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 .
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) .
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 .
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 .
The crossing numbers of the following node classes are explicitly known:
- The number of intersections of the - torus knot with is
- The intersection number of the twist knot is .
- The number of intersections of the 2-bridge node with twist parameters is .
It is an open conjecture that the number of intersections is additive under the connected sum of nodes. The inequality is proven .
Individual evidence
- ↑ Kunio Murasugi: Jones polynomials and classical conjectures in knot theory. Topology, 26 (1987) no. 2, 187-194.
- ↑ 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.
- ↑ Marc Lackenby: The crossing number of composite knots. J. Topol. 2 (2009), no. 4, 747-768.