• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1305-1321
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• 2015

We dene new link invariants which are called the quasitoric braid index and the cyclic length of a link and show that the quasitoric braid index of link with k components is the product of k and the cycle length of link. Also, we give bounds of Gordian distance between the (p,q)-torus knot and the closure of a braid of two specific quasitoric braids which are called an alternating quasitoric braid and a blockwise alternating quasitoric braid. We give a method of modication which makes a quasitoric presentation from its braid presentation for a knot with braid index 3. By using a quasitoric presentation of $10_{139}$ and $10_{124}$, we can prove that $u(10_{139})=4$ and $d^{\times}(10_{124},K(3,13))=8$.

• Kyungpook Mathematical Journal
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• v.57 no.1
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• pp.145-161
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• 2017

If a virtual knot diagram can be transformed to another virtual one by a finite sequence of crossing changes, Reidemeister moves and virtual moves then the two virtual knot diagrams are said to be homotopic. There are infinitely many homotopy classes of virtual knot diagrams. We give necessary conditions by using polynomial invariants of virtual knots for two virtual knots to be homotopic. For a sequence S of crossing changes, Reidemeister moves and virtual moves between two homotopic virtual knot diagrams, we give a lower bound for the number of crossing changes in S by using the affine index polynomial introduced in [13]. In [10], the first author gave the q-polynomial of a virtual knot diagram to find Reidemeister moves of virtually isotopic virtual knot diagrams. We find how to apply Reidemeister moves by using the q-polynomial to show homotopy of two virtual knot diagrams.