknotting number.

• \(\color{blue} g\) - the genus,

• \(\color{blue} Ord_v\) - the torsion order,

• \(\color{blue} \tau\) - the Ozsváth–Szabó’s tau,

• \(\color{blue} s\) - the Rasmussen’s s-invariant.

• \(\color{blue} sp\Delta_t\) - the span of the Alexander polynomial \(\Delta(t)\),

• \(\color{blue} spV_t\) - the span of the Jones polynomial \(V(t)\),

• \(\color{blue} degP_z\) - the \(z\)-degree of the HOMFLYPT polynomial \(P_K(a,z)\),

• \(\color{blue} spP_a\) - the \(a\)-spread of the HOMFLYPT polynomial \(P_K(a,z)\),

• \(\color{blue} spF_a\) - the \(a\)-spread of the Kauffman polynomial \(F_K(a,z)\).

The corresponding relations, numbered as the arrows labels, can be found in the References as follows. 1 in [1], 14 in [2], 2, 9 in [3], 3 in [4], 5 in [5], 6 in [6], 7 in [7], 8 in [8], 4, 11, 12, 19, 25, 29, 30, 35 in [9], 26 in [10], 13 in [11], 16, 17, 18, 44 in [12], 15 in [13], 20 in [14], 21 in [15], 31, 46 in [16], 27 in [17], 10, 28 in [18], 36 in [19], 38 in [20], 37 in [21], 45 in [22], 24, 32, 33 in [23], 23 in [7], [24], 22 in [25], 40 is immediate from definitions, 41 in [26], 42 in [27], 34 in [28], 39, 43 in [29].

2 Database↩︎

We create the new database NewDB of the \(33\) invariants from Figure 1 as follows. First, we extract the existing tables from KnotInfo [30], with values for the knots up to \(13\) crossings, for the \(19\) invariants (in the brackets, there are stated functions built from the mentioned invariant): (\(c\), \(2u\), \(2g\), \(2br-2\), \(2b-2\), \(|\sigma|\), \(\alpha-2\), \(sp\Delta_t\), \(\left\lceil \frac{spV_t}{2}\right\rceil\), \(spF_a\), \(2g_4\), \(|s|\), \(2|\tau|\), \(2cl_4\), \(2cl\), \(g_{ds}\), \(m-1\), \(spP_a\), \(degP_z\)).
Then, we add the \(6\) invariants (\(c_3\), \(2c_{\Delta}\), \(td\), \(tr\), \(2a\), \(2u_c\)) with the values extracted from the References of this paper. Finally we add the rest of \(8\) invariants (\(2g_c\), \(2g_f\), \(2g_r\), \(2u^*_r\), \(2u_s\), \(Ord_v\), \(u_b\), \(ul_b\)) as the not-known values for the knots (up to \(13\) crossings), to the database. We iterate the entries (using computations with [31]) of the base NewDB according to \(46\) inequalities from Figure 1, giving us bounds of the values of the invariants. In the iteration process, we also use the parity properties of some of the invariants. We repeat this process until no update is possible. We obtain the new database NewDB, which has many updates on the original exact values, bounds, or missing values.

The database NewDB is available from the author upon request.

By this method, we updated (by an integer or an interval) the non-empty cells in KnotInfo for Unknotting Number and Double Slice Genus by \(232\) in total. That includes \(139\) new exact values, which are as follows.

The following \(36\) knots have the Unknotting Number equal to \(2\): \(13n_{128}\), \(13n_{152}\), \(13n_{365}\), \(13n_{391}\), \(13n_{492}\), \(13n_{839}\), \(13n_{842}\), \(13n_{967}\), \(13n_{976}\), \(13n_{993}\), \(13n_{1082}\), \(13n_{1084}\), \(13n_{1146}\), \(13n_{1345}\), \(13n_{1478}\), \(13n_{1524}\), \(13n_{1846}\), \(13n_{2453}\), \(13n_{2456}\), \(13n_{2612}\), \(13n_{2681}\), \(13n_{2689}\), \(13n_{2729}\), \(13n_{3251}\), \(13n_{3263}\), \(13n_{3639}\), \(13n_{3729}\), \(13n_{3811}\), \(13n_{3853}\), \(13n_{3881}\), \(13n_{4029}\), \(13n_{4381}\), \(13n_{4623}\), \(13n_{4648}\), \(13n_{4672}\), \(13n_{4964}\).

The following \(88\) knots have the Double Slice Genus equal to \(4\): \(13a_{5}\), \(13a_{15}\), \(13a_{55}\), \(13a_{109}\), \(13a_{146}\), \(13a_{179}\), \(13a_{422}\), \(13a_{568}\), \(13a_{650}\), \(13a_{720}\), \(13a_{793}\), \(13a_{820}\), \(13a_{828}\), \(13a_{1020}\), \(13a_{1099}\), \(13a_{1332}\), \(13a_{1419}\), \(13a_{1476}\), \(13a_{1544}\), \(13a_{1581}\), \(13a_{1784}\), \(13a_{1901}\), \(13a_{2143}\), \(13a_{2151}\), \(13a_{2233}\), \(13a_{2372}\), \(13a_{2377}\), \(13a_{2436}\), \(13a_{2507}\), \(13a_{2604}\), \(13a_{2607}\), \(13a_{2875}\), \(13a_{2951}\), \(13a_{3105}\), \(13a_{3122}\), \(13a_{3157}\), \(13a_{3484}\), \(13a_{3513}\), \(13a_{3930}\), \(13a_{4005}\), \(13a_{4034}\), \(13a_{4122}\), \(13a_{4225}\), \(13a_{4295}\), \(13a_{4422}\), \(13a_{4458}\), \(13a_{4531}\), \(13a_{4727}\), \(13a_{4778}\), \(13a_{4807}\), \(13a_{4857}\), \(13n_{42}\), \(13n_{111}\), \(13n_{128}\), \(13n_{142}\), \(13n_{152}\), \(13n_{196}\), \(13n_{365}\), \(13n_{391}\), \(13n_{492}\), \(13n_{839}\), \(13n_{993}\), \(13n_{1118}\), \(13n_{1146}\), \(13n_{1345}\), \(13n_{1348}\), \(13n_{1478}\), \(13n_{1524}\), \(13n_{1846}\), \(13n_{2612}\), \(13n_{2681}\), \(13n_{2689}\), \(13n_{2729}\), \(13n_{2810}\), \(13n_{3251}\), \(13n_{3263}\), \(13n_{3639}\), \(13n_{3729}\), \(13n_{3733}\), \(13n_{3811}\), \(13n_{3853}\), \(13n_{3881}\), \(13n_{4029}\), \(13n_{4381}\), \(13n_{4623}\), \(13n_{4648}\), \(13n_{4672}\), \(13n_{4964}\).

The following \(15\) knots have the Double Slice Genus equal to \(6\): \(13a_{616}\), \(13a_{660}\), \(13a_{825}\), \(13a_{3212}\), \(13n_{246}\), \(13n_{259}\), \(13n_{277}\), \(13n_{305}\), \(13n_{616}\), \(13n_{669}\), \(13n_{709}\), \(13n_{982}\), \(13n_{1105}\), \(13n_{3390}\), \(13n_{4894}\).

3 Open problems↩︎

We state here the "interesting" open problems concerning inequalities. We choose the methodology of selection, as any "interesting" inequality must satisfy the following five conditions.

• it is an inequality between the invariants being the vertices of the graph from Figure 1,

• does not follow directly (by the transitive closure) from the relations being the edges of the graph from Figure 1,

• it is satisfied for each knot in our base NewDB,

• it is equality for at least one knot in our base NewDB,

• it is a strict inequality for at least one knot in our base NewDB,

We also rejected incomparable such relations where it is known that there exists a pair of knots (at lest one of them with the crossing number more than \(13\)) where the corresponding invariants are in a contradictory relation of strict inequality, such as: \((2g ,\; degP_z)\) by [7], \((degP_z,\; 2cl)\), \((2g_c,\; 2cl)\) by [32], \((2g_f,\; 2cl)\) by [6], \((2br-2,\; 2cl)\) by [33].

Using this method, we obtain (using computations with [31]) the set \({Conj}\) of conjectural inequalities. From \({Conj}\) we select the subset \({BasicConj}\) of those inequalities such that proving each one from \({BasicConj}\) that does not directly (by the relation transitivity) imply any other inequality from \({BasicConj}\). The set \({BasicConj}\) contains the following \(10\) inequalities:

1. \(2br-2 \leq spF_a\),

2. \(2br-2 \leq tr\),

3. \(2br-2 \leq 2c_{\Delta}\),

4. \(2g_f \leq td\),

5. \(2g_r \leq 2u_s\),

6. \(2cl_4 \leq 2c_{\Delta}\),

7. \(spP_a \leq spF_a\),

8. \(\left\lceil \frac{spV_t}{2}\right\rceil \leq td\),

9. \(2|\tau| \leq degP_z\),

10. \(|s| \leq degP_z\),

Appendix. Short definitions of the Invariants↩︎

Crossing-type and Diagrammatic Invariants↩︎

1. Crossing number \(c(K)\)↩︎

The crossing number \(c(K)\) of a knot \(K\) is the minimum number of ordinary double-crossings among all regular planar diagrams representing the knot. A regular diagram is obtained from a generic projection of the knot into the plane so that all singularities are transverse double points equipped with over-under crossing information.

2. Triple-crossing number \(c_3(K)\)↩︎

The triple-crossing number \(c_3(K)\) is the minimum number of triple-crossings among all triple-crossing projections of the knot \(K\). A triple-crossing is a singular point where exactly three strands intersect transversely at one point, together with a specified height ordering of top, middle, and bottom strands.

3. Delta-crossing number \(c_{\Delta}(K)\)↩︎

The delta-crossing number \(c_{\Delta}(K)\) is the minimum number of delta-crossings appearing in any delta-diagram representing the knot \(K\). A delta-crossing is a local configuration of three arcs arranged through three ordinary crossings in the pattern associated with the classical \(\Delta\)-move.

4. Skein tree depth \(td(K)\)↩︎

The skein tree depth \(td(K)\) is the minimum possible depth among all binary skein trees resolving the knot into unlinks through repeated applications of a skein relation. Each vertex of the tree corresponds to a knot or link diagram, while edges correspond to skein resolutions at selected crossings.

The depth of the tree is defined as the maximal number of skein resolutions occurring along any root-to-leaf path. Thus, the invariant measures the sequential complexity of reducing the knot diagram through skein-theoretic procedures. The invariant is closely connected with polynomial invariants such as the HOMFLYPT polynomial and gives a combinatorial measure of recursive computational complexity.

5. Trivializing number \(tr(K)\)↩︎

The trivializing number \(tr(K)\) is the minimum number of crossings in a knot diagram that must be replaced by precrossings so that every possible assignment of crossing information to those precrossings yields the unknot.

Equivalently, the invariant measures how many crossing choices determine the nontriviality of the knot. The notion originates from pseudo-diagram and parity-type theories in knot theory. Unlike the unknotting number, which records explicit crossing changes, the trivializing number records the number of crossings whose information must be forgotten in order to force triviality regardless of subsequent choices.

6. Free genus \(g_f(K)\)↩︎

The free genus \(g_f(K)\) of a knot \(K\) is the minimum genus among all Seifert surfaces whose complement in \(S^3\) has free fundamental group. Such spanning surfaces are called free Seifert surfaces.

7. Canonical genus \(g_c(K)\)↩︎

The canonical genus \(g_c(K)\) is the minimum genus among all Seifert surfaces obtained from Seifert’s algorithm applied directly to knot diagrams of \(K\).

8. Unknotting number \(u(K)\)↩︎

The unknotting number \(u(K)\) is the minimum number of crossing changes required to transform the knot into the unknot. A crossing change replaces a positive crossing with a negative crossing or vice versa.

9. Band-unknotting number \(u_b(K)\)↩︎

The band-unknotting number \(u_b(K)\) is the minimum number of oriented band surgeries required to transform the knot into the unknot. A band surgery removes two small arcs from the knot and reconnects them using the opposite sides of an embedded band.

10. Band-unlinking number \(ul_b(K)\)↩︎

The band-unlinking number \(ul_b(K)\) is the minimum number of oriented band surgeries required to transform the knot into a completely split unlink.

11. Clasp number \(cl(K)\)↩︎

The clasp number \(cl(K)\) is the minimum number of clasp singularities appearing on any immersed disk in \(S^3\) bounded by the knot \(K\). A clasp singularity is a transverse self-intersection locally modeled by two sheets intersecting in a standard clasp configuration.

12. Bridge number \(br(K)\)↩︎

The bridge number \(br(K)\) is the minimum number of local maxima of the knot with respect to a Morse height function on \(S^3\), minimized over all embeddings isotopic to the knot.

Equivalently, it is the minimum number of bridges appearing in a bridge decomposition of the knot.

13. Braid index \(b(K)\)↩︎

The braid index \(b(K)\) is the minimum number of strands required to represent the knot as the closure of an element of the Artin braid group.

Explicitly, \(b(K)=\min\{n : K \text{ is the closure of an } n\text{-braid}\}.\)

14. Arc index \(\alpha(K)\)↩︎

The arc index \(\alpha(K)\) is the minimum number of arcs needed in an arc presentation of the knot. Equivalently, it is the smallest number of half-planes in an open-book decomposition of \(S^3\) whose pages contain embedded arcs forming the knot.

15. Mosaic number \(m(K)\)↩︎

The mosaic number \(m(K)\) is the minimum integer \(n\) such that the knot admits a representation as an \(n\times n\) knot mosaic. A knot mosaic is a finite grid of prescribed tiles assembled according to compatibility rules that encode the knot diagram combinatorially.

16. Ascending number \(a(K)\)↩︎

The ascending number \(a(K)\) is the minimum number of crossing changes needed to transform a based oriented knot diagram into a descending diagram.

A descending diagram is one in which, after choosing a base point and orientation, every crossing is first encountered as an overcrossing. Such diagrams always represent the unknot.

Four-dimensional Invariants↩︎

17. Four-dimensional clasp number \(cl_4(K)\)↩︎

The four-dimensional clasp number \(cl_4(K)\) is the minimum number of transverse double points among all immersed disks in the four-ball \(B^4\) whose boundary is the knot \(K \subset \partial B^4=S^3.\) The immersed disks are required to be smooth except at isolated transverse self-intersections.

The invariant is a four-dimensional analogue of the classical clasp number. Instead of considering immersed disks inside \(S^3\), one allows the disk to move into the four-ball, where additional flexibility becomes available.

18. Slice genus \(g_4(K)\)↩︎

The slice genus \(g_4(K)\), also called the smooth four-ball genus, is the minimum genus of a smooth, compact, connected, orientable surface properly embedded in \(B^4\) whose boundary is the knot \(K\subset S^3=\partial B^4.\)

The invariant measures how efficiently the knot bounds a smooth surface in four dimensions rather than in the three-sphere.

19. Knot signature \(\sigma(K)\)↩︎

The knot signature \(\sigma(K)\) is the signature of the symmetrized Seifert form associated with a Seifert surface for the knot. If \(V\) is a Seifert matrix, then \(\sigma(K)=\operatorname{sign}(V+V^T),\) where the signature denotes the number of positive eigenvalues minus the number of negative eigenvalues.

The signature is an integer-valued concordance invariant independent of the chosen Seifert surface.

20. Doubly slice genus \(g_{ds}(K)\)↩︎

The doubly slice genus \(g_{ds}(K)\) is the minimum genus of an unknotted closed orientable surface embedded in \(S^4\) whose intersection with an equatorial copy of \(S^3\) is the knot \(K.\)

21. Ribbon genus \(g_r(K)\)↩︎

The ribbon genus \(g_r(K)\) is the minimum genus of a ribbon surface in \(B^4\) bounded by the knot \(K.\) A ribbon surface is a properly embedded surface obtained from immersed surfaces in \(S^3\) having only ribbon singularities and no local maxima with respect to the radial function.

22. Weak ribbon unknotting number \(u_r^*(K)\)↩︎

Let \(K \subset S^3\) be a knot. The weak ribbon unknotting number \(u_r^*(K)\) is the minimal number of ribbon singularities of an immersed ribbon disk in \(B^4\) bounded by \(K\).

Equivalently, \[u_r^*(K) = \min\{ d(D) : D \subset B^4 \text{ is a ribbon immersed disk with } \partial D = K \},\] where \(d(D)\) denotes the number of ribbon double points of \(D\).

A ribbon immersed disk is an immersed disk in \(B^4\) whose singularities are only ribbon singularities.

23. Slicing number \(u_s(K)\)↩︎

The slicing number \(u_s(K)\) is the minimum number of crossing changes required to transform the knot into a slice knot.

A slice knot is a knot bounding a smoothly embedded disk in the four-ball. Thus, the slicing number measures the distance from the knot to the slice concordance class rather than to the unknot itself.

24. Concordance unknotting number \(u_c(K)\)↩︎

The concordance unknotting number \(u_c(K)\) is defined by \(u_c(K)=\min\{u(J)\;:\; J\sim K\},\) where \(\sim\) denotes smooth knot concordance.

The invariant measures the smallest unknotting number achievable within the smooth concordance class of the knot rather than for the knot itself.

Quantities from Knot Homologies↩︎

25. Genus \(g(K)\)↩︎

The genus \(g(K)\) of a knot \(K\) is the minimum genus among all connected, compact, orientable Seifert surfaces whose boundary is the knot. A Seifert surface is an orientable surface smoothly embedded in \(S^3\) with boundary equal to \(K.\)

26. Torsion order \(\mathrm{Ord}_v(K)\)↩︎

Let \(K \subset S^3\) be a knot, and let \(HFK^-(K)\) denote the minus version of knot Floer homology, viewed as a finitely generated module over \(\mathbb{F}_2[v]\).

The \(v\)–torsion submodule of \(HFK^-(K)\) is \[\mathrm{Tors}_v(HFK^-(K)) = \{x \in HFK^-(K) : v^m x = 0 \text{ for some } m \ge 0\}.\]

The torsion order of \(K\) is defined by \[\mathrm{Ord}_v(K) = \min \left\{ n \ge 0 : v^n \cdot \mathrm{Tors}_v(HFK^-(K)) = 0 \right\}.\]

Equivalently, \(\mathrm{Ord}_v(K)\) is the smallest integer \(n\) such that multiplication by \(v^n\) annihilates all \(v\)–torsion elements in \(HFK^-(K)\).

27. Ozsváth–Szabó invariant \(\tau(K)\)↩︎

The invariant \(\tau(K)\) is an integer-valued concordance invariant arising from knot Floer homology as introduced by Ozsváth and Szabó. It is defined using the filtration structure on the Heegaard Floer chain complex associated with the knot.

28. Rasmussen invariant \(s(K)\)↩︎

The Rasmussen invariant \(s(K)\) is an integer-valued concordance invariant defined using Lee’s deformation of Khovanov homology. It is extracted from the filtered structure of the deformed homology theory and assigns an even integer to every knot.

Quantities from Knot Polynomials↩︎

29. Span of the Alexander polynomial \(sp\Delta_t(K)\)↩︎

The span of the Alexander polynomial is defined by \(sp\Delta_t(K)=\deg_{\max}\Delta_K(t)-\deg_{\min}\Delta_K(t),\) where \(\Delta_K(t)\) is the Alexander polynomial of the knot.

The invariant measures the total width of the exponents appearing in the Alexander polynomial and provides a numerical measure of algebraic complexity. Since the Alexander polynomial is symmetric up to multiplication by powers of \(t,\) the span is independent of normalization choices.

30. Span of the Jones polynomial \(spV_t(K)\)↩︎

The span of the Jones polynomial is defined by \(spV_t(K)=\deg_{\max}V_K(t)-\deg_{\min}V_K(t),\) where \(V_K(t)\) denotes the Jones polynomial of the knot.

31. HOMFLYPT \(z\)-degree \(degP_z(K)\)↩︎

The invariant \(degP_z(K)\) denotes the maximal degree in the variable \(z\) appearing in the HOMFLYPT polynomial \(P_K(a,z).\)

The HOMFLYPT polynomial is characterized by its skein relation and simultaneously generalizes both the Alexander and Jones polynomials.

32. HOMFLYPT \(a\)-spread \(spP_a(K)\)↩︎

The HOMFLYPT \(a\)-spread is defined by
\(spP_a(K)=\deg_{\max}^{a}P_K(a,z)-\deg_{\min}^{a}P_K(a,z).\)

33. Kauffman polynomial \(a\)-spread \(spF_a(K)\)↩︎

The Kauffman polynomial \(a\)-spread is defined by \(spF_a(K)=\deg_{\max}^{a}F_K(a,z)-\deg_{\min}^{a}F_K(a,z),\) where \(F_K(a,z)\) is the Kauffman polynomial of the knot.

References↩︎