(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 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 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 apart from the knot 8_9:
In[14]:=

Jones[K][q] == Jones[Knot[8,9]][q]

Out[14]=

True

But 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
