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Gyo Taek Jin

Publications and source records attributed to Gyo Taek Jin.

16 recordsLinked to original sources
arXiv2025

Minimal grid diagrams of the prime knots with crossing number 14 and arc index 13, 14

There are 46,972 prime knots with crossing number 14. Among them 19,536 are alternating and have arc index 16. Among the non-alternating knots, 17, 477, and 3,180 have arc index 10, 11, and 12, respectively. The remaining 23,762 have arc index 13 or 14. There are none with arc index smaller than 10 or larger than 14. We obtained 8,027 knots having arc index 13 and 15,735 knots having arc index 14. We show them by their minimal grid diagrams.

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arXiv2024

Minimal grid diagrams of the prime alternating knots with 13 crossings

A knot is a closed loop in space without self-intersection. Two knots are equivalent if there is a self homeomorphism of space bringing one onto the other. An arc presentation is an embedding of a knot in the union of finitely many half planes with a common boundary line such that each half plane contains a simple arc of the knot. The minimal number of such half planes among all arc presentations of a given knot is called the arc index of the knot. A knot is usually presented as a planar diagram with finitely many crossings of two strands where one of the strands goes over the other. A grid diagram is a planar diagram which is a non-simple rectilinear polygon such that vertical edges always cross over horizontal edges at all crossings. It is easily seen that an arc presentation gives rise to a grid diagram and vice versa. It is known that the arc index of an alternating knot is two plus its minimal crossing number. There are 4878 prime alternating knots with minimal crossing number 13. We obtained minimal arc presentations of them in the form of grid diagrams having 15 vertical segments. This is a continuation of the works on prime alternating knots of 11 crossings and 12 crossings.

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arXiv2024

Minimal grid diagrams of the prime knots with crossing number 13 and arc index 13

We give a list of minimal grid diagrams of the 13 crossing prime nonalternating knots which have arc index 13. There are 9,988 prime knots with crossing number 13. Among them 4,878 are alternating and have arc index 15. Among the other nonalternating knots, 49, 399, 1,412 and 3,250 have arc index 10, 11, 12, and 13, respectively. We used the Dowker-Thistlethwaite code of the 3,250 knots provided by the program Knotscape to generate spanning trees of the corresponding knot diagrams to obtain minimal arc presentations in the form of grid diagrams.

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arXiv2020

Prime knots with arc index 12 up to 16 crossings

As a continuation of the previous works to tabulate the prime knots up to arc index 11, we provide the list of prime knots with arc index 12 up to 16 crossings and their minimal grid diagrams. There are 19,513 prime knots of arc index 12 up to 16 crossings.

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arXiv2017

Mutation invariance of the arc index for some Montesinos knots

For the alternating knots or links, mutations do not change the arc index. In the case of nonalternating knots, some semi-alternating knots or links have this property. We mainly focus on the problem of mutation invariance of the arc index for nonalternating knots which are not semi-alternating. In this paper, we found families of infinitely many mutant pairs/triples of Montesinos knots with the same arc index.

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arXiv2014

Arc index of pretzel knots of type $(-p,q,r)$

We computed the arc index for some of the pretzel knots $K=P(-p,q,r)$ with $p,q,r\ge2$, $r\geq q$ and at most one of $p,q,r$ is even. If $q=2$, then the arc index $α(K)$ equals the minimal crossing number $c(K)$. If $p\ge3$ and $q=3$, then $α(K)=c(K)-1$. If $p\ge5$ and $q=4$, then $α(K)=c(K)-2$.

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arXiv2011

Prime knots whose arc index is smaller than the crossing number

It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of prime nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is strictly less than the minimal crossing number. We also give minimal grid diagrams of some prime nonalternating knots with 13 crossings and 14 crossings whose arc index is the minimal crossing number minus one.

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arXiv2010

A tabulation of prime knots up to arc index 11

As a supplement to the authors' article "Prime knots with arc index up to 11 and an upper bound of arc index for non-alternating knots", to appear in the Journal of Knot Theory and its Ramifications, we present minimal arc presentations of the prime knots up to arc index 11.

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arXiv2010

Quadrisecant approximation of hexagonal trefoil knot

It is known that every nontrivial knot has at least two quadrisecants. Given a knot, we mark each intersection point of each of its quadrisecants. Replacing each subarc between two nearby marked points with a straight line segment joining them, we obtain a polygonal closed curve which we will call the quadrisecant approximation of the given knot. We show that for any hexagonal trefoil knot, there are only three quadrisecants, and the resulting quadrisecant approximation has the same knot type.

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arXiv2010

Prime knots with arc index up to 11 and an upper bound of arc index for non-alternating knots

Every knot can be embedded in the union of finitely many half planes with a common boundary line in such a way that the portion of the knot in each half plane is a properly embedded arc. The minimal number of such half planes is called the arc index of the knot. We have identified all prime knots with arc index up to 11. We also proved that the crossing number is an upperbound of arc index for non-alternating knots. As a result the arc index is determined for prime knots up to twelve crossings.

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arXiv2001

Superbridge index of composite knots

An upper bound of the superbridge index of the connected sum of two knots is given in terms of the braid index of the summands. Using this upper bound and minimal polygonal presentations, we give an upper bound in terms of the superbridge index and the bridge index of the summands when they are torus knots. In contrast to the fact that the difference between the sum of bridge indices of two knots and the bridge index of their connected sum is always one, the corresponding difference for the superbridge index can be arbitrarily large.

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arXiv2001

Theta-curve polynomials and finite-type invariants

The normalized Yamada polynomial is a polynomial invariant in variable A for theta-curves. In this work, we show that the coefficients of the power series obtained from this polynomial by the substitution A=e^x=1+x+x^2/2+x^3/6+... are finite-type invariants for theta-curves although the coefficients of original polynomial are not. A similar result can be obtained in the case of Yokota polynomial for theta-curves.

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arXiv2000

P^2-reducing and toroidal Dehn fillings

We study the situation where we have two exceptional Dehn fillings on a given hyperbolic 3-manifold. We consider two cases that one filling creates a projective plane, and the other creates an essential torus or a Klein bottle, and give the best possible upper bound on the distance between two fillings for each case.

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