Unknotting Number (KNOT THEORY)

Unknotting Number (knot theory)

Introduction To Unkotting Number

The Unknotting Number

Calculating the Unknotting Number (Trefoil)

Calculating the Unknotting Number ( Figure 8)

Sources

Conclusion

The Knot Book By: Colin C. Adams

The Unknotting Number

• The Number - The number included in the unknot tells us how many moves (crossings) it takes to untangle the knot, but calculating the number can be very difficult depending on the knot's complexity
• For simpler knots, it could be untangled with one change (crossing) at a time
• The Complexity - More complex knots require more advanced math tools such as Knot Invariants.
• Knot Invariants - Using Knot Invariants such as Alexander Polynomials, could help us estimate the unknotting number
• The advanced tools that we use for complex knots offer shortcuts when the number is becoming difficult to find

Introduction to Unknotting Number (Knot Theory) :

• Unknotting Theory - The fewest changes (crossings) needed to turn a knot into an unknot, with each change made and altered how the knot is projected.
• Equivalence to the Methods - Changing crossings in one projection or multiple often leads to the same result
• Examples of Unknotting Numbers - The trefoil and figure 8 knot has an unknotting number of 1.
• Context - the number helps us identify how many changes (crossings) are needed to untangle the projected knot
• Context - With the trefoil and figure 8 knots only having an unknotting number of 1, you only need 1 change (crossing) to untangle the knot.

Conclusion

The Unknotting Number serves as a helpful shortcut/tool when dealing with knot theory, showing how the complexity of a knot is by helping us count the least number of changes needed to untangle it. It's easier to figure out for more simple knots, but for more complex knots, you will require a more advanced math strategy (TOOL) such as the knot invariants to calculate it. Knot Theory and Unknotting Numbers could also help us in real-world predicaments such as abstract math. Understanding a concept such as the Unknotting Number can make the connection between math and real-world educational problems clearer and easier to grasp.

Sources

Calculating The Unkotting Number (Trefoil)

• Step 1 - First we Identify how many crossings it has, with that being said the trefoil knot has exactly 3 crossings
• Step 2 - Look for ways to simplify the knot by changing one of the crossings
• Step 3 - Flip one of the over-crossings over the under-crossing across from it to decrease the complexity of the knot (FIRST CHANGE)
• Step 4 - Flip the other crossing to completely untangle the knot
• Step 5 - Since we identified that only one change was needed to untangle the knot, the unknotting number for the Trefoil is 1

Calculating the Unknotting Number ( Figure 8)

• Step 1 - The Figure 8 Knot has exactly 4 crossings
• Step 2 - Start with changing a crossing that could make it slightly easier to simplify the knot
• Step 3 - Flip the first over-strand crossing over the under-strand to reduce the knot's complexity
• Step 4 - Flip the second crossing to continue the simplifying, bringing it closer to the unknot
• Step 5 - Since we flipped the crossings, we can identify that it took 2 crossings to untangle the knot. Therefore the unknotting number is 2