Slice knot

A slice knot is a type of mathematical knot. A slice knot is a type of mathematical knot. In knot theory, a 'knot' means an embedded circle in the 3-sphere and that the 3-sphere can be thought of as the boundary of the four-dimensional ball A knot K ⊂ S 3 {displaystyle Ksubset S^{3}} is slice if it bounds a nicely embedded 2-dimensional disk D in the 4-ball. What is meant by 'nicely embedded' depends on the context, and there are different terms for different kinds of slice knots. If D is smoothly embedded in B4, then K is said to be smoothly slice. If D is only locally flat (which is weaker), then K is said to be topologically slice. The following is a list of all slice knots with 10 or fewer crossings; it was compiled using the Knot Atlas:61, 8 8 {displaystyle 8_{8}} , 8 9 {displaystyle 8_{9}} , 8 20 {displaystyle 8_{20}} , 9 27 {displaystyle 9_{27}} , 9 41 {displaystyle 9_{41}} , 9 46 {displaystyle 9_{46}} , 10 3 {displaystyle 10_{3}} , 10 22 {displaystyle 10_{22}} , 10 35 {displaystyle 10_{35}} , 10 42 {displaystyle 10_{42}} , 10 48 {displaystyle 10_{48}} , 10 75 {displaystyle 10_{75}} , 10 87 {displaystyle 10_{87}} , 10 99 {displaystyle 10_{99}} , 10 123 {displaystyle 10_{123}} , 10 129 {displaystyle 10_{129}} , 10 137 {displaystyle 10_{137}} , 10 140 {displaystyle 10_{140}} , 10 153 {displaystyle 10_{153}} and 10 155 {displaystyle 10_{155}} . Every ribbon knot is smoothly slice.An old question of Fox asks whether every slice knot is actually a ribbon knot. The signature of a slice knot is zero. The Alexander polynomial of a slice knot factors as a product f ( t ) f ( t − 1 ) {displaystyle f(t)f(t^{-1})} where f ( t ) {displaystyle f(t)} is some integral Laurent polynomial. This is known as the Fox–Milnor condition.