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The dynamics of two slender Hopf-linked vortex rings at vortex Reynolds numbers (
$Re \equiv \varGamma /\nu, \mathrm {circulation/viscosity}$) 
$2000$, 
$3000$ and 
$4000$ is studied using direct numerical simulations of the incompressible Navier–Stokes equations. Under self-induction, the initially perpendicularly placed vortex rings approach each other and reconnect to form two separate vortex rings. The leading ring is closely cuddled and further undergoes secondary reconnection to form two even smaller rings. At high 
$Re$, the leading ring and the subsequent smaller rings are unstable and break up into turbulent clouds consisting of numerous even smaller-scale structures. Although the global helicity 
$H$ remains constant before reconnection, it increases and then rapidly decays during reconnection – both the growth and decay rates increase with 
$Re$. In the two higher 
$Re$ (i.e. 3000 and 4000) cases, 
$H$ further rises after the first reconnection and reaches a quasi-plateau with the asymptotic value continuously increasing with 
$Re$ – suggesting that 
$H$ for viscous flows is not conserved at very high 
$Re$. Further flow analysis demonstrates that significant numbers of positive and negative helical structures are simultaneously generated before and during reconnection, and their different decay rates is the main reason for the complex evolution of 
$H$. By examining the topological aspects of the helicity dynamics, we find that, different from 
$H$, the sum of link and writhe (
$L_k+W_r$) continuously drop during reconnection. Our results also clearly demonstrate that the twist, which increases with 
$Re$, plays a significant role in the helicity dynamics, particularly at high 
$Re$.
A slender trefoil knotted vortex is studied using direct numerical simulation of the Navier–Stokes equations for vortex Reynolds numbers (
$Re \equiv \varGamma /\nu$, circulation/viscosity) up to 12 000. For initially zero twist (
$T_{w,0}=0$), neither the writhe 
$W_r$ nor the global helicity 
$H$ is conserved. Initially 
$W_r$ slowly decreases, then suddenly drops during reconnection and becomes almost constant thence; its evolution is almost 
$Re$ independent. Before reconnection, 
$H$ also gradually decreases but sharply increases during reconnection. The evolution of 
$H$ after reconnection strongly depends on 
$Re$. While steadily decreasing at low 
$Re$, 
$H$ significantly increases before eventually decaying at high 
$Re$. Flow visualization, helicity decomposition and helical wave decomposition reveal that significant amounts of positive and negative twist helicities are simultaneously generated before and during reconnection. Also, the small leading and large trailing rings resulting from asymmetric reconnection have respectively negative and positive twists, which then decay differently due to different initial values, geometries and mutual induction. In particular, at high 
$Re$, the twist in the small ring, under stretching by the large trailing ring, decays much faster and even switches sign to become positive by the writhe-to-twist conversion – the main reason for the ‘transient growth’ of 
$H$. Simulations with non-zero initial twists (
$T_{w,0}=7.48$ and 
$22.48$) reveal that the overall dynamics is similar to the 
$T_{w,0}=0$ case. Hence, the evolution of the trefoil knotted vortex is mainly governed by 
$W_r$, not 
$T_w$, although the latter is found to play an essential role in enstrophy growth as well as energy cascade.
