Special alternating links of minimal unlinking number

Abstract

For any link in the 3-sphere, there is a natural lower bound for the unlinking number in terms of the classical signature. We prove that if this lower bound is sharp for a special alternating link $L$, then the unlinking number of $L$ is necessarily realized by crossing changes in any alternating diagram for $L$. As an application, we compute new values of the unknotting numbers for some special alternating knots with crossing number 11 and 12.

Figures (4)

• Figure 1: The incidence number of a crossing.
• Figure 2: A diagram of the special alternating knot $8_{15}$, which has signature $-4$. The labellings in the white regions correspond to an embedding of the Goeritz form into $\mathbb{Z}^8$ satisfying the conclusions of Theorem \ref{['thm:lattice_obstruction']}. Since $p=2$, we have $v\cdot e_1 = v\cdot e_3$ and $v\cdot e_2 = v\cdot e_4$ for each $v$ in the image of $\Lambda_D$. Note that $8_{15}$ can be unknotted by changing crossings between the regions labelled by the vectors $e_1$ and $e_2$.
• Figure 3: A shaded positive crossing
• Figure 4: Changing a crossing in a clasp corresponds to the deletion of two bands from the white surface $S_-$.

Theorems & Definitions (13)

• Theorem 1
• Corollary 2
• Corollary 3
• Proposition 4
• proof
• Theorem 5
• proof
• Claim A
• proof : Proof of Claim
• Claim B