arXiv 1111.33332011-11-14math.GT

Minimal Degree Parameterizations of the Trefoil and Figure-Eight Knots

Samantha Pezzimenti

英文摘要：

This paper determines the minimal degree sequence for two compact rational knots, namely the trefoil and figure-eight knots. We find explicit projections with the minimal degree sequence of each knot. This is done by modifying a non-compact rational minimal-degree parameterization of the trefoil and figure-eight knots to make it compact. This research was conducted at Ramapo College of New Jersey during the summer of 2011 with Dr. Donovan McFeron.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

The Dehn surgery characterization of the trefoil and the figure eight knot

We give a Dehn surgery characterization of the trefoil and the figure eight knots. These results are gotten by combining surgery formulas in Heegaard Floer homology from an earlier paper with the characterization of these knots in terms of their knot Floer homology given in a recent paper of Ghiggini.

2006-04-04math.GTmath.SG

The colored HOMFLY-PT polynomials of the trefoil knot, the figure-eight knot and twist knots

We give a rigorous proof of the colored HOMFLY-PT polynomials of the trefoil knot, the figure-eight knot and twist knots. For the trefoil knot and the figure-eight knot, it is expressed by a single sum, and for a twist knot, it is expressed by a double sum.

2021-07-19hep-thmath.GTmath.QA

Geometric conemanifold structures on $\mathbb{T}_{p/q}$, the result of $p/q$ surgery in the left-handed trefoil knot $\mathbb{T}$

As an example of the transitions between some of the eight geometries of Thurston, investigated before, we study the geometries supported by the cone-manifolds obtained by surgery on the trefoil knot with singular set the core of the surgery. The geometric structures are explicitly constructed. The most interesting phenomenon is the transition from $SL(2,\mathbb{R})$-geometry to $S^{3}$-geometry through Nil-geometry. A plot of the different geometries is given, in the spirit of the analogous plot of Thurston for the geometries supported by surgeries on the figure-eight knot.

2014-11-11math.GT

Minimal Flat Knotted Ribbons

This paper gives mathematical models for flat knotted ribbons, and makes specific conjectures for the least length of ribbon (for a given width) needed to tie the trefoil knot and the figure eight knot. The first conjecture states that (for width one) the least length of ribbon needed to tie an open-ended trefoil knot is L = 4(F + 1)/Sqrt[2 + F] where F = (1 + Sqrt[5])/2 is the golden ratio. The second conjecture states that the least length of the figure eight knot (in the same sense) is 32/Sqrt[15].

2004-11-25math.GT

Differential expansion and rectangular HOMFLY for the figure eight knot

Differential expansion (DE) for a Wilson loop average in representation $R$ is built to respect degenerations of representations for small groups. At the same time it behaves nicely under some changes of the loop, e.g. of some knots in the case of $3d$ Chern-Simons theory. Especially simple is the relation between the DE for the trefoil $3_1$ and for the figure eight knot $4_1$. Since arbitrary colored HOMFLY for the trefoil are known from the Rosso-Jones formula, it is therefore enough to find their DE in order to make a conjecture for the figure eight. We fulfil this program for all rectangu…

2016-06-05hep-thmath-phmath.GT

On homology 3-spheres defined by two knots

We show that if each of $K_1$ and $K_2$ is a trefoil knot or figure eight knot, the homology 3-sphere defined by the Kirby diagram which is a simple link of $K_1$ and $K_2$ with framing (0, n) is represented by an n-twisted Whitehead double of $K_2$ .

2014-10-28math.GT

更多相关论文收起更多论文

Knots with small lattice stick numbers

The lattice stick number of a knot type is defined to be the minimal number of straight line segments required to construct a polygon presentation of the knot type in the cubic lattice. In this paper, we mathematically prove that the trefoil knot $3_1$ and the figure-8 knot $4_1$ are the only knot types of lattice stick number less than 15, which verifies the result from previous numerical estimations on this quantity.

2015-12-11math.GT

On 3-super bridge knots

It is known that there are only finitely many knots with super bridge index 3. Jin and Jeon have provided a list of possible such candidates. However, they conjectured that the only knots with super bridge index 3 are trefoil and the figure eight knot. In this paper, we prove that the $5_2$ knot and the $6_2$ knot are also 3-super bridge knots by providing a polynomial representation of these knots in degree $6.$ This also answers a question asked by Durfee and O'Shea in their paper on polynomial knots: is there any 5-crossing knot in degree 6?

2012-09-14math.GT

Rectangular superpolynomials for the figure-eight knot

We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot $4_1$ in arbitrary rectangular representation $R=[r^s]$ as a sum over all Young sub-diagrams $λ$ of $R$ with extraordinary simple coefficients $D_{λ^{tr}}(r)\cdot D_λ(s)$ in front of the $Z$-factors. Somewhat miraculously, these coefficients are made from quantum dimensions of symmetric representations of the groups $SL(r)$ and $SL(s)$ and restrict summation to diagrams with no more than $s$ rows and $r$ columns. They possess a natural $β$-deformation to Macdonald dimensions and produces positive…

2016-09-10hep-thmath-phmath.GT

结在最小亏格Seifert面上的托管与友谊

Amicable Knots on Minimal Genus Seifert Surfaces

Komazawa University（驹泽大学）

研究结在最小亏格Seifert面上的托管关系及其友谊，揭示了结的托管现象和友谊现象的框架，指出不存在普遍托管的单个结。

2026-08-05math.GT

Dunfield-Gukov-Rasmussen 猜想的一个证明

A proof of Dunfield-Gukov-Rasmussen Conjecture

本文通过构造从 gl_0 同调到纽结 Floer 同调的 Bockstein 型双分次谱序列，证明了 Dunfield-Gukov-Rasmussen 猜想，并推出相关同调可识别若干经典纽结。

2025-01-17math.ATmath.GT