(For In[1] see Setup)
In[1]:=
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?Crossings
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Crossings[L] returns the number of crossings of a knot/link L (in its given presentation).
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In[2]:=
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?PositiveCrossings
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PositiveCrossings[L] returns the number of positive (right handed) crossings in a knot/link L (in its given presentation).
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In[3]:=
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?NegativeCrossings
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NegativeCrossings[L] returns the number of negaitve (left handed) crossings in a knot/link L (in its given presentation).
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Thus here's one tautology and one easy example:
In[4]:=
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Crossings /@ {Knot[0, 1], TorusKnot[11,10]}
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Out[4]=
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{0, 99}
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And another easy example:
In[5]:=
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K=Knot[6, 2]; {PositiveCrossings[K], NegativeCrossings[K]}
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Out[5]=
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{2, 4}
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In[6]:=
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?PositiveQ
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PositiveQ[xing] returns True if xing is a positive (right handed) crossing and False if it is negative (left handed).
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In[7]:=
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?NegativeQ
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NegativeQ[xing] returns True if xing is a negative (left handed) crossing and False if it is positive (right handed).
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For example,
In[8]:=
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PositiveQ /@ {X[1,3,2,4], X[1,4,2,3], Xp[1,3,2,4], Xp[1,4,2,3]}
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Out[8]=
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{False, True, True, True}
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In[9]:=
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?ConnectedSum
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ConnectedSum[K1, K2] represents the connected sum of the knots K1 and K2 (ConnectedSum may not work with links).
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The connected sum of the knot 4_1 with itself has 8 crossings (unsurprisingly):
In[10]:=
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K = ConnectedSum[Knot[4,1], Knot[4,1]]
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Out[10]=
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ConnectedSum[Knot[4, 1], Knot[4, 1]]
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In[11]:=
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Crossings[K]
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Out[11]=
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8
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It is also nice to know that, as expected, the Jones polynomial of is the square of the Jones polynomial of 4_1:
In[12]:=
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Jones[K][q] == Expand[Jones[Knot[4,1]][q]^2]
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Out[12]=
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True
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It is less nice to know that the Jones polynomial cannot tell apart from the knot 8_9:
In[13]:=
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Jones[K][q] == Jones[Knot[8,9]][q]
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Out[13]=
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True
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But isn't equivalent to 8_9; indeed, their Alexander polynomials are different:
In[14]:=
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{Alexander[K][t], Alexander[Knot[8,9]][t]}
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Out[14]=
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-2 6 2 -3 3 5 2 3
{11 + t - - - 6 t + t , 7 - t + -- - - - 5 t + 3 t - t }
t 2 t
t
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