whether there is a "knot theory" for graphs...

 

or can it be essentially reduced to the study of knots (and links)?

As Dror Bar-Natan points out in his interesting answer, it can, "if you totally understand the theory of tangles". If you don't, but you're very generous as to what amounts to a reduction, then it "almost can" (up to about one integer invariant) by a theorem of Roberston, Seymour and Thomas: two knotless, linkless embeddings $f,g$ of a graph $G$ in $\Bbb R^3$ are equivalent (by an isotopy of $\Bbb R^3$) if and only if the restictions of $f$ and $g$ to every subgraph of $G$, homeomorphic to $K_5$ or $K_{3,3}$ are equivalent. Here "knotless" means that every cycle (a subgraph homeomorphic to $S^1$) in $G$ is unknotted, and "linkless" means that every two disjoint subgraphs are separated by an embedded $2$-sphere. To be precise, Robertson, Seymour and Thomas had a slightly different formulation (with "panelled" in place of "knotless and linkless") and the above version is proved in http://arxiv.org/abs/math/0612082.

What is the "about one integer invariant"? As Ryan Budney points out in his interesting answer, it helps to study graphs up to weaker equivalence relation than ambient equivalence or non-ambient isotopy (which, incidentally, already kills all local knots). Taniyama (Topol. Appl. 65 (1995), 205-228) has shown that two embeddings of a graph $G$ in $\Bbb R^3$ are "homologous" (=cobound an embedded $G\times I$+(handles) in $\Bbb R^3\times I$, where each handle is a torus attached by a tube to a $2$-cell, (edge)$\times I$) if and only if they have the same Wu invariant (this integer invariant is really just the $1$-parameter version of the van Kampen obstruction). On the other hand, Shinjo and Taniyama (Topol. Appl. 134 (2003), 53-67) have shown that the vanishing of the Wu invariant of a graph is determined by the vanishing of its restriction to subgraphs homeomorphic to $K_5$, $K_{3,3}$ and $S^1\sqcup S^1$.

Another interesting relation on embedded graphs in link homotopy, i.e. arbitrary self-intersections of connected components are allowed, but distinct components may not intersect. The link homotopy classification of embeddings in $\Bbb R^3$ of a disjoint union of two $S^1$'s and a wedge of $S^1$ is already pretty nontrivial.

whether there is a "knot theory" for graphs...

or can it be essentially reduced to the study of knots (and links)?

As Dror Bar-Natan points out in his interesting answer, it can, "if you totally understand the theory of tangles". If you don't, but you're very generous as to what amounts to a reduction, then it "almost can" (up to about one integer invariant) by a theorem of Roberston, Seymour and Thomas: two knotless, linkless embeddings $f,g$ of a graph $G$ in $\Bbb R^3$ are equivalent (by an isotopy of $\Bbb R^3$) if and only if the restictions of $f$ and $g$ to every subgraph of $G$, homeomorphic to $K_5$ or $K_{3,3}$ are equivalent. Here "knotless" means that every cycle (a subgraph homeomorphic to $S^1$) in $G$ is unknotted, and "linkless" means that every two disjoint subgraphs are separated by an embedded $2$-sphere. To be precise, Robertson, Seymour and Thomas had a slightly different formulation (with "panelled" in place of "knotless and linkless") and the above version is proved in http://arxiv.org/abs/math/0612082.

What is the "about one integer invariant"? As Ryan Budney points out in his interesting answer, it helps to study graphs up to weaker equivalence relation than ambient equivalence or non-ambient isotopy (which, incidentally, already kills all local knots). Taniyama (Topol. Appl. 65 (1995), 205-228) has shown that two embeddings of a graph $G$ in $\Bbb R^3$ are "homologous" (=cobound an embedded $G\times I$+(handles) in $\Bbb R^3\times I$, where each handle is a torus attached by a tube to a $2$-cell, (edge)$\times I$) if and only if they have the same Wu invariant (this integer invariant is really just the $1$-parameter version of the van Kampen obstruction). On the other hand, Shinjo and Taniyama (Topol. Appl. 134 (2003), 53-67) have shown that the vanishing of the Wu invariant of a graph is determined by the vanishing of its restriction to subgraphs homeomorphic to $K_5$, $K_{3,3}$ and $S^1\sqcup S^1$.

Another interesting relation on embedded graphs in link homotopy, i.e. arbitrary self-intersections of connected components are allowed, but distinct components may not intersect. The link homotopy classification of embeddings in $\Bbb R^3$ of a disjoint union of two $S^1$'s and a wedge of $S^1$ is already pretty nontrivial.

whether there is a "knot theory" for graphs...

or can it be essentially reduced to the study of knots (and links)?

As Dror Bar-Natan points out in his interesting answer, it can, "if you totally understand the theory of tangles". If you don't, but you're very generous as to what amounts to a reduction, then it "almost can" (up to about one integer invariant) by a theorem of Roberston, Seymour and Thomas: two knotless, linkless embeddings $f,g$ of a graph $G$ in $\Bbb R^3$ are equivalent (by an isotopy of $\Bbb R^3$) if and only if the restictions of $f$ and $g$ to every subgraph of $G$, homeomorphic to $K_5$ or $K_{3,3}$ are equivalent. Here "knotless" means that every cycle (a subgraph homeomorphic to $S^1$) in $G$ is unknotted, and "linkless" means that every collection of disjoint cycles is unlinked. In fact, pairs of disjoint cycles suffice here, and their unlinking can be weakened to zero linking number (this of course leads to inequivalent definitions of "linkless", but either one suffices for the theorem to hold). To be precise, Robertson, Seymour and Thomas had a slightly different formulation (with "panelled" in place of "knotless and linkless") and the above version is proved in http://arxiv.org/abs/math/0612082.

What is the "about one integer invariant"? As Ryan Budney points out in his interesting answer, it helps to study graphs up to weaker equivalence relation than ambient equivalence or non-ambient isotopy (which, incidentally, already kills all local knots). Taniyama (Topol. Appl. 65 (1995), 205-228) has shown that two embeddings of a graph $G$ in $\Bbb R^3$ are "homologous" (=cobound an embedded $G\times I$+(handles) in $\Bbb R^3\times I$, where each handle is a torus attached by a tube to a $2$-cell, (edge)$\times I$) if and only if they have the same Wu invariant (this integer invariant is really just the $1$-parameter version of the van Kampen obstruction). On the other hand, Shinjo and Taniyama (Topol. Appl. 134 (2003), 53-67) have shown that the vanishing of the Wu invariant of a graph is determined by the vanishing of its restriction to subgraphs homeomorphic to $K_5$, $K_{3,3}$ and $S^1\sqcup S^1$.

Another interesting relation on embedded graphs in link homotopy, i.e. arbitrary self-intersections of connected components are allowed, but distinct components may not intersect. The link homotopy classification of embeddings in $\Bbb R^3$ of a disjoint union of two $S^1$'s and a wedge of $S^1$ is already pretty nontrivial.

whether there is a "knot theory" for graphs...

or can it be essentially reduced to the study of knots (and links)?

As Dror Bar-Natan points out in his interesting answer, it can, "if you totally understand the theory of tangles". If you don't, but you're very generous as to what amounts to a reduction, then it "almost can" (up to about one integer invariant) by a theorem of Roberston, Seymour and Thomas: two knotless, linkless embeddings $f,g$ of a graph $G$ in $\Bbb R^3$ are equivalent (by an isotopy of $\Bbb R^3$) if and only if the restictions of $f$ and $g$ to every subgraph of $G$, homeomorphic to $K_5$ or $K_{3,3}$ are equivalent. Here "knotless" means that every cycle (a subgraph homeomorphic to $S^1$) in $G$ is unknotted, and "linkless" means that every two disjoint subgraphs are separated by an embedded $2$-sphere. To be precise, Robertson, Seymour and Thomas had a slightly different formulation (with "panelled" in place of "knotless and linkless") and the above version is proved in http://arxiv.org/abs/math/0612082.

What is the "about one integer invariant"? As Ryan Budney points out in his interesting answer, it helps to study graphs up to weaker equivalence relation than ambient equivalence or non-ambient isotopy (which, incidentally, already kills all local knots). Taniyama (Topol. Appl. 65 (1995), 205-228) has shown that two embeddings of a graph $G$ in $\Bbb R^3$ are "homologous" (=cobound an embedded $G\times I$+(handles) in $\Bbb R^3\times I$, where each handle is a torus attached by a tube to a $2$-cell, (edge)$\times I$) if and only if they have the same Wu invariant (this integer invariant is really just the $1$-parameter version of the van Kampen obstruction). On the other hand, Shinjo and Taniyama (Topol. Appl. 134 (2003), 53-67) have shown that the vanishing of the Wu invariant of a graph is determined by the vanishing of its restriction to subgraphs homeomorphic to $K_5$, $K_{3,3}$ and $S^1\sqcup S^1$.

Another interesting relation on embedded graphs in link homotopy, i.e. arbitrary self-intersections of connected components are allowed, but distinct components may not intersect. The link homotopy classification of embeddings in $\Bbb R^3$ of a disjoint union of two $S^1$'s and a wedge of $S^1$ is already pretty nontrivial.