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        Questions tagged [knot-theory]

        Knot theory is dealing with embedding of curves in manifolds of dimension 3. A knot is a single circle embedded in the affine space of dimension 3 as a smooth curve not crossing itself. Many knot invariants are known and can be used to distinguish knots.

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        Motivic Knot Embedding

        I've been trying to be more diligent about reading motivic homotopy theory, and have been reading Levine's 'An Overview of Motivic Homotopy Theory.' I think the subject is fascinating, and I've ...
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        0answers
        53 views

        Classification of surface curves? [duplicate]

        Having a genus $g$ closed orientable surface $\Sigma$ (connected sum of $g$ tori), how do I encode any embedded closed curve up to planar isotopy? For $g=1$ we can just state the coefficient $k \in \...
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        42 views

        Dimension of the skein module of a closed manifold?

        I'm looking for a reference to Witten's conjecture that the free part of the (Kauffman bracket) skein module of a closed 3-manifold is finitely generated, i.e. the dimension of $K(M)$, where $M$ is a ...
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        5answers
        1k views

        Connection Between Knot Theory and Number Theory

        Is there any connection between knot theory and number theory in any aspects? Does anybody know any book that is about knot theory and number theory?
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        1answer
        199 views

        Is the quantum $\mathfrak{sl}_3$ invariant stronger than the quantum $\mathfrak{sl}_2$ invariant?

        Both the $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ quantum framed link invariants can be computed using linear skeins. The first being computed using the Kauffman bracket and the second using a similar ...
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        1answer
        111 views

        Signature/nullity function for a link obtained by parallel pushoffs of a knot?

        Let $K$ be an oriented knot in $S^3$ together with a framing $n$. Let $K(a,b)$ be the oriented link obtained by taking $a$ copies of the $n$-pushoff of $K$ with the same same orientation as $K$ and $...
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        2answers
        235 views

        About the commutativity of the $1^{st}$ homotopy group of the space of knots

        I would like to know if the fundamental group of the connected component of a knot space could be non commutative. I am specially interested in the case of $\mathbb{R}^3$, $\mathbb{S^3}$ or some other ...
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        72 views

        Is there a notion of tunnel number for 2-knots?

        Given an embedded circle $K$ in $S^3$, the tunnel number of $K$ is the minimum number of embedded arcs one needs to add to $K$ so that the complement of $K$ and the arcs is a handlebody. For an ...
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        0answers
        200 views

        Chromatic polynomial and the circle

        In https://arxiv.org/pdf/1208.5781.pdf It is proved that there is spectral sequence converging to $H^*(M^G,R)$ with the E1 page given by the graph cohomology complex $C_A(G)$ where $A:=H^*(M,R)$. My ...
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        0answers
        29 views

        Subalgebras and ideals of algebra of Vassiliev invariants

        Let $\mathcal{K}$ be the commutative monoid whose elements are (isotopy) equivalence classes of knots with composition under the connected knot sum, and $\mathbb{Z}\mathcal{K}$ be the corresponding ...
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        3answers
        503 views

        SL(2, C)-representation of a knot

        When studying knot theory I often encounter $SL(2, \mathbb{C})$-representation of knots (of the knot group) or the $SL(2, \mathbb{C})$ character variety of a knot group. But I just don't seem to ...
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        1answer
        213 views

        A question about dimension of SL(2,C) character variety of knot group

        It is known that if there isn't a closed essential surface in $S^3 \setminus K$, the dimension of $SL(2,\mathbb C)$ character variety is $1$. (In fact, it holds for a general 3-manifold, not only for ...
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        votes
        0answers
        65 views

        Handlesliding a two component, linking number 1 link

        Let $L = K_1 \cup K_2 \subset S^3$ be a two component framed link with $lk(K_1,K_2) = 1$. Let $\hat{L}$ denote the set of all links obtained by handlesliding $L$ around an an arbitrary number of ...
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        1answer
        72 views

        Proving knot polynomial dependencies and skein relations

        I have two questions: From the definition of the Jones polynomial as the normalization of the Kauffman bracket $(-A^3)^{-w(D)} \langle D\rangle$ and substituting $A\rightarrow t^{-1/4}$, how does one ...
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        2answers
        201 views

        Is there a known invariant for knotted surfaces defined by skein relations?

        Is there a known invariant for knotted surfaces in $\mathbb{R}^4$ (possibly with additional structures, e.g. colored, framed, etc.) which can be defined using skein relations? By skein relations for ...

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