The Homfly polynomial of double crossover links

Abstract

Motivated by double crossover DNA polyhedra (He et al. in Nature 452:198, 2008; Lin et al. in Biochemistry 48:1663, 2009; Zhang et al. in J Am Chem Soc 131:1413, 2009; Zhang et al. in Proc Natl Acad Sci USA 105:10665, 2008; He et al. in Angew Chem Int Ed 49:748, )2010, in this paper, we construct a new type of link, called the double crossover link, formed by utilizing the “$n$-point star” to cover each vertex of a connected graph $G$. The double crossover link can be used to characterize the topological properties of double crossover DNA polyhedra. We show that the Homfly polynomial of the double crossover link can be obtained from the chain polynomial of the truncated graph of $G$ with two distinct labels. As an application, by using computer algebra (Maple) techniques, the Homfly polynomial of a double crossover tetrahedral link is obtained. Our result may be used to characterize and analyze the topological structure of DNA polyhedra.

Appendix

Let $GG$ be a graph, $MM$ the labeled edge matrix of $GG,\, ww$ algebraic number, so the program of the labled graph $GG$ is written as follows, and the content in “[ ]” explains the program.

• ChainPolynomial := proc ($GG$::GRAPHLN, $MM$::Matrix, $ww$::algebraic)    [make a function in order transfer the datas]

• local $G,\, chp,\, AAM,\, m$;

• global $w$;

• $G$:= ‘ if ‘ (op(2, $GG$) = ’unweighted’, MakeWeighted(GG), $GG$);    [if $GG$ is an unweighted graph, then can let $GG$ be weighted graph denoted by $G$]

• $AAM$ := $MM$;

• $w$ := $ww$;

• $m$ := nops(op(3, $G$));    [$m$ equals the vertex number of the graph $G$]

• $chp$ := proc ($G$, AAM)    [make a function in order to compulate the chain polynomial of $G$]

• local $i, j, k, ii, jj, e, ee, E, LL, CCM, DDM, n, x, y, z, GDel, GCon, chpp, GGC, GGD$;

• option remember;[record the program]

• $GGC$ := $G$;

• $GGD$ := $G$;

• $E$ := Edges($G$, weights);    [$E$ is a list of the weighted edges]

• if $nops(E) = 0$ then return 1 end if;[if $G$ is a null graph, return 1]

• $e {:=} E[1]$;    [$e$ denotes the first edge of $E$]

• $ee {:=} e[1]$;    [$ee$ are related vertics of $e$]

• $ii {:=} op(1, sort(ee))$;    [$ii$ is the first vertex of $e$]

• $jj {:=} op(2, sort(ee))$;    [$jj$ is the first vertex of $e$]

• [the following is the recursive of chain polynomial,where parallel edges are deleted and contracted at a time]

• $LL {:=} AAM[ii, jj];$

• $n {:=} nops(LL);$

• $x {:=} 1$;

• for $i$ to $n$ do

• $x {:=} x*(LL[i]-1)$

• end do;

• $y {:=} 0$;

• for $k$ to $n$ do $z {:=} 1;$

• for $i$ to $k-1$ do

• $z {:=} z*(LL[i]-w)$

• end do;

• for $i$ from $k+1$ to $n$ do

• $z {:=} z*(LL[i]-1)$

• end do;

• $y {:=} y+z$ end do;

• $GCon$ := Contract($GGC,\, ee$, multi = true);

• $GDel$ := DeleteEdge($GGD,\, ee$);

• $DDM$ := matrix($m,\, m$);

• $CCM$:= matrix($m,\, m$);

• for $i$ to $m$ do for $j$ to $m$ do

• if $i = ii$ and $j = jj$ then

• $DDM[i, j] {:=} [~]$

• elif $i = jj$ and $j = ii$

• then $DDM[j, i] {:=} [~]$

• else $DDM[i, j] {:=} AAM[i, j]$

• end if

• end do

• end do;

• for $i$ to $m$ do

• for $j$ to $m$ do

• if $i\ne ii$ and $i\ne jj$ and $j\ne ii$ and $j\ne jj$

• then $CCM[i, j] {:=} AAM[i, j]$

• elif $i = ii$ and $j\ne ii$ and$j\ne jj$

• then $CCM[i, j] {:=} [op(AAM[ii, j]), op(AAM[jj, j])]$

• elif $j = ii$ and $i\ne ii$ and $j\ne jj$

• then $CCM[i, j] {:=} [op(AAM[i, ii]), op(AAM[i, jj])]$

• else $CCM[i, j] {:=} [~]$

• end if

• end do

• end do;

• $chpp$ := $x*$procname($GDel, DDM$)+$y*$procname($GCon, CCM$);

• simplify(chpp)

• end proc;

• chp($G,\, AAM$)

• end proc