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I am doing project in Quandles . And I am reading Thesis by David joyce An Algebraic Approach to Symmetry with Applications to Knot Theory . In its Section 4.5 Knot quandles page number 46 there is a line " when a loop in X-K links once with K " , I am not getting what it means . And after that knot quandle is defined as nooses linking once with K upto homotopy which is also not clear to me . I have not done course in Algebraic topology and knot theory but I have read fundamental groups myself . If possible please explain through figure.

Any type of help will be appreciated.

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    $\begingroup$ Maybe helpful: www.ams.org/publications/journals/notices/201604/rnoti-p378.pdf $\endgroup$ – user940 Apr 10 '17 at 18:26
  • $\begingroup$ @byron : thanks for your efforts but the in above pdf the construction of knot quandle is not there . $\endgroup$ – Manpreet Singh Apr 13 '17 at 9:00
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First, $X-K$ is the knot complement, I will denote this $X(K)$. A loop $\gamma$ in $X(K)$ is a piece-wise-linear (or smooth, choose your approach) embedding of the circle $S^1$. Then, you can consider $\gamma$ in $X$, and $K \cup \gamma$ is now a two-component link in $X$.

The criteria is then that the linking number of $K$ and $\gamma$ is 1. In particular, if you look at the knot in $S^3$ (or $\mathbb{R}^3$) as a diagram in the plane, you can choose your loop to be based at your nose, then going down through the page, under one arc of the diagram, and then back up to your nose.

In the reference given by Byron*, the construction of the knot quandle is given by Figure 3 and the associated discussion. You need to be familiar with the Wirtinger presentation of the knot group to appreciate why Sam Nelson's presentation is equivalent to David Joyce's.

* Nelson, Sam, What is … a quandle?, Notices Am. Math. Soc. 63, No. 4, 378-380 (2016). ZBL1338.57001.

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