The concordance classification of low crossing number knots

Abstract

We present the complete classification of the subgroup of the classical knot concordance group generated by knots with eight or fewer crossings. Proofs are presented in summary. We also describe extensions of this work to the case of nine crossing knots.

Branched covers bounding rational homology balls

“…Considerably less seems to be known with regards to finite order elements in ker ϕ. Kirk and Livingston showed that the knot 8 17 , which is negative-amphichiral, is not concordant to its reverse; hence 8 17 #8 r 17 represents a nontrivial element of order two in ker ϕ [26]; see also, [7]. In the present article, we extend this result by showing that there exists a subgroup H of ker ϕ such that H is isomorphic to (Z 2 ) 5 ; see Theorem 1.2 below.…”

The nonorientable 4–genus for knots with 8 or 9 crossings

“…and use it to provide another proof of the unboundedness of γ 4 by demonstrating that γ 4 (# n T (3,4) ) ≥ n, where # n T (3,4) is the n-fold connected sum of the (3, 4) torus knot T (3,4) with itself. The converse inequality γ 4 (# n T (3,4) ) ≤ n is easy to verify by finding an explicit Möbius band bounded by T (3,4) , leading to γ 4 (# n T (3,4) ) = n. γ 4 (K) = 1 for K = 9 1 , 9 3 , 9 4 , 9 5 , 9 6 , 9 7 , 9 8 , 9 9 , 9 13 , 9 15 , 9 17 , 9 19 , 9 21 , 9 22 , 9 23 , 9 25 , 9 26 , 9 27 , 9 28 , 9 29 , 9 31 , 9 32 , 9 35 , 9 36 , 9 41 , 9 42 , 9 43 , 9 44 , 9 45 , 9 46 , , 9 47 , 9 48 . γ 4 (K) = 2 for K = 9 2 , 9 10 , 9 11 , 9 12 , 9 14 , 9 16 , 9 18 , 9 20 , 9 24 , 9 30 , 9 33 , 9 34 , 9 37 , 9 38 , 9 39 , 9 40 , 9 49 .…”

“…Note that g top 4 (K) ≤ g 4 (K) ≤ g 3 (K). In another direction, Clark [3] defined the non-orientable 3-genus or 3-dimensional crosscap number γ 3 (K) as the smallest first Betti number of any non-orientable surface Σ ⊂ S 3 with ∂Σ = K. The non-orientable (smooth) 4-genus or 4-dimensional crosscap number γ 4 (K) was defined by Murakami and Yasuhara [20] as the minimal first Betti number of any non-orientable surface Σ smoothly and properly embedded in D 4 and with ∂Σ = K. Some authors additionally define γ 4 (K) = 0 for any slice knot K, but in the interest of a more unifying treatment we adopt the definition from the previous sentence. Just as in the case of oriented surfaces, so too for non-orientable surfaces there is a topological version of this invariant denoted by γ top 4 (K).…”

“…Having been introduced relatively recently, the literature available on γ 4 is relatively sparse too. First results go back to work of Viro [30] who uses Witt classes of intersection forms of 4-manifolds to obstruct a knot K from bounding a smoothly and properly embedded Möbius band in D 4 . He uses his findings to demonstrate that γ 4 (4 1 ) > 1.…”

A note on the concordance of fibered knots

“…Hedden-Kirk-Livingston have shown that there are many pairs of tight fibered knots (sums of algebraic knots) that are algebraically concordant and yet not topologically locally-flat concordant [HKL12]. Meier also points out that there are many pairs of positive non-fibered knots that are algebraically concordant and yet not topologically locally-flat concordant [Mei14]; as reported by KnotInfo and its Concordance Calculator, the positive knots 3 1 #9 2 and 9 23 give an example [CL14,CKL13].…”