Findings on Tangle Embeddings

Tangle 6_3

Knots containing this tangle:

(8_10) = N( T(6_3) + R(2,1) )* (Orientation: NW In, SW Out) (MS reported.)

(8_20) = N( T(6_3) + R(2) )* (Orientation: NW In, SW Out) (MS reported. Look at the braid form [-1,-1,-1,2,1,1,1,2], the mirror of 8_20.)

(9_24) = N( T(6_3) + R(2,2) )* (Orientation: NW In, SW Out) (MS reported. Look at the braid form [1,1,-2,1,3,-2,-2,-2,3] of 9_24.)

Knots that do not contain this tangle:

(1) For Orientation (NW In, SW Out):

(1-1) In this case, the tangle consists of a trefoil diagram of parallel orientations and its mirror image. Hence it colors non-trivially by Z_p[t]/(t^2-t+1). (KA, MS reported.)

(1-2) The source region for these two copies of trefoil diagrams coincide. The signs of the crossings are opposite. Hence the invariant is trivial, (p^2)^3 for Z_p[t]/(t^2-t+1).

For p=2, the tangle has invariant value 64. From the table this does not embed in knots up to 9 crossings except for: 8_5, 8_10, 8_15, 8_18, 8_19, 8_20, 8_21, 9_16, 9_22, 9_24, 9_25, 9_28, 9_29, 9_30, 9_36, 9_38, 9_39, 9_40, 9_41, 9_42, 9_43, 9_44, 9_45, 9_49, and their mirrors. (It was incorrectly listed as 6_1, 8_10, 8_11, 8_20, 9_1, 9_27, 9_24, 9_35, 9_37, 9_46 - 9_48 earlier.)

For p=3, the tangle has invariant value 729. From the table this does not embed in knots up to 9 crossings except for: 7_7, 8_10, 8_11, 8_18, 8_20, 9_1, 9_6, 9_10, 9_11, 9_15, 9_23, 9_24, 9_35, 9_37, 9_46, 9_47, 9_48.

For p=5, the tangle has invariant value 15625. From the table this does not embed in knots up to 9 crossings except for: 8_10, 8_12, 8_18, 8_20, 9_24.

For p=7, the tangle has invariant value 117649. From the table this does not embed in knots up to 9 crossings except for: 3_1, 8_5, 8_10, 8_11, 8_15, 8_18, 8_19, 8_20, 8_21, 9_1, 9_6, 9_16, 9_23, 9_24, 9_28, 9_29, 9_38, 9_40.

For R_3, with the Mochizuki cocycle of the dihedral quandle, the tangle has the trivial value 27 again, but 8_18 has the value 9+36*u+36*u^2, so that the tangle does not embed in 8_18.

All combined, it does not embed except for: 8_10, 8_20, 9_24.

Note that all these embed the tangle. So in this case the cocycle invariants determines the embeddings completely. (KA, MS reported.)

(2) For Orientation (NW In, SW In):

(2-1) There are non-trivial colorings by R_3. (MS reported, KA, ME confirmed.) (Ealier report had error: There are non-trivial colorings by Alexander quandles of the form Z_p[t]/(t-2) where p is non-divisible by 2. (KA reported, ME confirmed.))

(2-2) Theorem 3.1 in [Mochi03] says that the third cohomology group of a linear Alexander quandle vanishes except for dihedral quandles. We look at the dihedral quandle, in this case R_3. From the symmetry, the cocycle invariant of this tangle is 27. By our Theorem, if this embeds in a knot, the knot must have at least 27 as the constant term of the cocycle invariant with R_3. From the table of invariants, up to 9 crossings, this does not embed in knots in the table excluding: 6_1, 8_10, 8_11, 8_20, 9_1, 9_27, 9_24, 9_35, 9_37, 9_46 - 9_48. For these knots excluded, the invariant does not show whether the tangle embeds. The list for this dihedral quandle in our web site contains knots up to 12 crossings, and it is seen that many do not embed this tangle. (KA, MS reported.)

(3) For Orientation (NW Out, SW Out): Colored only trivially by Alexander quandles. (KA reported.)

Tangle 6_4

Knots containing this tangle:

(6_2) = N(6_4) (NW In, SW In, or opposite) (TR reported, ME confirmed.)

(6_3) = D(6_4) (NW In, SW Out, or opposite) (TR reported.)

Knots that do not contain this tangle:

For all orientations (4 cases), the tangle has only the trivial coloring by any Alexander quandle, so that we are not able to use the quandle cocycle invariants with Alexander quandles. (KA reported, ME confirmed.)

Tangle 7_4

Knots containing this tangle:

(4_1) = N( T(7_4) + R(-1) ) (Orientation: NW In, NE In) (TR reported, KA, ME confirmed.)

Knots that do not contain this tangle:

(1) For Orientation (NW In, NE In):

(1-1) This is non-trivially colored by Z_2[t]/(t^2+t+1). (KA reported.)

(1-2) The tangle has the cocycle invariant 16+48*u^t by calculation, with the 3-cocycle f(x,y,z)=(x-y)(y-z)^2. By Theorem, if this embeds in a knot, the knot must contain this factor (in a multiset). From the table of invariants, this tangle does not embed in knots in the table excluding: 3_1, 4_1, 7_2, 7_3, 8_1, 8_4, 8_11, 8_13, 8_18, 9_1, 9_6, 9_12, 9_13, 9_14, 9_21, 9_23, 9_35, 9_37, 9_40. (KA reported.)

(2) For Orientation (NW In, NE Out):

(2-1) Colored trivially by Alexander quandles (ME reported).

Tangle 7_5

Knots containing this tangle:

(7_3)* = N( T(7_5) + R(-1) ) (Orientation: NW In, NE In) (KA, ME, MS reported and confirmed.)

O (unknot) = N( T(7_5) ) (NW In, NE Out) (TR reported, KA, ME confirmed.)

Knots that do not contain this tangle:

(1) For Orientation (NW In, NE In):

(1-1) There are non-trivial colorings by Alexander quandles of the form Z_2[t]/(t^2+t+1). (ME reported.)

(1-2) From the above fact (7_3)* = N( T(7_5) + R(-1) ), the number of colorings of 7_3 and the fact that the value of 7_3, 16 + 48*u^t, has non-trivial contributions from non-trivial colorings, the invariant of the tangle must be also 16 + 48*u^t. From the table of invariants, this tangle does not embed in knots in the table excluding: 3_1, 4_1, 7_2, 7_3, 8_1, 8_4, 8_11, 8_13, 8_18, 9_1, 9_6, 9_12, 9_13, 9_14, 9_21, 9_23, 9_35, 9_37, 9_40. (ME reported.)

(2) For Orientation (NW In, NE Out):

The tangle has only the trivial coloring by any Alexander quandle, so that we are not able to use the quandle cocycle invariants with Alexander quandles. (ME reported.) This is computed earlier but confirmed by the above fact O (unknot) = N( T(7_5) ).

Tangle 7_6

Knots containing this tangle:

Knots that do not contain this tangle:

(1) For Orientation (NW In, NE In):

(1-1) The left half of the tangle being trefoil, we take the quandle Z_p[t]/(t^2-t+1). Then the color x_2 must be y. Then the right half of the tangle can be closed to form the figure-eight knot with this coloring. Since the Alexander polynomial of the figure-eight is (t^2-3t+1), we take p=2 to color the tangle non-trivially. (KA, MS reported.)

(1-2) We use the 3-cocycle invariant with the 3-cocycle f(x,y,z)=(x-y)(y-z)^2. From the table of cocycle invariant, 3_1 and 4_1 have the invariant 16+48*u^t. This implies that any non-trivial coloring gives non-trivial contribution for both 3_1 and 4_1. 3_1 and 4_1 are invertible, and their mirrors with opposite orientations have the same invariant value since -t = t for p=2. Hence any non-trivial coloring of this tangle contributes u^(t+t)=1, so the invariant is 64.

If this embeds in a knot, the knot must have at least 64 as the constant term. From the table of invariants, this tangle does not embed in knots in the table excluding:

8_5, 8_10, 8_15, 8_18 - 8_21, 9_16, 9_22, 9_24, 9_25, 9_28 - 9_30, 9_36, 9_38, 9_39, 9_41 - 9_45, 9_49. (KA, MS reported.)

(2) For Orientation (NW In, NE Out):

(2-1) Colored trivially by Alexander quandles (TR reported).

Tangle 7_7

Knots containing this tangle:

O (unknot) = N( T(7_7) ) (NW In, NE Out) (TR reported, KA, ME confirmed.)

Knots that do not contain this tangle:

(1) For Orientation (NW In, NE In):

(1-1) By the same argument as for 7_6, we have to use the same quandle, Z_p[t]/(t^2-t+1) for p=2. (KA, MS reported, TR confirmed.)

(1-2) We obtain the same conclusions as 7_6. (KA, MS reported.)

(2) For Orientation (NW In, NE Out):

The tangle has only the trivial coloring by any Alexander quandle, so that we are not able to use the quandle cocycle invariants with Alexander quandles. (TR reported, ME confirmed.) This is computed earlier but confirmed by the above fact O (unknot) = N( T(7_7) ).

Tangle 7_10

Knots containing this tangle:

(3_1) = N( T(7_10) + R(-1) ) (NW In, NE In) (TR reported.)

Knots that do not contain this tangle:

For all orientations (4 cases), the tangle has only the trivial coloring by any Alexander quandle, so that we are not able to use the quandle cocycle invariants with Alexander quandles. (KA reported.)

Tangle 7_11

Knots containing this tangle:

(3_1) = N( (7_11) + R(-1) ) (NW In, SE Out, or opposite) (KA reported, TR confirmed.)

Knots that do not contain this tangle:

(1) For Orientation (NW In, SE In):

Colors trivially. (KA reported.)

(2) For Orientation (NW In, SE Out):

Colors trivially. (KA reported.)

(3) For Orientation (NW Out, SE In):

Colors trivially. (KA reported.)

(4) For Orientation (NW Out, SE Out):

Colors trivially. (KA reported.)

Tangle 7_12

O (unknot) = N( T(7_12) ) (NW In, NE Out) (TR reported, KA, ME confirmed.)

Knots containing this tangle:

Knots that do not contain this tangle:

(1) For Orientation (NW In, NE In):

Colors trivially. (KA reported.)

(2) For Orientation (NW In, NE Out):

Colors trivially. (KA reported.)

(3) For Orientation (NW Out, NE In):

Colors trivially. (TR reported, ME confirmed.)

(4) For Orientation (NW Out, NE Out):

Colors trivially. (KR reported, KA confirmed.)

Tangle 7_13

Knots containing this tangle:

(4_1) = N( T(7_13) + R(-1) ) (Orientation: NW In, NE In) (TR reported, KA, ME, MS confirmed.)

(7_4) = N( T(7_13) ) (Orientation: NW In, NE Out) (KA, MS reported.)

(8_16) = N( T(7_13) + R(1) ) (Orientation: NW In, NE Out) (KA, MS reported.)

(9_39) = N( T(7_13) + R(1,1) ) (Orientation: NW In, NE Out) (KA, MS reported.)

(9_49) = N( T(7_13) + R(-1,-1) ) (Orientation: NW In, NE Out) (KA, MS reported.)

Knots that do not contain this tangle:

(1) For Orientation (NW In, NE In): At first it was reported here that the tangle has only the trivial colorings by any Alexander quandle,but in fact it colors non-trivially by R_5. (KA reported.)

(2) For Orientation (NW In, NE Out):

(2-1) The tangle is colored non-trivially by R_5. (KA reported.)

(2-2) The tangle has the invariant value 25+50*u^2+50*u^3. This is proved as follows. The invariant of the knot 7_4 is as above. Since we see that N( T(7_13) )=7_4, and the number of colorings are the same for the tangle T(7_13) and 7_4, the invariant value of T(7_13) must be the same as above. From the table for R_5, This tangle does not embed in knots in the table (up to 9 crossings) excluding: 4_1, 7_4, 8_16, 9_24, 9_37, 9_39, 9_40, 9_49 and their mirrors. Those we do not know yet whether it embeds are: 4_1, 9_24, 9_37, 9_40, (and possibly mirrors).

Tangle 7_14

Knots containing this tangle:

O (unknot) = N(T(7_14) + R(-1) )(NW In, NE In) (TR reported.)

Knots that do not contain this tangle:

(1) For Orientation (NW In, NE In): The tangle has only the trivial colorings by any Alexander quandle. (TR reported.)

(2) For Orientation (NW In, NE Out):

(2-1) The tangle is non-trivially colored by Z_5[t]/(t-3). (TR reported, ME confirmed.)

Although the number of colorings by this quandle can be used, the cocycle invariants will not give stronger results, as cohomology groups vanish for this linear quandle by [Mochi03].

(3) For Orientation (NW Out, NE In):

Colored trivially. (KA reported.)

(4) For Orientation (NW Out, NE Out):

Colored trivially. (KA reported.)

Tangle 7_15

Knots containing this tangle:

(5_2) = N( T(7_15) + R(-1) ) (Orientation: NW In, SW In) (TR reported, Others confirmed.)

Knots that do not contain this tangle:

(1) For Orientation (NW In, SW In):

(1-1) The tangle is colored non-trivially by R_7. (ME reported, KA confirmed.)

(1-2) The tangle has the invariant value 49+98*u^3+98*u^5+98*u^6. This is proved as follows. The tangle can be non-trivially colored by R_7, and hence so does (5_2) = N( T(7_15) + R(-1) ). By Chad's table, the invariant for 5_2 is 49+98*u^3+98*u^5+98*u^6. The number of colorings is 343. The tangle has non-trivial colorings, hence there are at least 7^3=343 colorings. Hence every coloring of 5_2 comes from a coloring of the tangle, so that they share the same invariant value.

From the table for R_7, This tangle does not embed in knots in the table (up to 9 crossings) excluding: 5_2, 8_16, 9_41, 9_42. (All reported and confirmed.) It embeds in 5_2. N( T(7_15) + R(1) ) is a reduced alternating diagram with 8 crossings, and 8_16 is the only possibility, therefore this must be 8_16 (although we have not been able to diagrammtically match them). We do not know whether it embeds in 9_41 or 9_42.

(2) For Orientation (NW In, SW Out):

(2-1) The tangle is colored non-trivially by R_7. (ME reported, KA confirmed.)

(2-2) The tangle has the invariant value 49+98*u+98*u^2+98*u^4. This is proved as follows. The tangle can be non-trivially colored by R_7, and hence so does its denominator (a knot K). Since K is a reduced alternating diagram, the crossing number of K is 7. By Chad's table, there are only two 7 crossing knots that are colored non-trivially by R_7, 7_1 and 7_7. Both have the same (above) invariant value. Hence the tangle has this invariant value. (The knot K is presumably 7_7.)

From the table for R_7, This tangle does not embed in knots in the table (up to 9 crossings) excluding: 7_1, 7_7, 8_5, 9_4, 9_12, 9_41. (KA, MS reported.)

D( T(7_15) ) is a reduced alternating diagram, and has 7 crossings, therefore it must be either 7_1 or 7_7. Determinants are Det(7_1)=7, Det(7_7)=21, and D( T(7_15) ) is 3-colorable by checking, hence it must be 7_7. Determinants of 9_4, 9_12, 9_41 are 21, 35, and 49, respectively. N( T(7_15) + R(2) ) has determinant 49 (check! MS reported). Hence it is 9_41. So far we know that it embeds in 7_7 and 9_41. Other possibilities should be checked by the determinants.

Tangle 7_16

Knots containing this tangle:

(7_7) or (7_7)* = D( T(7_16) ) (Orientation: NW In, NE In) The denominator of T(7_16) is colored by R_21 by calculation. The only knot less that 8 crossings with deteminant divisible by 21 is 7_7 or its mirror, therefore it must be 7_7 or its mirror.(Reported by all.) By Kodama's program, D( T(7_16) ) has the Jones polynomial equal to that of (7_7)*, hence it must be (7_7)*. (Reported by KA.)

(7_4) = N( T(7_16) ) (Orientation: NW In, NE Out) (Reported by all.)

Knots that do not contain this tangle:

(1) For Orientation (NW In, NE In):

(1-1) The tangle is colored non-trivially by R_3. (TR reported, ME, KA confirmed.)

(1-2) Since (7_7)* = D( T(7_16) ) and the invariant value of 7_7 is 9+18*u, T(7_16) has the invariant value 9+18*u^2. Hence T(7_16) does not embed in those less than or equal to 9 crossings except for: 8_5, 8_15, 8_18, 8_19, 8_21, 9_2, 9_4, 9_11, 9_15, 9_16, 9_28, 9_34, 9_37, 9_40, 9_46, 9_47.

(2) For Orientation (NW In, NE Out):

(2-1) The tangle is colored non-trivially by R_3. (TR reported, ME, KA confirmed.)

From the number of colorings, we conclude that the invariant value of T(7_17) with this orientation is that of (7_4), which is 9+18*u. Hence T(7_19) does not embed except: 3_1, 7_4, 7_7, 8_18, 9_10, 9_17, 9_29, 9_35, 9_37, 9_38, 9_46, 9_48.

Tangle 7_17

Knots containing this tangle:

(3_1)* = N( T(7_17) + R(-1) ) (NW In, SW In) (TR reported.)

(8_18) = N( T(7_17) + R(1) ) (NW In, SW In) (KA, MS, TR reported.)

Knots that do not contain this tangle:

(1) For Orientation (NW In, SW In):

(1-1) The tangle is colored non-trivially by Z_p[t]/(t^2-t+1). (KA reported.)

(1-2) From (3_1)* = N( T(7_17) + R(-1) ), and the numbers of the colorings by the quandle Z_p[t]/(t^2-t+1) are the same for (3_1)* and T(7_17), their invariants agree.

We use 3-cocycle invariants. For p=2, the invariant for (3_1)* is the same as that of 3_1 which is 16 + 48*u^t. Hence it does not embed in knots except for: 3_1, 4_1, 7_2, 7_3, 8_1, 8_4, 8_11, 8_13, 8_18, 9_1, 9_6, 9_12, 9_13, 9_14, 9_21, 9_23, 9_35, 9_37, 9_40.

For p=3, the invariant for 3_1 is 243+486*u^(2*t+2). Its mirror has the invariant 243+486*u^(t+1), which is the invariant value for this tangle. Hence it does not embed in knots except for: 6_1, 8_5, 8_15, 8_18, 8_19, 8_21, 9_4, 9_16, 9_17, 9_28, 9_40, and mirrors of 3_1, 8_18, 9_2, 9_4, 9_29, 9_34, 9_38.

For p=5, the table gives 625+3750*u^(t+3)+3750*u^(4*t+2)+3750*u^(3*t+4)+3750*u^(2*t+1) as the invariant for trefoil. The tangle has the invariant value its negatives, 625+3750*u^(4*t+2)+3750*u^(t+3)+3750*u^(2*t+1)+3750*u^(3*t+4). This is the same as trefoil. From the table this does not embed in knots up to 9 crossings except for: 3_1, 8_3, 8_5, 8_11, 8_15, 8_18, 8_19, 8_21, 9_1, 9_5, 9_6, 9_16, 9_19, 9_23, 9_28, 9_29, 9_38, 9_40, and the same list for mirrors (by the symmetry of the invariant values).

For p=7, the trefoil has 117649 as the invariant value, and so does the tangle. From the table this does not embed in knots up to 9 crossings except for: 3_1, 8_5, 8_10, 8_11, 8_15, 8_18, 8_19, 8_20, 8_21, 9_1, 9_6, 9_16, 9_23, 9_28, 9_29, 9_38, 9_40, and the same list for mirrors.

All combined, this does not embed in knots up to 9 crossings except for: 8_18, 9_40, and mirrors of 3_1. We verified that it embeds in these except 9_40.

R_3 does not exclude further.

If we realize the embedding for 8_18 and 9_40, then we will completeley determine the embedding for this tangle to be the above three.

(2) For Orientation (NW In, SW Out):

(2-1) The tangle is colored non-trivially by R_3. (KA reported.)

(2-2) The invariant does not depend on the orientation. The tangle has the same invariant value as the mirror of the trefoil. The 3-cocycle invariant for trefoil is 9 + 18*u from the table. Its mirror has contribution 2 (u^2) from any non-trivial coloring. The tangle has the invariant value 9 + 18*u^2. From the table of invariants for dihedral quandles, this does not embed in knots up to 9 crossings except for: 8_5, 8_15, 8_18, 8_19, 8_21, 9_2, 9_4, 9_11, 9_15,9_16, 9_28, 9_34, 9_37, 9_40, 9_46, 9_47and mirrors of: 3_1, 7_4, 7_7, 8_18, 9_10, 9_29, 9_35, 9_37, 9_38, 9_46, 9_48. (KA, MS reported.)

Tangle 7_18

Knots containing this tangle:

D( T(7_18) ) = 5_1 (NW In, SW Out) (KA, MS reported.)

N( T(7_18) + R(1) ) (NW In, SW In) has determinant 15 and 8 crossings. There are only two knots with determinant 15 in the table less than 9 crossings, 7_4 and 8_21. Hence it must be one of these. Then from the calculation below, the cocycle invariant agrees with 8_21, so it must be 8_21.

Knots that do not contain this tangle:

(1) For Orientation (NW In, SW In):

(1-1) The tangle is colored non-trivially by R_5. (KA reported.)

(1-2) The invariant does not depend on the choice of orientation for the dihedral quandle with the Mochizuki cocycle (Satoh). Hence the invariant is the same as the case below (2-2) and 25+50*u+50*u^4. The invariant for 7_4 is 25+50*u^2+50*u^3 and that of 8_21 is 25+50*u+50*u^4. Hence we conclude that N( T(7_18) + R(1) ) = 8_21. Then the same conclusion works for this case, this tangle does not embed in knots in the table (up to 9 crossings) excluding: 5_1, 8_18, 8_21, 9_2, 9_12, 9_23, 9_31, 9_40, 9_49, and their mirrors.

(2) For Orientation (NW In, SW Out):

(2-1) The tangle is colored non-trivially by R_5. (KA reported.)

(2-2) The tangle has the invariant value 25+50*u+50*u^4. This is proved as follows. The invariant of the knot 5_1 is as above. Since we see that D( T(7_18) )=5_1, and the number of colorings are the same for the tangle T(7_18) and 5_1, the invariant value of T(7_18) must be the same as above. From the table for R_5, This tangle does not embed in knots in the table (up to 9 crossings) excluding: 5_1, 8_18, 8_21, 9_2, 9_12, 9_23, 9_31, 9_40, 9_49, and their mirrors.