Arc Presentations

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An Arc Presentation $A$ of a knot $K$ (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 alway "above" the horizontal arcs, and in which no two horizontal arcs have the same $y$-coordinate and no two vertical arcs have the same $x$-coordinate (read more at [1]). Without loss of generality, the $x$-coordinates of the vertical arcs in $A$ are the integers $1$ through $n$ for some $n$, and the $y$-coordinates of the horizontal arcs in $A$ are (also!) the integers $1$ through $n$.

[{5,2},{1,3},{2,4},{3,5},{4,1}]

Thus for example, on the left is an arc presentation of the trefoil knot. It can be represented numerically

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-

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-

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