Twist knot (math)

In the mathematical field of knot theory , a twist knot is a knot created by repeatedly twisting an unknot . For every number of half-twists there is a twist knot . The twist nodes thus form an infinite family of nodes. In addition to the torus nodes , the twist nodes are considered to be the simplest family of nodes. ${\displaystyle n}$ ${\displaystyle T_{n}}$

• 

  A half-twist
  ( shamrock loop )

• 

  Two half-twists
  ( figure eight knots )

• 

  Three half-twists
  ( 5 ₂ knots )

• 

  Four Half Twists
  ( Stevedore Knots )

• 

  Five half twists
  (7 ₂ knots)

• 

  Six half-twists
  (8 ₁ knots)

So twist knots are the whitehead doubles of the unknot.

properties

The stevedore knot is made from an unknot with four half-twists by intertwining the two ends.

All twist knots have an untying number because the knot can be untied by untying the two ends (as in the picture on the right). ${\displaystyle u(K)=1}$

Twist knots are special 2-bridge knots .

With the exception of the shamrock loop, all twist knots are hyperbolic .

Only the unknot and the stevedore knot are disc knots .

The intersection number of the twist knot is . ${\displaystyle T_{n}}$ ${\displaystyle n+2}$

All twist nodes can be inverted .

Only the unknot and the figure eight knot are amphichiral .

The node group of has the presentation with . ${\displaystyle T_{n}}$ ${\displaystyle \langle a,b\mid aw^{n}=w^{n}b\rangle }$ ${\displaystyle w=(ba^{-1}b^{-1}a)^{-1}}$

Invariants

The Alexander polynomial of the twist knot is ${\displaystyle T_{n}}$

${\displaystyle \Delta (t)={\begin{cases}{\frac {n+1}{2}}t-n+{\frac {n+1}{2}}t^{-1}&{\text{wenn }}n{\text{ ungerade ist}}\\-{\frac {n}{2}}t+(n+1)-{\frac {n}{2}}t^{-1}&{\text{wenn }}n{\text{ gerade ist,}}\\\end{cases}}}$

and the Conway polynomial is

${\displaystyle \nabla (z)={\begin{cases}{\frac {n+1}{2}}z^{2}+1&{\text{wenn }}n{\text{ ungerade ist}}\\1-{\frac {n}{2}}z^{2}&{\text{wenn }}n{\text{ gerade ist.}}\\\end{cases}}}$

For odd , the Jones polynomial is ${\displaystyle n}$

${\displaystyle V(q)={\frac {1+q^{-2}+q^{-n}-q^{-n-3}}{q+1}},}$

and for just it is ${\displaystyle n}$

${\displaystyle V(q)={\frac {q^{3}+q-q^{3-n}+q^{-n}}{q+1}}.}$

literature

Dale Rolfsen : Knots and links. Corrected reprint of the 1976 original. Mathematics Lecture Series, 7th Publish or Perish, Inc., Houston, TX, 1990. ISBN 0-914098-16-0

Web links

Twist Knot (MathWorld)