Knots with propretyR+

If we consider the set of manifolds that can be obtained by surgery on a fixed knotK, then we have an associated set of numbers corresponding to the Heegaard genus of these manifolds. It is known that there is an upper bound to this set of numbers. A knotKis said to have PropertyR+if longitudinal surgery yields a manifold of highest possible Heegaard genus among those obtainable by surgery onK. In this paper we show that torus knots,2-bridge knots, and knots which are the connected sum of arbitrarily many(2,m)-torus knots have PropertyR+It is shown that ifKis constructed from the tangles(B1,t1),(B2,t2),…,(Bn,tn)thenT(K)≤1+∑i=1nT(Bi,ti)whereT(K)is the tunnel ofKandT(Bi,ti)is the tunnel number of the tangle(Bi,ti). We show that there exist prime knots of arbitrarily high tunnel number that have PropertyR+and that manifolds of arbitrarily high Heegaard genus can be obtained by surgery on prime knots.

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SUPERBRIDGE INDEX OF COMPOSITE KNOTS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216500000360 ◽

2000 ◽

Vol 09 (05) ◽

pp. 669-682

Author(s):

GYO TAEK JIN

Keyword(s):

Upper Bound ◽

Connected Sum ◽

Torus Knots ◽

Braid Index ◽

The Difference

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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The Heegaard genus of manifolds obtained by surgery on links and knots

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171280000440 ◽

1980 ◽

Vol 3 (3) ◽

pp. 583-589 ◽

Cited By ~ 9

Author(s):

Bradd Clark

Keyword(s):

Upper Bound ◽

Heegaard Genus ◽

Tunnel Number

LetL⊂S3be a fixed link. It is shown that there exists an upper bound on the Heegaard genus of any manifold obtained by surgery onL. The tunnel number ofL,T(L), is defined and used as an upper bound. IfK′is a double of the knotK, it is shown thatT(K′)≤T(K)+1. IfMis a manifold obtained by surgery on a cable link aboutKwhich hasncomponents, it is shown that the Heegaard genus ofMis at mostT(K)+n+1.

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Incoherent nullification of torus knots and links

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216519500330 ◽

2019 ◽

Vol 28 (05) ◽

pp. 1950033

Author(s):

Zac Bettersworth ◽

Claus Ernst

Keyword(s):

Upper Bound ◽

Torus Knots ◽

Number Formula ◽

Knots And Links

In the paper, we study the incoherent nullification number [Formula: see text] of knots and links. We establish an upper bound on the incoherent nullification number of torus knots and links and conjecture that this upper bound is the actual incoherent nullification number of this family. Finally, we establish the actual incoherent nullification number of particular subfamilies of torus knots and links.

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A SHARP UPPER BOUND FOR REGION UNKNOTTING NUMBER OF TORUS KNOTS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216513500193 ◽

2013 ◽

Vol 22 (05) ◽

pp. 1350019 ◽

Cited By ~ 1

Author(s):

SIWACH VIKASH ◽

MADETI PRABHAKAR

Keyword(s):

Large Class ◽

Upper Bound ◽

Torus Knots ◽

Torus Links

Region crossing change for a knot or a proper link is an unknotting operation. In this paper, we provide a sharp upper bound on the region unknotting number for a large class of torus knots and proper links. Also, we discuss conditions on torus links to be proper.

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AN APPLICATION OF TQFT: DETERMINING THE GIRTH OF A KNOT

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216508006774 ◽

2008 ◽

Vol 17 (12) ◽

pp. 1539-1547 ◽

Cited By ~ 1

Author(s):

LISA HERNÁNDEZ ◽

XIAO-SONG LIN

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Upper Bound ◽

Topological Quantum Field Theory ◽

Quantum Field ◽

Heegaard Genus ◽

Kauffman Bracket ◽

Branched Covering ◽

Topological Quantum ◽

Intersection Points

A knot diagram can be divided by a circle into two parts, such that each part can be coded by a planar tree with integer weights on its edges. A half of the number of intersection points of this circle with the knot diagram is called the girth. The girth of a knot is the minimal girth of all diagrams of this knot. The girth of a knot minus one is an upper bound of the Heegaard genus of the 2-fold branched covering of that knot. We will use Topological Quantum Field Theory (TQFT) coming from the Kauffman bracket to determine the girth of some knots. Consequently, our method can be used to determine the Heegaard genus of the 2-fold branched covering of some knots.

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THE ADDITIVITY OF CROSSING NUMBERS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216504003524 ◽

2004 ◽

Vol 13 (07) ◽

pp. 857-866 ◽

Cited By ~ 6

Author(s):

YUANAN DIAO

Keyword(s):

Knot Theory ◽

Difficult Problem ◽

Crossing Number ◽

The Other ◽

Connected Sum ◽

Torus Knots ◽

Crossing Numbers ◽

Weaker Conditions ◽

Alternating Links ◽

Alternating Link

It has long been conjectured that the crossing numbers of links are additive under the connected sum of links. This is a difficult problem in knot theory and has been open for more than 100 years. In fact, many questions of much weaker conditions are still open. For instance, it is not known whether Cr(K1#K2)≥Cr(K1) or Cr(K1#K2)≥Cr(K2) holds in general, here K1#K2 is the connected sum of K1 and K2 and Cr(K) stands for the crossing number of the link K. However, for alternating links K1 and K2, Cr(K1#K2)=Cr(K1)+Cr(K2) does hold. On the other hand, if K1 is an alternating link and K2 is any link, then we have Cr(K1#K2)≥Cr(K1). In this paper, we show that there exists a wide class of links over which the crossing number is additive under the connected sum operation. This class is different from the class of all alternating links. It includes all torus knots and many alternating links. Furthermore, if K1 is a connected sum of any given number of links from this class and K2 is a non-trivial knot, we prove that Cr(K1#K2)≥Cr(K1)+3.

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On composite types of tunnel number two knots

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216515500133 ◽

2015 ◽

Vol 24 (02) ◽

pp. 1550013 ◽

Cited By ~ 2

Author(s):

Kanji Morimoto

Keyword(s):

Point Of View ◽

Connected Sum ◽

Tunnel Number

Let K be a tunnel number two knot. Then, by considering the (g, b)-decompositions, K is one of (3, 0), (2, 1), (1, 2) or (0, 3)-knots. In this paper, we analyze the connected sum summands of composite tunnel number two knots and give a complete table of those summands from the point of view of (g, b)-decompositions.

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TWISTED TORUS KNOTS T(p, q; 3, s) ARE TUNNEL NUMBER ONE

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216511009121 ◽

2011 ◽

Vol 20 (06) ◽

pp. 807-811 ◽

Cited By ~ 3

Author(s):

JUNG HOON LEE

Keyword(s):

Torus Knots ◽

Tunnel Number ◽

Unknotting Tunnel

We show that twisted torus knots T(p, q; 3, s) are tunnel number one. A short spanning arc connecting two adjacent twisted strands is an unknotting tunnel.

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