Intersection number (knot theory)

The number of crossings or crossings of a knot or a link is an elementary invariant from the mathematical field of knot theory .

Table of all prime nodes with number of intersections . (The mirror image is omitted for chiral knots .) ${\displaystyle cr(K)\leq 7}$

The number of crossings of a knot (or a link) is defined as the minimum number of crossings in a knot diagram representing the knot (or the link) . ${\displaystyle cr(K)}$ ${\displaystyle K}$ ${\displaystyle K}$

In node tables , nodes are usually arranged according to their intersection number and each node is designated by two numbers , where the intersection number is the node and the nodes are numbered with the same intersection number. For example, is trefoil knot the only node with crossing number 3 and the eighth node is the only node with crossing number 4. There are 1,701,936 prime knots with crossing number . ${\displaystyle p_{q}}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle 3_{1}}$ ${\displaystyle 4_{1}}$ ${\displaystyle cr(K)\leq 16}$

In general, the number of intersections of a node is difficult to calculate. But if you find a reduced alternating diagram for a node , then its number of intersections calculates the number of intersections . ${\displaystyle K}$ ${\displaystyle cr(K)}$

The crossing numbers of the following node classes are explicitly known:

• The number of intersections of the - torus knot with is ${\displaystyle (p,q)}$ ${\displaystyle p,q>0}$ ${\displaystyle \min((p-1)q,(q-1)p).}$
• The intersection number of the twist knot is . ${\displaystyle T_{n}}$ ${\displaystyle n+2}$
• The number of intersections of the 2-bridge node with twist parameters is . ${\displaystyle a_{1},\ldots ,a_{n}}$ ${\displaystyle a_{1}+\ldots +a_{n}}$

It is an open conjecture that the number of intersections is additive under the connected sum of nodes. The inequality is proven . ${\displaystyle {\frac {1}{152}}(cr(K_{1})+cr(K_{2}))\leq cr(K_{1}\sharp K_{2})\leq cr(K_{1})+cr(K_{2})}$

Individual evidence

1. ↑ Kunio Murasugi: Jones polynomials and classical conjectures in knot theory. Topology, 26 (1987) no. 2, 187-194.
2. ↑ This is the number of crossings in the standard diagram of the 2-bridge node, which is reduced and alternating and therefore calculates the number of crossings.
3. ↑ Marc Lackenby: The crossing number of composite knots. J. Topol. 2 (2009), no. 4, 747-768.