Tutorial

Step-by-step tutorial introducing KnotPy, from installation to constructing knots, links, and computing invariants.

Basic operations

Create the (unoriented) trefoil knot and compute the Jones polynomial.

import knotpy as kp
from knotpy import from_pd_notation, from_knotpy_notation

K = kp.knot("3_1")
kp.jones(K)

\[\displaystyle - t^{4} + t^{3} + t\]

kp.is_knot(K)

True

A = kp.knot("3_1")
B = kp.knot("4_1")
AB = kp.connected_sum(A, B)
kp.is_connected_sum(AB)

True

A_, B_ = kp.connected_sum_decomposition(AB)
kp.identify(A_), kp.identify(B_)

('3_1', '4_1')

D = kp.disjoint_union(A, B)
kp.is_disjoint_union(D)

True

A_, B_ = kp.disjoint_union_decomposition(D)
kp.identify(A_), kp.identify(B_)

('3_1', '4_1')

K = kp.knot("trefoil")
K.name

'3_1'

K = kp.knot("miller institute knot")
K.name

'6_2'

K = kp.knot("unknot")
K.name

'0_1'

K = kp.knot("+3_1")
K.is_oriented()

True

K_ = kp.knot("-3_1")
kp.identify(kp.knot("-3_1"))

'+3_1'

K = kp.knot("3_1")
print("Knot:", K.name)
print("Jones polynomial:", kp.jones(K))
print("Alexander polynomial:", kp.alexander(K))
print("Bracket polynomial:", kp.bracket(K, normalize=True))
print("HOMFLYPT polynomial:", kp.homflypt(K))
print("HOMFLYPT polynomial (az):", kp.homflypt(K, variables="az"))
print("HOMFLYPT polynomial (lm):", kp.homflypt(K, variables="lm"))
print("HOMFLYPT polynomial (xyz):", kp.homflypt(K, variables="xyz"))
print("Kauffman 2-variable polynomial:", kp.kauffman(K))

Knot: 3_1
Jones polynomial: -t**4 + t**3 + t
Alexander polynomial: t**2 - t + 1
Bracket polynomial: A**(-4) + A**(-12) - 1/A**16
HOMFLYPT polynomial: -v**4 + v**2*z**2 + 2*v**2
HOMFLYPT polynomial (az): z**2/a**2 + 2/a**2 - 1/a**4
HOMFLYPT polynomial (lm): m**2/l**2 - 2/l**2 - 1/l**4
HOMFLYPT polynomial (xyz): -2*y/x - y**2/x**2 + z**2/x**2
Kauffman 2-variable polynomial: a**5*z + a**4*z**2 - a**4 + a**3*z + a**2*z**2 - 2*a**2

K = kp.knot("+3_1")
G = kp.fundamental_group(K)
G.generators, G.relators

((x0, x1, x2), [x0*x1*x2**-1*x1**-1, x2*x0*x1**-1*x0**-1, x1*x2*x0**-1*x2**-1])

K = kp.from_pd_notation("[[1,5,2,4],[3,1,4,6],[5,3,6,2]]")
kp.identify(K)

'3_1'

K = kp.from_pd_notation("PD[X[1,9,2,8], X[3,10,4,11], X[5,3,6,2],X[7,1,8,12], X[9,4,10,5], X[11,7,12,6]]")
kp.identify(K)

'6_2'

kp.to_pd_notation(kp.knot("3_1"))

'X[0,1,2,3],X[3,4,5,0],X[1,5,4,2]'

# EM, CONDENSED EM,... PLANTRI

K = kp.knot("3_1")
kp.identify(kp.orient(K))

'+3_1'

[kp.identify(K) for K in kp.orientations(kp.knot("9_32"))]

['+9_32', '-9_32']

K = kp.knot("3_1")
kp.identify(kp.mirror(K))

'3_1*'

K = kp.knot("+9_32")
kp.identify(kp.reverse(K))

'-9_32'

K = kp.knot("+9_32")
K = kp.mirror(kp.reverse(K))
kp.identify(K)