Arc Presentations

An Arc Presentation ${\displaystyle A}$ of a knot ${\displaystyle K}$ (in "grid form", to be precise) is a planar (toroidal, to be precise) picture of the knot in which all arcs are either horizontal or vertical, in which the vertical arcs are always "above" the horizontal arcs, and in which no two horizontal arcs have the same ${\displaystyle y}$-coordinate and no two vertical arcs have the same ${\displaystyle x}$-coordinate (read more at [1]). Without loss of generality, the ${\displaystyle x}$-coordinates of the vertical arcs in ${\displaystyle A}$ are the integers ${\displaystyle 1}$ through ${\displaystyle n}$ for some ${\displaystyle n}$, and the ${\displaystyle y}$-coordinates of the horizontal arcs in ${\displaystyle A}$ are (also!) the integers ${\displaystyle 1}$ through ${\displaystyle n}$.

((5,2), (1,3), (2,4), (3,5), (4,1))

Thus for example, on the left is an arc presentation ${\displaystyle A}$ of the trefoil knot. It can be represented numerically by the sequence of ordered pairs shown below it. This sequence reads: the lowest horizontal arc in ${\displaystyle A}$ connects the 5th vertical arc with the 2nd; the next horizontal arc in ${\displaystyle A}$ connects the 1st vertical with the 3rd, and so on. In general, an arc presentation involving ${\displaystyle n}$ horizontal and ${\displaystyle n}$ vertical arcs will be described in this way by a sequence of ${\displaystyle n}$ ordered pairs of integers in the range between ${\displaystyle 1}$ and ${\displaystyle n}$.

Arc presentations are used extensively in the computation of Heegaard Floer Knot Homologies.

KnotTheory knows about arc presentations:

(For In[1] see Setup)

 In[1]:= ?ArcPresentation ArcPresentation[{a1,b1}, {a2, b2}, ..., {an,bn}] is an arc presentation of a knot (as often used in the realm of Heegaard-Floer homologies), where the horizontal arc at row i connects column ai to column bi. ArcPresentation[K] returns an arc presentation of the knot K. ArcPresentation[K, Reduce -> r] attemps at most r reduction steps (using a naive reduction algorithm) following a naive creation of some arc presentation for K.
 In[2]:= ap = ArcPresentation["K11n11"] Out[2]= ArcPresentation[{12, 2}, {1, 10}, {3, 9}, {5, 11}, {9, 12}, {4, 8}, {2, 5}, {11, 7}, {8, 6}, {7, 4}, {10, 3}, {6, 1}]
 In[4]:= Draw[ap] Out[4]= -Graphics-
 In[5]:= ap0 = ArcPresentation["K11n11", Reduce -> 0] Out[5]= ArcPresentation[{13, 19}, {20, 23}, {19, 22}, {15, 14}, {14, 2}, {1, 13}, {3, 12}, {2, 4}, {16, 18}, {17, 15}, {8, 16}, {12, 17}, {5, 7}, {4, 6}, {7, 11}, {6, 8}, {18, 10}, {11, 9}, {10, 21}, {9, 20}, {21, 5}, {22, 3}, {23, 1}]
 In[6]:= ?Draw Draw[ap] draws the Arc Presentation ap. Draw[ap, OverlayMatrix -> M] overlays the matrix M on top of that draw.
 In[8]:= Draw[ap0] Out[8]= -Graphics-
 In[9]:= Reflect[ap_ArcPresentation] := ArcPresentation @@ ( (Last /@ Sort[Reverse /@ Position[ap, #]]) & /@ Range[Length[ap]] )
 In[11]:= Reflect[ap] // Draw Out[11]= -Graphics-

The Minesweeper Matrix ${\displaystyle M_{A}}$ (name not generally accepted) of an arc presentation ${\displaystyle A}$ of ${\displaystyle n}$ rows/columns is the ${\displaystyle n\times n}$ matrix whose ${\displaystyle (ij)}$ entry is the rotation number of ${\displaystyle A}$ around a point placed between the ${\displaystyle i}$ and ${\displaystyle i+1}$ rows of ${\displaystyle A}$ and between the ${\displaystyle j}$ and ${\displaystyle j+1}$ column of ${\displaystyle A}$. Here's a little program to compute the minesweeper matrix of a given arc presentation, along with its output on the arc presentation of K11n11 that we have been studying above:

 In[12]:= MinesweeperMatrix[ap_ArcPresentation] := Module[ {l, CurrentRow, c1, c2, k, s}, l = Length[ap]; CurrentRow = Table[0, {l}]; Table[ {c1, c2} = Sort[ap[[k]]]; s = Sign[{-1, 1}.ap[[k]]]; Do[ CurrentRow[[c]] += s, {c, c1, c2 - 1} ]; CurrentRow, {k, l} ] ]; 
 In[14]:= Draw[ap, OverlayMatrix -> MinesweeperMatrix[ap]] Out[14]= -Graphics-

If ${\displaystyle M_{A}=(m_{ij})}$, it is known that the determinant of the matrix ${\displaystyle (t^{m_{ij}})}$ is the Alexander polynomial of the knot presented by ${\displaystyle A}$, up to signs and powers of ${\displaystyle t}$ and ${\displaystyle (t-1)}$. Let us check this in our case:

 In[15]:= {Det[t^MinesweeperMatrix[ap]], Alexander[ap][t]} // Factor Out[15]=  11 2 2 3 4 5 6 {(-1 + t) t (1 - 5 t + 13 t - 17 t + 13 t - 5 t + t ), 2 3 4 5 6 1 - 5 t + 13 t - 17 t + 13 t - 5 t + t -------------------------------------------} 3 t`