A General Method for Computing the Homfly Polynomial of DNA Double Crossover 3-Regular Links

<div><p>In the last 20 years or so, chemists and molecular biologists have synthesized some novel DNA polyhedra. Polyhedral links were introduced to model DNA polyhedra and study topological properties of DNA polyhedra. As a very powerful invariant of oriented links, the Homfly polynomial of some of such polyhedral links with small number of crossings has been obtained. However, it is a challenge to compute Homfly polynomials of polyhedral links with large number of crossings such as double crossover 3-regular links considered here. In this paper, a general method is given for computing the chain polynomial of the truncated cubic graph with two different labels from the chain polynomial of the original labeled cubic graph by substitutions. As a result, we can obtain the Homfly polynomial of the double crossover 3-regular link which has relatively large number of crossings.</p></div

The labeled hexahedral graph <i>H</i> and its truncation.

<p>The labeled hexahedral graph <i>H</i> and its truncation.</p

Right-handed (+) and left-handed crossings (−).

<p>Right-handed (+) and left-handed crossings (−).</p

The <i>Y</i> − △ transformation.

<p>The <i>Y</i> − △ transformation.</p

Assembly of DNA 4-turn hexahedra from two different component three-point-star tiles (A and B).

<p>Assembly of DNA 4-turn hexahedra from two different component three-point-star tiles (A and B).</p

The planar graph of a double crossover hexahedral link with 16 × 12 = 192 crossings.

<p>The planar graph of a double crossover hexahedral link with 16 × 12 = 192 crossings.</p

The construction of the labeled graph <i>G</i>* from the labeled graph <i>G</i>.

<p>The construction of the labeled graph <i>G</i>* from the labeled graph <i>G</i>.</p

The double crossover tangle <i>T</i>₁ (left) and the vertical integer tangle <i>T</i>₂ (right).

<p>The double crossover tangle <i>T</i>₁ (left) and the vertical integer tangle <i>T</i>₂ (right).</p