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Apply the given skein relation to a positive and a negative kink in a diagram of the unknot. Both crossed link diagrams represent the unknot, whereas the oriented smoothing represents the two-component unlink. ThusFactor the left-hand side and cancel the nonzero factor in the Laurent polynomial ring:The positive sign follows from the skein relation as printed in this paper. It differs from the frequently used Jones polynomial convention whose two-component unlink has value $-(q^{1/2}+q^{-1/2})$. With the printed convention, an $r$-component link has invariant $(-1{)}^{r-1}$ times that convention's Jones polynomial.

Reflection through the projection plane reverses the orientation of $S^3$ and exchanges the overpassing and underpassing strands at every crossing. Thus switching all crossings gives the mirror of a link, with the component orientations transported by the reflection.

Define $W(L)(q)=V(\overline{L})(q^{-1})$. Reflection exchanges $D_{-}$ with $D_{+}$ and preserves the oriented smoothing. Apply the printed skein relation to the reflected triple at parameter $q^{-1}$:Multiplication by $-1$ gives the original skein relation for $W$. Also $W(U_1)=1$. The stated uniqueness of the Jones polynomial now gives $W=V$, hence

Let $H_{+}$ be the positively oriented Hopf link obtained by closing a two-strand braid with two positive crossings. Switching one crossing gives the two-component unlink; the oriented smoothing gives the unknot. The printed skein relation givesRepresent the right-handed trefoil knot by the closure of the two-strand braid with three positive crossings. Switching one crossing gives the unknot, while its oriented smoothing gives $H_{+}$. Consequentlyso the two Jones polynomials areThe second expression follows from the mirror of a link identity. They are unequal Laurent polynomials, so invariance under link isotopy proves that the right-handed and left-handed trefoils are not isotopic.

Apply the Kauffman bracket skein relation to every crossing in the two-string tangle $D_1$. Each complete smoothing is a collection of closed circles together with one of the two crossingless pairings of the four boundary points. Rotation through $180^{\circ }$ about the indicated vertical axis preserves both crossingless pairings and the number of closed circles. Gluing the unchanged tangle $D_2$ to corresponding smoothings therefore gives equal terms, including equal loop factors, so $⟨D⟩=⟨D^{\prime }⟩$.

Choose an orientation of $L$, and transport the induced orientations of the four ends of $D_1$ through the rotation to orient $L^{\prime }$. Crossing signs inside the rotated tangle and outside it then have the same total, so the writhe of a link diagram satisfies $w(D)=w(D^{\prime })$. Applying the writhe normalization of the Jones polynomial gives

The component knot types can change under this Conway mutation. To construct an example, place a knotted arc carrying a trefoil knot summand in one strand of $D_1$ and a knotted arc carrying a figure-eight knot summand in one strand of $D_2$, leaving the other two strand portions trivial. Choose the outside pairing so that before rotation the two knotted portions lie on different components, while after rotation they lie on the same component. Then the component multisets arerespectively. A link isotopy preserves the unordered multiset of component knot types, so these links are not isotopic, although part (a) shows that suitable orientations give them the same Jones polynomial.

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

The unknot $U$ is the knot type represented by the planar circleEquivalently, an unknot is a knot that bounds a smoothly embedded disk in the three-sphere. The disk must itself be embedded; merely filling a loop with a map that is allowed to intersect itself does not establish that the knot is an unknot. Flattening a neighborhood of an embedded spanning disk explains why this criterion recovers the planar circle.

A complicated drawing can still represent an unknot. Starting with a crossing-free knot diagram of $U$, insert a small kink by a type I Reidemeister move. The resulting knot diagram has one crossing, but removing the kink exhibits the same unknot. Repeating in separate small regions gives knot diagrams of the unknot with arbitrarily many crossings, although the crossing number of a knot is $c(U)=0$. A drawing's apparent complexity is therefore not a certificate of knotting.

The normalized Alexander polynomial of a knot and Jones polynomial of the unknot are both $1$, and its knot group is the infinite cyclic group $\mathbb{Z}$, because the knot exterior is a solid torus. These are useful comparison values. In particular, a nonconstant Alexander polynomial of a knot certifies that a knot is not an unknot, but the converse fails: nontrivial untwisted Whitehead doubles also have Alexander polynomial of a knot $1$.

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 $(-2{)}^l$.