Link

A link is a smooth embedding of a finite disjoint union of circles in $S^3$, considered up to ambient isotopy. A one-component link is a knot.

A link invariant assigns the same value to links related by link isotopy. The Jones polynomial applies to oriented links, while its bracket precursor retains a framing dependence. An invariant need not distinguish all link types.

A split link admits an embedded sphere in its complement with nonempty portions of the link on both sides. A split diagram admits a simple closed curve in the projection sphere separating nonempty portions of the diagram. A split diagram always represents a split link; the converse need not hold for an arbitrary diagram.

A split diagram has two nonempty parts separated by a simple closed curve disjoint from the diagram. For a two-component link, this is equivalent to disconnectedness of its projection graph. An alternating link diagram representing a split link cannot have a connected projection graph, as its Jones value at minus one detects a nonzero spanning-tree count.

The Borromean rings are a three-component link whose components are unknotted, which is nontrivial although removing any one component leaves an unlink. Its complement has a complete finite-volume hyperbolic three-manifold structure obtained from two regular ideal octahedra.

With group commutator convention $[a,b]=a^{-1}{b}^{-1}ab$, the displayed group presentation describes the fundamental group of the Borromean rings complement. Its abelianization is ${\mathbb{Z}}^3$, since each relator is a group commutator and hence imposes no abelian relation. It embeds discretely in the Gaussian Bianchi group by sending the meridians to $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$, $\left(\begin{array}{cc}1& 0\\ 2i& 1\end{array}\right)$ and $\left(\begin{array}{cc}i& 1\\ 2i& 2-i\end{array}\right)$.

For a link with at least two components, this polynomial is the greatest common divisor of the appropriate maximal minors of an abelianized Alexander matrix, with one variable per oriented component meridian. It is defined up to multiplication by a signed monomial. For a deficiency-one presentation with $g$ generators and $g-1$ relators, use its $(g-1)$-row minors. This normalization gives one for a Hopf link.

The exterior of a link is the complement of the interior of a closed tubular neighborhood of the entire link. Its boundary consists of one torus per component; its first homology has the component meridians as a free basis.

The Hopf link consists of two unknots with linking number $\pm 1$. Each component is a meridian of a knot of the other.

The Hopf link exterior is $T^2\times I$. Each component's Seifert longitude is the other's meridian. Filling the two boundary tori along these longitudes therefore attaches solid tori with meridians intersecting once on a common torus, recovering the standard genus-one splitting of the three-sphere. Merely specifying two linked unknots without specifying their link type does not force this answer.

An unlink is a link whose components bound pairwise disjoint embedded disks in $S^3$.

The linking number of two disjoint oriented closed curves in $S^3$ is the signed intersection number of one curve with a Seifert surface for the other. In an oriented diagram it is half the signed sum of crossings between the two components.

A link isotopy is an isotopy of the ambient three-sphere carrying one link to another. It preserves the number and individual knot types of the components.

Here ambient isotopy is applied to the embedded link inside the three-sphere.

Apply an orientation-reversing homeomorphism of $S^3$ to a link and transport its component orientations. Reflection through the projection plane produces a link diagram by switching every crossing.

For an oriented closed curve, geometric writhe is the signed crossing number averaged over generic viewing directions. For an individual link diagram, writhe of a link diagram is its signed crossing sum. Both depend on geometry or projection rather than only the link isotopy class.

A link diagram is a generic planar projection of a link with overcrossing and undercrossing data.

A link diagram is alternating if overpassing and underpassing crossings alternate along every component. Its checkerboard smoothing choices are consistent across all crossings of a connected projection. An alternating knot diagram is the one-component case.

A reduced alternating link diagram is an alternating link diagram with no nugatory crossing. Its Tait graph and dual have no loops, so it is an adequate link diagram. If connected, its Jones polynomial breadth equals its crossing count.

// Target: module-theory.bigb

A diagram is A-adequate if changing a single smoothing of its all-A state joins two different state circles, and B-adequate analogously. It is adequate if both conditions hold. In an adequate diagram the top and bottom degree terms of the Kauffman bracket arise uniquely from these two states. Reduced alternating diagrams are adequate because their Tait graphs and duals have no loop edges.

Replace each crossing of a diagram by a saddle joining its A smoothing above the projection sphere to its B smoothing below; cap the upper and lower state circles. For a connected classical link diagram this constructs a connected closed orientable surface. Its Euler characteristic is $s_A+s_B-c(D)$, so $s_A+s_B\le c(D)+2$. For an alternating diagram the surface has genus zero.

Choose a checkerboard coloring of the complementary regions of a link diagram. The Tait graph has one vertex for each black region and one edge through each crossing joining its adjacent black regions. The other coloring gives its planar dual. For a connected alternating link diagram, all-A and all-B smoothing circles correspond to the regions of its two checkerboard colors; one-circle smoothing states correspond to spanning trees.

A skein relation relates the invariants of link diagrams agreeing outside a disk and differing by a positive crossing, a negative crossing, and a smoothing within it. Its signs and normalization determine the particular invariant convention.

The oriented smoothing replaces a crossing by two disjoint arcs whose orientations agree with the four oriented ends. It is the zero-resolution in an oriented skein relation.

The writhe is the sum of the signs of all crossings in an oriented link diagram. It changes under the first Reidemeister move and is unchanged under the second and third moves.

Diagram writhe depends on the chosen projection; averaging it over projection directions gives the geometric writhe.

A tangle is a proper embedding of arcs and circles in a three-ball with prescribed arc endpoints on its boundary. A two-string tangle has four boundary endpoints.

Objects are finite words of strand orientations. Morphisms are boundary-fixed ambient-isotopy classes of oriented tangles in a slab, composed by stacking compatible boundary words; horizontal juxtaposition is the tensor product. Cups, caps, identity strands and crossings generate the category. Ordinary unframed isotopy includes all three Reidemeister moves; framed variants impose different first-move relations.

A Conway mutation cuts a link along a sphere meeting it in four points, rotates the enclosed two-string tangle through $180^{\circ }$, and reglues it.

The Kauffman bracket is the unoriented diagram invariant determined by resolving each crossing into its two smoothings and replacing every additional disjoint circle by the loop factor $-A^2-A^{-2}$. It is invariant under the second and third Reidemeister moves.

For an oriented 3-manifold, take the linear span of isotopy classes of framed links, including the empty link, and impose the local bracket smoothing relation and the trivial-circle relation with factor $-A^2-A^{-2}$. Fixing framed boundary endpoints gives its relative version. Gluing manifolds or stacking cylinders induces maps between skein modules; framed isotopy, rather than unframed Reidemeister-I invariance, is built into this construction.

The relative skein module fixes finitely many framed endpoints on the boundary of a 3-manifold and uses properly embedded framed tangles with those endpoints. Local Kauffman bracket relations are imposed away from the boundary. Gluing compatible endpoint sets composes tangles; a rectangle cylinder with n endpoints on each horizontal side gives the Temperley-Lieb diagram algebra.

A state with $k$ B smoothings has at most $s_A+k$ circles, so its largest A exponent is at most $c(D)+2s_A-2$. The dual argument bounds the smallest exponent below by $-c(D)-2s_B+2$. For a connected link diagram, the Turaev surface inequality bounds the resulting breadth by $4c(D)$. For a connected reduced alternating diagram, adequacy prevents cancellation at both extremes and planar Euler characteristic gives equality. A disconnected projection needs the additional term in the Jones polynomial breadth bound for a disconnected diagram.

Here $n$ is the crossing count and $k$ is the number of connected components of the planar link diagram, not necessarily the number of link components. The all-$A$ and all-$B$ state counts satisfy $s_A+s_B\le n+2k$, by summing the Turaev surface Euler characteristic bound over projection components. The Kauffman bracket breadth bound and $t=A^{-4}$ give the displayed bound. It is sharp for an unlink diagram with $n=0$, whose Jones polynomial has breadth of a Laurent polynomial $k-1$.

The Temperley-Lieb diagram algebra is generated by planar cap-cup tangles, with composition by stacking and each closed circle replaced by $\delta$. Its local relations $e_i^2=\delta e_i$ and $e_i{e}_{i+1}{e}_i=e_i$ make the second and third Reidemeister moves direct algebraic identities for $R_i=AI+A^{-1}{e}_i$ when $\delta =-A^2-A^{-2}$.

The Jones-Wenzl idempotent has identity-diagram coefficient one, satisfies $f_n^2=f_n$, and annihilates each adjacent cap-cup generator on either side. With $\delta =-A^2-A^{-2}$ and ${\Delta }_0=1$, ${\Delta }_1=\delta$, ${\Delta }_n=\delta {\Delta }_{n-1}-{\Delta }_{n-2}$, it is constructed recursively by $f_n=F-({\Delta }_{n-2}/{\Delta }_{n-1})F{e}_{n-1}F$, where $F=f_{n-1}\otimes 1$. The required denominators must be nonzero. Its closed bracket trace is ${\Delta }_n=(-1{)}^n[n+1]$.

Close the last strand of the projector recursion. Closing $f_{n-1}\otimes 1$ gives $\delta f_{n-1}$, while closing its cap-cup correction gives $f_{n-1}$. The coefficient is $\delta -{\Delta }_{n-2}/{\Delta }_{n-1}={\Delta }_n/{\Delta }_{n-1}$. Iteration proves $\mathrm{tr}(f_n)={\Delta }_n$ and supplies the normalization needed to split the two fusion of Jones-Wenzl colors channels in $f_{n-1}\otimes 1$.

At $A=e^{\pi i/(2r)}$, $r\ge 3$, the projector $f_{r-1}$ exists but its closed trace is zero. Quotient by the skein ideal generated by this projector. The surviving simple colors are $0,\dots ,r-2$, with nonzero quantum dimensions $d_j=(-1{)}^j\sin ((j+1)\pi /r)/\sin (\pi /r)$. This truncation prevents zero-trace channels from interfering with nondegenerate fusion and surgery normalization.

A surviving color denotes $j$ parallel framed strands equipped with the Jones-Wenzl idempotent $f_j$. Its closed trace is its quantum dimension. The semisimplicity of the reduced Temperley-Lieb category identifies these colors with the simple objects; their tensor products decompose according to fusion of Jones-Wenzl colors.

The Jones-Wenzl idempotent recursion decomposes $f_j\otimes 1$ into the orthogonal $f_{j+1}$ and $f_{j-1}$ channels. The Jones-Wenzl partial trace supplies the nonzero scalar normalizing the cap-cup channel. At the highest surviving color the $f_{r-1}$ channel is zero. Induction therefore decomposes every tensor power into the surviving colors. A planar matching between different projected colors has a turnback on the larger side and vanishes; between equal colors only the identity matching remains. Their endomorphism spaces are consequently one-dimensional, and the resulting finite matrix blocks split. Taking direct sums and splitting these idempotents produces a semisimple category, supplying the complete resolutions needed in fusion of Jones-Wenzl colors and the Kirby color handle-slide identity.

In the reduced theory the product of colors a and b decomposes with multiplicity one into colors c satisfying $|a-b|\le c\le \min (a+b,2r-4-a-b)$ and $a+b+c$ even. These rules follow by iterating $1\otimes j=(j-1)\oplus (j+1)$, with color $r-1$ set to zero. The closure traces give $d_a{d}_b={\sum }_c{N}_{ab}^c{d}_c$, and self-duality makes the multiplicities symmetric in all three colors.

In the reduced Jones-Wenzl theory, bend a fixed self-dual color a in the mutually inverse inclusion/projection maps of $c\subset b\otimes a$, obtaining $U_c^b:b\to c\otimes a$ and $V_c^b:c\otimes a\to b$. Cyclic closure trace and straightening cups and caps give $V_c^b{U}_c^b=(d_c/d_b)1_b$. Completeness of the normalized maps for all b then proves the weighted resolution. The dimensions are nonzero in the reduced theory. This local identity gives the precise weight preservation in the Kirby color handle-slide identity.

The full bracket of a zero-framed Hopf link colored i and j is the displayed pairing. Its normalized matrix is the finite sine transform with parity signs. Sine orthogonality gives $H^2={\mathcal{D}}^{2}I$, where ${\mathcal{D}}^2={\sum }_j{d}_j^2=r/(2{\sin }^2(\pi /r))$. Thus the pairing is nondegenerate in the reduced theory.

A positive framing twist on an n-strand Jones-Wenzl cable has n positive individual curls and $n(n-1)$ braid crossings. Each curl contributes $-A^3$, and each braid crossing contributes $A$ because its turnback term annihilates the Jones-Wenzl idempotent. Their product is the displayed scalar. This is the colored framing factor used in surgery invariants.

Close a three-strand planar tangle and evaluate its reduced Kauffman bracket. The basis $1,e_1,e_2,e_1{e}_2,e_2{e}_1$ has the displayed trace values, with the last value also valid for $e_2{e}_1$. Writing braid generators as $A1+A^{-1}{e}_i$ turns knot-polynomial computations into a five-dimensional algebra calculation.

A bracket smoothing state chooses one of the two unoriented smoothings at every crossing of a link diagram. Its weight is $A^{a-b}{\delta }^{s-1}$ in the reduced Kauffman bracket, where $a,b$ count the smoothing types and $s$ counts resulting circles. The all-$A$ and all-$B$ states choose the same smoothing type at every crossing.

The Jones polynomial is obtained from the Kauffman bracket of an oriented diagram by the writhe normalization

This asks whether the Jones polynomial equal to one characterizes the unknot among knots. A zero breadth of a Laurent polynomial makes $V_K$ a monomial; the identities $V_K(1)=1$ and $V_K^{\prime }(1)=0$ then force that monomial to be one. Thus using Jones breadth as a strictly positive complexity for nontrivial knots requires this detection property. To prove the derivative identity, differentiate the Jones polynomial skein relation at one for a crossing change in a knot. Both crossed diagrams have value one, while the smoothed two-component link has value minus two by Jones polynomial evaluation at one, so $V_{K_{+}}^{\prime }(1)=V_{K_{-}}^{\prime }(1)$. Crossing changes reduce to the unknot, whose derivative is zero. The corresponding Alexander polynomial detection property is false, as nontrivial untwisted Whitehead doubles demonstrate. The Jones question is discussed in research on closed four-strand braids.

Use the Conway-normalized Alexander polynomial. Choosing $t^{1/2}=i$ in the Jones polynomial skein relation gives $V_{L_{+}}(-1)-V_{L_{-}}(-1)=-2i{V}_{L_0}(-1)$. The quantities $(-1{)}^{\ell -1}{\nabla }_L(2i)$ obey the same relation, because smoothing changes the component count by one. Their unlink values agree, so crossing-change induction proves equality for every link. For a knot, the sign is one and ${\nabla }_K(2i)={\Delta }_K(-1)$. This proves the signed identity, not only equality of knot determinants.

Reflecting a link diagram interchanges its two bracket smoothings and negates its writhe of a link diagram. This replaces $A$ by $A^{-1}$ in the normalized Kauffman bracket, hence $t$ by $t^{-1}$. A knot whose Jones polynomial differs from its reciprocal cannot be an amphichiral knot.

For a nonempty link with $\ell$ components, the Jones polynomial skein relation at $t=1$ makes a crossing change preserve the value. Crossing changes turn the link into an unlink with the same number of components. Repeated Jones polynomial of a split union gives the formula. In particular the Jones polynomial is never the zero polynomial.

Placing the two link diagrams in disjoint disks keeps all state circles separate. The reduced Kauffman bracket therefore acquires one extra loop factor $-A^2-A^{-2}$. Correcting writhe of a link diagram and setting $t=A^{-4}$ gives the displayed formula. It concerns a split link, not an arbitrary union of linked components.

For knots, matching bracket smoothing states across a connected sum of knots joins one state circle from each side. Their total circle count is therefore the sum of the two counts minus one. The reduced Kauffman bracket state weights multiply, and writhe of a link diagram adds, proving the formula.

Resolve the two crossings in a skein relation using the Kauffman bracket, then correct by the writhe of a link diagram. The other smoothing cancels, leaving the displayed coefficient of the oriented smoothing. This relation and the value one on the unknot determine the Jones polynomial: crossing changes give descending diagrams, which are unlinks, while the smoothing terms have fewer crossings.

Use the Kauffman bracket assigning the empty diagram value $1$ and a disjoint circle factor $-A^2-A^{-2}$. Correct by the writhe of a link diagram and substitute $t=A^{-4}$. Relative to the Jones polynomial normalized to be $1$ on the unknot, this unreduced invariant has an additional factor $-(t^{1/2}+t^{-1/2})$. For a nonempty $l$-component link, its value at $t=1$ is $(-2{)}^l$.

The torus link $T_{p,q}$ lies on an unknotted torus and winds $p$ times meridionally and $q$ times longitudinally. It has $\gcd (p,q)$ components; distinct components of the positively oriented $T_{n,n}$ have linking number one.

A framed link is a link together with a homotopy class of nonzero normal vector field on every component. Relative to the Seifert framing, each component framing is encoded by an integer.

The framing of a link diagram obtained by pushing a component slightly in the plane of the diagram. Diagram curls change this framing; arbitrary removal of curls changes an anyon diagram by its topological spin.

The symmetric matrix with the component framing integers on the diagonal and the pairwise linking numbers off the diagonal. Its signature enters the Witten-Reshetikhin-Turaev invariant.

A surface framing pushes a curve in the direction tangent to an oriented topological surface containing it and normal to the curve. For a two-handle attached along a curve on a boundary topological surface, this specifies the attaching annulus.

The Seifert framing pushes a knot in the direction tangent to a Seifert surface and normal to its boundary. The knot and its push-off then have linking number zero.

Integral Dehn surgery removes a tubular neighborhood of every component and reglues a solid torus so that its meridian follows the slope specified by the framing.

The framing specifies the integer slope in this special case of Dehn surgery.

Relative to the meridian of a knot $\mu$ and Seifert longitude $\lambda$, an integer surgery coefficient $n$ means that the filling meridian of a solid torus is identified with $n\mu +\lambda$.

The surgery trace $W(L)$ is obtained from $B^4$ by attaching one two-handle along every component of a framed link $L\subset S^3$. Its boundary is the three-manifold produced by surgery on $L$.

The surgery linking matrix has the framing coefficients on its diagonal and pairwise linking numbers off the diagonal. It represents the intersection form on the second homology of the surgery trace, presents the first homology of its boundary, and has kernel isomorphic to the boundary's second homology.

Kirby calculus changes framed-link diagrams by handle slides and creation or cancellation of standard handle pairs without changing the represented four-manifold up to diffeomorphism.

For an m-component integral framed surgery link, normalize its full colored bracket by the displayed expression, where $\sigma (L)$ is the signature of its linking matrix. The Kirby color handle-slide identity handles slides; $p_{+}{p}_{-}={\mathcal{D}}^2$ cancels disjoint plus- or minus-one-framed unknots. Kirby calculus therefore makes this an oriented 3-manifold invariant. This convention has ${\tau }_r(S^3)=1$.

// Target: module-theory.bigb

A Kirby diagram depicts a handle decomposition of a four-dimensional manifold using a framed link for the two-handles and dotted circles for the one-handles. The framing specifies the attaching normal directions; it controls the resulting intersection form and the boundary Dehn surgery.

If a meridian of a knot component in a surgery link links only one other component, filling it with coefficient $b\ne 0$ removes it and replaces the other coefficient $a$ by $a-1/b$. This gives the negative continued-fraction chain description of rational Dehn filling.

For coprime integers $p>0$ and $q$, the lens space $L(p,q)$ is the quotient of $S^3\subset {\mathbb{C}}^2$ by $(z_1,z_2)\mapsto (e^{2\pi i/p}{z}_1,e^{2\pi iq/p}{z}_2)$. Equivalently, it is obtained by an appropriate rational surgery on the unknot.

The quotient model of a lens space splits into two solid tori. In the boundary angle lattice ${\mathbb{Z}}^2+\mathbb{Z}(1/p,q/p)$, choose ${\mu }_1=(0,1)$ and ${\lambda }_1=(1/p,q/p)$. The other meridian is $(1,0)=-q{\mu }_1+p{\lambda }_1$. A second column completing this primitive slope to a determinant-one matrix specifies a gluing map after reversing one boundary-basis convention. Different completions differ by a meridional Dehn twist extending across the second solid torus, so they give the same three-manifold.

With a consistent surgery orientation convention, $L(p,q)$ and $L(p,q^{\prime })$ are orientation-preservingly related by a homeomorphism exactly when $q^{\prime }\equiv q^{\pm 1}(\mathrm{mod}p)$. An orientation-reversing homeomorphism instead requires $q^{\prime }\equiv -q^{\pm 1}(\mathrm{mod}p)$.