Ask Your Question
1

Knot Theory Gauss Code?

asked 2025-08-01 11:46:48 +0200

Marcus Collins gravatar image

updated 2025-08-02 19:16:10 +0200

Max Alekseyev gravatar image

.

from sage.all import * 
W = Knots() 
K1 = W.from_table(5, 2)
K2 = W.from_gauss_code([-1, 2, -3, 4, -5, 1, -2, 5, -4, 3])   
print(K1.is_isotopic(K2))   
print(K1.gauss_code(),   K2.gauss_code())

I have the following code. I'm troubled because K1 and K2 are the same knot, yet they return Gauss codes with different numbers of crossings. Specifically, K1 as 5_2 has 5 crossings, and since the Gauss code is just labelling crossings I don't see why there would possibly be any 6's in the output, yet K1.gauss_code() returns [-1, 2, -3, -4, 6, 1, -2, 3, 5, -6, 4, -5]. Thoughts? Am I misunderstanding something?

edit retag flag offensive close merge delete

1 Answer

Sort by ยป oldest newest most voted
0

answered 2025-08-07 15:43:41 +0200

rburing gravatar image

In the table at src/sage/knots/knot_table.py, knots are described as closures of braids. The (minimum) braid representative of 5_2 in the table has 3 strands and 6 crossings, which explains the output.

This could indeed be documented better, or the implementation of the table could be improved. It's related to the existing issue #31188: knot genus outputs wrong value for 5_2.

edit flag offensive delete link more

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

Stats

Asked: 2025-08-01 11:44:24 +0200

Seen: 10,677 times

Last updated: Aug 07