# Structure and Operations

(For In[1] see Setup)

 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 negative (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 $K=4_1\#4_1$ 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`
 4_1 8_9

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 $K=4_1\#4_1$ 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```