# Nakanishi Index

The Alexander module of a knot is the first homology of its infinite cyclic cover,
viewed as a **Z**[t,t^{-1}] module. The Nakanishi Index is the minimal m such that
this module is presented by an m by m matrix. (One presentation is given by v - tV^{t},
where V is a Seifert matrix, but this might not be the minimal size square presentation.)

The values given here are taken from the table in Kawauchi's book on knot theory
and were originally calculated by Nakanishi. Three values (for knots of fewer than 10 crossings) were unknown
and were since computed to all be 2 in [1].

Many values of the Nakanishi Index for 11 and 12 crossing knots were computed with Knotorious [2]. Felipe Castellano-Macia and Nicholas Owad [3] computed another 82 values that had not be determined by Knotorious.

Ana Wright used computations of the algebraic unknotting number from Knottorious to derive upper bounds on the Nakanishi index. The replaced the value [1,2] with the value 1 for 77 knots.

## Specific Knots

9_{38}, 10_{69}, 10_{101}

Ref. [1]

10_{65}

Killian O'Brian informed us that the correct value is 2, not 1 as listed in earlier tables.

## References

[1] Kearton, C. and Wilson, S. M. J., "Knot modules and the Nakanishi index," Proc. Amer. Math. Soc. **131** no. 2
(2003), 655-663.

[2] Borodzik, M. and Friedl, S., "Knotorious"
World Wide Web page.

[3] Castellano-Macias, F. and Owad, N., "The tunnel number of all 11 and 12 crossing alternating knots" Arxiv preprint.