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Fixed garbled statement of def of DW path integral
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Kevin Walker
Kevin Walker
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Short answer: Untwisted Dijkgraaf-Witten theories with non-abelian gauge groups (e.g. $S_3$) distinguish most knots from the unknot and from each other.

Longer answer:

Here's my understanding of your question. (Please correct me if I've misinterpreted it.) For any 3d TQFT, the path integral $Z(M)$ of a 3-manifold $M$ should give a vector in the Hilbert space $H(\partial M)$ of the boundary $\partial M$. Given 3-manifolds $M_1$ and $M_2$ with the same boundary $Y$, we say that the TQFT distinguishes $M_1$ from $M_2$ if $Z(M_1) \ne Z(M_2)$ as vectors in $H(Y)$. You are looking for a discrete gauge theory (e.g. untwisted DW theory) which distinguishes the complement of the figure-8 knot from the complement of the unknot, both of which have boundary the torus $T^2$. (Both knots must be framed in order to identify the boundaries of their complements with $T^2$.)

For an untwisted DW theory with group $G$, one basis of the Hilbert space $H(Y)$ is the set of group homomorphisms $\rho: \pi_1(Y) \to G$, modulo group conjugation. (So when $Y = T^2$, this is the set of $(m, l) \in G\times G$ such that $mlm^{-1}l^{-1} = 1$, modulo the relation $(m, l) \sim (g^{-1}mg, g^{-1}lg$.)

Let $\partial M = Y$. The component of $Z(M)$ at the basis vector corresponding to a homomorphism $\rho : \pi_1(Y) \to G$ is equal to the number of extensions of $\rho$ to $\rho':\pi_1(M) \to G$, divided byeach counted with weight $1/r$, where $r$ is the number of elements in the stabilizer of $\rho'$ (group elements $g$ such that $\rho'$ conjugated by $g$ is equal to $\rho'$).

In particular, for a framed knot $K$ in $S^3$, and for $G$ the symmetric group $S_k$, the DW path integral $Z(S^3 \setminus ndb(K))$ determines the number of homomorphisms from $\pi_1(S^3 \setminus ndb(K))$ to $S_k$ such that the meridian is taken to the transposition $(1,2)$ and the longitude is taken to an element which does not commute with $(1,2)$.

A simple calculation shows that the above homomorphism-counting invariant distinguishes the figure-8 knot from the unknot for $k=3$.

More generally, in the early, pre-Jones-polynomial days of knot tabulation, one of the most powerful tools for distinguishing knots from each other was to count homomorphisms of $\pi_1(S^3 \setminus ndb(K))$ to $S_k$ for small values of $k$ (e.g. 3, 4, 5). It was observed that these invariants nearly always distinguished non-equivalent knots. It follows that untwisted DW TQFTs do a very good job of distinguishing knots.

A good reference for untwisted DW theories is a paper by Freed and Quinn from the early 1990s. http://de.arxiv.org/abs/hep-th/9111004 .

Short answer: Untwisted Dijkgraaf-Witten theories with non-abelian gauge groups (e.g. $S_3$) distinguish most knots from the unknot and from each other.

Longer answer:

Here's my understanding of your question. (Please correct me if I've misinterpreted it.) For any 3d TQFT, the path integral $Z(M)$ of a 3-manifold $M$ should give a vector in the Hilbert space $H(\partial M)$ of the boundary $\partial M$. Given 3-manifolds $M_1$ and $M_2$ with the same boundary $Y$, we say that the TQFT distinguishes $M_1$ from $M_2$ if $Z(M_1) \ne Z(M_2)$ as vectors in $H(Y)$. You are looking for a discrete gauge theory (e.g. untwisted DW theory) which distinguishes the complement of the figure-8 knot from the complement of the unknot, both of which have boundary the torus $T^2$. (Both knots must be framed in order to identify the boundaries of their complements with $T^2$.)

For an untwisted DW theory with group $G$, one basis of the Hilbert space $H(Y)$ is the set of group homomorphisms $\rho: \pi_1(Y) \to G$, modulo group conjugation. (So when $Y = T^2$, this is the set of $(m, l) \in G\times G$ such that $mlm^{-1}l^{-1} = 1$, modulo the relation $(m, l) \sim (g^{-1}mg, g^{-1}lg$.)

Let $\partial M = Y$. The component of $Z(M)$ at the basis vector corresponding to a homomorphism $\rho : \pi_1(Y) \to G$ is equal to the number of extensions of $\rho$ to $\rho':\pi_1(M) \to G$, divided by the number of elements in the stabilizer of $\rho'$ (group elements $g$ such that $\rho'$ conjugated by $g$ is equal to $\rho'$).

In particular, for a framed knot $K$ in $S^3$, and for $G$ the symmetric group $S_k$, the DW path integral $Z(S^3 \setminus ndb(K))$ determines the number of homomorphisms from $\pi_1(S^3 \setminus ndb(K))$ to $S_k$ such that the meridian is taken to the transposition $(1,2)$ and the longitude is taken to an element which does not commute with $(1,2)$.

A simple calculation shows that the above homomorphism-counting invariant distinguishes the figure-8 knot from the unknot for $k=3$.

More generally, in the early, pre-Jones-polynomial days of knot tabulation, one of the most powerful tools for distinguishing knots from each other was to count homomorphisms of $\pi_1(S^3 \setminus ndb(K))$ to $S_k$ for small values of $k$ (e.g. 3, 4, 5). It was observed that these invariants nearly always distinguished non-equivalent knots. It follows that untwisted DW TQFTs do a very good job of distinguishing knots.

A good reference for untwisted DW theories is a paper by Freed and Quinn from the early 1990s. http://de.arxiv.org/abs/hep-th/9111004 .

Short answer: Untwisted Dijkgraaf-Witten theories with non-abelian gauge groups (e.g. $S_3$) distinguish most knots from the unknot and from each other.

Longer answer:

Here's my understanding of your question. (Please correct me if I've misinterpreted it.) For any 3d TQFT, the path integral $Z(M)$ of a 3-manifold $M$ should give a vector in the Hilbert space $H(\partial M)$ of the boundary $\partial M$. Given 3-manifolds $M_1$ and $M_2$ with the same boundary $Y$, we say that the TQFT distinguishes $M_1$ from $M_2$ if $Z(M_1) \ne Z(M_2)$ as vectors in $H(Y)$. You are looking for a discrete gauge theory (e.g. untwisted DW theory) which distinguishes the complement of the figure-8 knot from the complement of the unknot, both of which have boundary the torus $T^2$. (Both knots must be framed in order to identify the boundaries of their complements with $T^2$.)

For an untwisted DW theory with group $G$, one basis of the Hilbert space $H(Y)$ is the set of group homomorphisms $\rho: \pi_1(Y) \to G$, modulo group conjugation. (So when $Y = T^2$, this is the set of $(m, l) \in G\times G$ such that $mlm^{-1}l^{-1} = 1$, modulo the relation $(m, l) \sim (g^{-1}mg, g^{-1}lg$.)

Let $\partial M = Y$. The component of $Z(M)$ at the basis vector corresponding to a homomorphism $\rho : \pi_1(Y) \to G$ is equal to the number of extensions of $\rho$ to $\rho':\pi_1(M) \to G$, each counted with weight $1/r$, where $r$ is the number of elements in the stabilizer of $\rho'$ (group elements $g$ such that $\rho'$ conjugated by $g$ is equal to $\rho'$).

In particular, for a framed knot $K$ in $S^3$, and for $G$ the symmetric group $S_k$, the DW path integral $Z(S^3 \setminus ndb(K))$ determines the number of homomorphisms from $\pi_1(S^3 \setminus ndb(K))$ to $S_k$ such that the meridian is taken to the transposition $(1,2)$ and the longitude is taken to an element which does not commute with $(1,2)$.

A simple calculation shows that the above homomorphism-counting invariant distinguishes the figure-8 knot from the unknot for $k=3$.

More generally, in the early, pre-Jones-polynomial days of knot tabulation, one of the most powerful tools for distinguishing knots from each other was to count homomorphisms of $\pi_1(S^3 \setminus ndb(K))$ to $S_k$ for small values of $k$ (e.g. 3, 4, 5). It was observed that these invariants nearly always distinguished non-equivalent knots. It follows that untwisted DW TQFTs do a very good job of distinguishing knots.

A good reference for untwisted DW theories is a paper by Freed and Quinn from the early 1990s. http://de.arxiv.org/abs/hep-th/9111004 .

Source Link
Kevin Walker
Kevin Walker
  • 12.8k
  • 2
  • 42
  • 91

Short answer: Untwisted Dijkgraaf-Witten theories with non-abelian gauge groups (e.g. $S_3$) distinguish most knots from the unknot and from each other.

Longer answer:

Here's my understanding of your question. (Please correct me if I've misinterpreted it.) For any 3d TQFT, the path integral $Z(M)$ of a 3-manifold $M$ should give a vector in the Hilbert space $H(\partial M)$ of the boundary $\partial M$. Given 3-manifolds $M_1$ and $M_2$ with the same boundary $Y$, we say that the TQFT distinguishes $M_1$ from $M_2$ if $Z(M_1) \ne Z(M_2)$ as vectors in $H(Y)$. You are looking for a discrete gauge theory (e.g. untwisted DW theory) which distinguishes the complement of the figure-8 knot from the complement of the unknot, both of which have boundary the torus $T^2$. (Both knots must be framed in order to identify the boundaries of their complements with $T^2$.)

For an untwisted DW theory with group $G$, one basis of the Hilbert space $H(Y)$ is the set of group homomorphisms $\rho: \pi_1(Y) \to G$, modulo group conjugation. (So when $Y = T^2$, this is the set of $(m, l) \in G\times G$ such that $mlm^{-1}l^{-1} = 1$, modulo the relation $(m, l) \sim (g^{-1}mg, g^{-1}lg$.)

Let $\partial M = Y$. The component of $Z(M)$ at the basis vector corresponding to a homomorphism $\rho : \pi_1(Y) \to G$ is equal to the number of extensions of $\rho$ to $\rho':\pi_1(M) \to G$, divided by the number of elements in the stabilizer of $\rho'$ (group elements $g$ such that $\rho'$ conjugated by $g$ is equal to $\rho'$).

In particular, for a framed knot $K$ in $S^3$, and for $G$ the symmetric group $S_k$, the DW path integral $Z(S^3 \setminus ndb(K))$ determines the number of homomorphisms from $\pi_1(S^3 \setminus ndb(K))$ to $S_k$ such that the meridian is taken to the transposition $(1,2)$ and the longitude is taken to an element which does not commute with $(1,2)$.

A simple calculation shows that the above homomorphism-counting invariant distinguishes the figure-8 knot from the unknot for $k=3$.

More generally, in the early, pre-Jones-polynomial days of knot tabulation, one of the most powerful tools for distinguishing knots from each other was to count homomorphisms of $\pi_1(S^3 \setminus ndb(K))$ to $S_k$ for small values of $k$ (e.g. 3, 4, 5). It was observed that these invariants nearly always distinguished non-equivalent knots. It follows that untwisted DW TQFTs do a very good job of distinguishing knots.

A good reference for untwisted DW theories is a paper by Freed and Quinn from the early 1990s. http://de.arxiv.org/abs/hep-th/9111004 .