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6 Structure and Operations

In[1]

In[2]:= ?Crossings
Crossings[L] returns the number of crossings of a knot/link L (in its given presentation).

In[3]:= ?PositiveCrossings
PositiveCrossings[L] returns the number of positive (right handed) crossings in a knot/link L (in its given presentation).

In[4]:= ?NegativeCrossings
NegativeCrossings[L] returns the number of negaitve (left handed) crossings in a knot/link L (in its given presentation).

Thus here's one tautology and one easy example: 

In[5]:=  
Crossings /@ {Knot[0, 1], TorusKnot[11,10]}
Out[5]=
{0, 99}

And another easy example: 
In[6]:=  
K=Knot[6, 2]; {PositiveCrossings[K], NegativeCrossings[K]}
Out[6]=
{2, 4}

In[7]:= ?PositiveQ
PositiveQ[xing] returns True if xing is a positive (right handed) crossing and False if it is negative (left handed).

In[8]:= ?NegativeQ
NegativeQ[xing] returns True if xing is a negative (left handed) crossing and False if it is positive (right handed).

For example, 

In[9]:=  
PositiveQ /@ {X[1,3,2,4], X[1,4,2,3], Xp[1,3,2,4], Xp[1,4,2,3]}
Out[9]=
{False, True, True, True}

In[10]:= ?ConnectedSum
ConnectedSum[K1, K2] represents the connected sum of the knots K1 and K2 (ConnectedSum may not work with links).

The connected sum $ K=4_1\char93 4_1$ of the knot $ 4_1$ with itself has 8 crossings (unsurprisingly): 

In[11]:=  
K = ConnectedSum[Knot[4,1], Knot[4,1]]
Out[11]=
ConnectedSum[Knot[4, 1], Knot[4, 1]]
In[12]:=  
Crossings[K]
Out[12]=
8

It is also nice to know that, as expected, the Jones polynomial of $ K$ is the square of the Jones polynomial of $ 4_1$: 

In[13]:=  
Jones[K][q] == Expand[Jones[Knot[4,1]][q]^2]
Out[13]=
True

It is less nice to know that the Jones polynomial cannot tell $ K$ apart from the knot $ 8_9$: 

In[14]:=  
Jones[K][q] == Jones[Knot[8,9]][q]
Out[14]=
True

But $ 4_1\char93 4_1$ isn't equivalent to $ 8_9$; indeed, their Alexander polynomials are different: 

In[15]:=  
{Alexander[K][t], Alexander[Knot[8,9]][t]}
Out[15]=
       -2   6          2       -3   3    5            2    3
{11 + t   - - - 6 t + t , 7 - t   + -- - - - 5 t + 3 t  - t }
            t                        2   t
                                    t


Dror Bar-Natan 2005-09-14