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Input data and context
% 8_3-2
s.r1.x0.x0.x0.x0.y1.y1.u2
,1
% Symmetry 2 of a 2-bridge knot corresponding to the fraction 17/4 = [4,4] (Sakuma, Prop. 3.6, Case (1), symmetry 2).
% Simplifies with options '-qs' to l1.l2.y4.y3.x2.x1.x1.x2.x1.x2.x1.x1.x1.x2.x1.x1.x1.x2.x2.y3.x1.y2.y4.x1.y3.x2.u3.x0.x1.u2.x0.x0
Bar-Natan multicurve \(\mathrm{\widetilde{BN}}(T)\)
options -c2-q
(diagram)
(metadata)
% file 'examples/strong_inversions/2-bridge_knots/8_3-2/cxBNr-c2-q', generated by kht, version v0.1, on Sun Mar 7 16:05:13 2021 GMT
1) h^ 0 q^ 0 δ^ 0 ⬯
2) h^10 q^14 δ^-3 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
3) h^ 6 q^ 8 δ^-2 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
4) h^ 2 q^ 2 δ^-1 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
5) h^-2 q^-4 δ^ 0 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
options -c3-q
(diagram)
(metadata)
% file 'examples/strong_inversions/2-bridge_knots/8_3-2/cxBNr-c3-q', generated by kht, version v0.1, on Sun Mar 7 16:05:13 2021 GMT
1) h^ 0 q^ 0 δ^ 0 ⬯
2) h^10 q^14 δ^-3 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
3) h^ 6 q^ 8 δ^-2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
4) h^ 2 q^ 2 δ^-1 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
5) h^-2 q^-4 δ^ 0 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
options -c5-q
(diagram)
(metadata)
% file 'examples/strong_inversions/2-bridge_knots/8_3-2/cxBNr-c5-q', generated by kht, version v0.1, on Sun Mar 7 16:05:13 2021 GMT
1) h^ 0 q^ 0 δ^ 0 ⬯
2) h^10 q^14 δ^-3 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
3) h^ 6 q^ 8 δ^-2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
4) h^ 2 q^ 2 δ^-1 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
5) h^-2 q^-4 δ^ 0 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
options -c7-q
(diagram)
(metadata)
% file 'examples/strong_inversions/2-bridge_knots/8_3-2/cxBNr-c7-q', generated by kht, version v0.1, on Sun Mar 7 16:05:13 2021 GMT
1) h^ 0 q^ 0 δ^ 0 ⬯
2) h^10 q^14 δ^-3 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
3) h^ 6 q^ 8 δ^-2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
4) h^ 2 q^ 2 δ^-1 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
5) h^-2 q^-4 δ^ 0 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
options -c11-q
(diagram)
(metadata)
% file 'examples/strong_inversions/2-bridge_knots/8_3-2/cxBNr-c11-q', generated by kht, version v0.1, on Sun Mar 7 16:05:13 2021 GMT
1) h^ 0 q^ 0 δ^ 0 ⬯
2) h^10 q^14 δ^-3 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
3) h^ 6 q^ 8 δ^-2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
4) h^ 2 q^ 2 δ^-1 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
5) h^-2 q^-4 δ^ 0 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
Khovanov multicurve \(\mathrm{\widetilde{Kh}}(T)\)
options -c2-q
(diagram)
(metadata)
% file 'examples/strong_inversions/2-bridge_knots/8_3-2/cxKhr-c2-q', generated by kht, version v0.1, on Sun Mar 7 16:05:13 2021 GMT
1) h^-1 q^-1 δ^ 1/2 ⬯~~S^2~>⬯<— r(∞) q^0 δ_^1/2
2) h^ 9 q^13 δ^-5/2 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^18 δ|^-3
3) h^10 q^15 δ^-5/2 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^20 δ|^-3
4) h^ 5 q^ 7 δ^-3/2 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^12 δ|^-2
5) h^ 6 q^ 9 δ^-3/2 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^14 δ|^-2
6) h^ 1 q^ 1 δ^-1/2 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^6 δ|^-1
7) h^ 2 q^ 3 δ^-1/2 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^8 δ|^-1
8) h^-3 q^-5 δ^ 1/2 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^0 δ|^0
9) h^-2 q^-3 δ^ 1/2 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^2 δ|^0
options -c3-q
(diagram)
(metadata)
% file 'examples/strong_inversions/2-bridge_knots/8_3-2/cxKhr-c3-q', generated by kht, version v0.1, on Sun Mar 7 16:05:13 2021 GMT
1) h^-1 q^-1 δ^ 1/2 ⬯~~(-S^2)~>⬯<— r(∞) q^0 δ_^1/2
2) h^ 9 q^13 δ^-5/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^18 δ|^-3
3) h^10 q^15 δ^-5/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^20 δ|^-3
4) h^ 5 q^ 7 δ^-3/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^12 δ|^-2
5) h^ 6 q^ 9 δ^-3/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^14 δ|^-2
6) h^ 1 q^ 1 δ^-1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^6 δ|^-1
7) h^ 2 q^ 3 δ^-1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^8 δ|^-1
8) h^-3 q^-5 δ^ 1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^0 δ|^0
9) h^-2 q^-3 δ^ 1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^2 δ|^0
options -c5-q
(diagram)
(metadata)
% file 'examples/strong_inversions/2-bridge_knots/8_3-2/cxKhr-c5-q', generated by kht, version v0.1, on Sun Mar 7 16:05:13 2021 GMT
1) h^-1 q^-1 δ^ 1/2 ⬯~~(-S^2)~>⬯<— r(∞) q^0 δ_^1/2
2) h^ 9 q^13 δ^-5/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^18 δ|^-3
3) h^10 q^15 δ^-5/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^20 δ|^-3
4) h^ 5 q^ 7 δ^-3/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^12 δ|^-2
5) h^ 6 q^ 9 δ^-3/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^14 δ|^-2
6) h^ 1 q^ 1 δ^-1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^6 δ|^-1
7) h^ 2 q^ 3 δ^-1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^8 δ|^-1
8) h^-3 q^-5 δ^ 1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^0 δ|^0
9) h^-2 q^-3 δ^ 1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^2 δ|^0
options -c7-q
(diagram)
(metadata)
% file 'examples/strong_inversions/2-bridge_knots/8_3-2/cxKhr-c7-q', generated by kht, version v0.1, on Sun Mar 7 16:05:13 2021 GMT
1) h^-1 q^-1 δ^ 1/2 ⬯~~(-S^2)~>⬯<— r(∞) q^0 δ_^1/2
2) h^ 9 q^13 δ^-5/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^18 δ|^-3
3) h^10 q^15 δ^-5/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^20 δ|^-3
4) h^ 5 q^ 7 δ^-3/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^12 δ|^-2
5) h^ 6 q^ 9 δ^-3/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^14 δ|^-2
6) h^ 1 q^ 1 δ^-1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^6 δ|^-1
7) h^ 2 q^ 3 δ^-1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^8 δ|^-1
8) h^-3 q^-5 δ^ 1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^0 δ|^0
9) h^-2 q^-3 δ^ 1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^2 δ|^0
options -c11-q
(diagram)
(metadata)
% file 'examples/strong_inversions/2-bridge_knots/8_3-2/cxKhr-c11-q', generated by kht, version v0.1, on Sun Mar 7 16:05:13 2021 GMT
1) h^-1 q^-1 δ^ 1/2 ⬯~~(-S^2)~>⬯<— r(∞) q^0 δ_^1/2
2) h^ 9 q^13 δ^-5/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^18 δ|^-3
3) h^10 q^15 δ^-5/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^20 δ|^-3
4) h^ 5 q^ 7 δ^-3/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^12 δ|^-2
5) h^ 6 q^ 9 δ^-3/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^14 δ|^-2
6) h^ 1 q^ 1 δ^-1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^6 δ|^-1
7) h^ 2 q^ 3 δ^-1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^8 δ|^-1
8) h^-3 q^-5 δ^ 1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^0 δ|^0
9) h^-2 q^-3 δ^ 1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^2 δ|^0
Khovanov type D structure \(\mathrm{\widetilde{CKh}}(L)\)
options -c0
(diagram)
(metadata)
% file 'examples/strong_inversions/2-bridge_knots/8_3-2/cxCKh-c0', generated by kht, version v0.1, on Sun Mar 7 15:45:53 2021 GMT
1) h^ 0 q^ 0 δ^0 ⬮
2) h^-4 q^-8 δ^0 ⬮——H—>⬮
3) h^-2 q^-4 δ^0 ⬮——H—>⬮
4) h^-2 q^-4 δ^0 ⬮——H—>⬮
5) h^-1 q^-2 δ^0 ⬮——H—>⬮
6) h^ 0 q^ 0 δ^0 ⬮——H—>⬮
7) h^ 1 q^ 2 δ^0 ⬮——H—>⬮
8) h^ 1 q^ 2 δ^0 ⬮——H—>⬮
9) h^ 3 q^ 6 δ^0 ⬮——H—>⬮
options -c2
(diagram)
(metadata)
% file 'examples/strong_inversions/2-bridge_knots/8_3-2/cxCKh-c2', generated by kht, version v0.1, on Sun Mar 7 15:58:53 2021 GMT
1) h^ 0 q^ 0 δ^0 ⬮
2) h^-4 q^-8 δ^0 ⬮——H—>⬮
3) h^-2 q^-4 δ^0 ⬮——H—>⬮
4) h^-2 q^-4 δ^0 ⬮——H—>⬮
5) h^-1 q^-2 δ^0 ⬮——H—>⬮
6) h^ 0 q^ 0 δ^0 ⬮——H—>⬮
7) h^ 1 q^ 2 δ^0 ⬮——H—>⬮
8) h^ 1 q^ 2 δ^0 ⬮——H—>⬮
9) h^ 3 q^ 6 δ^0 ⬮——H—>⬮
options -c3
(diagram)
(metadata)
% file 'examples/strong_inversions/2-bridge_knots/8_3-2/cxCKh-c3', generated by kht, version v0.1, on Sun Mar 7 15:58:53 2021 GMT
1) h^ 0 q^ 0 δ^0 ⬮
2) h^-4 q^-8 δ^0 ⬮——H—>⬮
3) h^-2 q^-4 δ^0 ⬮——H—>⬮
4) h^-2 q^-4 δ^0 ⬮——H—>⬮
5) h^-1 q^-2 δ^0 ⬮——H—>⬮
6) h^ 0 q^ 0 δ^0 ⬮——H—>⬮
7) h^ 1 q^ 2 δ^0 ⬮——H—>⬮
8) h^ 1 q^ 2 δ^0 ⬮——H—>⬮
9) h^ 3 q^ 6 δ^0 ⬮——H—>⬮
options -c5
(diagram)
(metadata)
% file 'examples/strong_inversions/2-bridge_knots/8_3-2/cxCKh-c5', generated by kht, version v0.1, on Sun Mar 7 15:58:53 2021 GMT
1) h^ 0 q^ 0 δ^0 ⬮
2) h^-4 q^-8 δ^0 ⬮——H—>⬮
3) h^-2 q^-4 δ^0 ⬮——H—>⬮
4) h^-2 q^-4 δ^0 ⬮——H—>⬮
5) h^-1 q^-2 δ^0 ⬮——H—>⬮
6) h^ 0 q^ 0 δ^0 ⬮——H—>⬮
7) h^ 1 q^ 2 δ^0 ⬮——H—>⬮
8) h^ 1 q^ 2 δ^0 ⬮——H—>⬮
9) h^ 3 q^ 6 δ^0 ⬮——H—>⬮
options -c7
(diagram)
(metadata)
% file 'examples/strong_inversions/2-bridge_knots/8_3-2/cxCKh-c7', generated by kht, version v0.1, on Sun Mar 7 15:58:53 2021 GMT
1) h^ 0 q^ 0 δ^0 ⬮
2) h^-4 q^-8 δ^0 ⬮——H—>⬮
3) h^-2 q^-4 δ^0 ⬮——H—>⬮
4) h^-2 q^-4 δ^0 ⬮——H—>⬮
5) h^-1 q^-2 δ^0 ⬮——H—>⬮
6) h^ 0 q^ 0 δ^0 ⬮——H—>⬮
7) h^ 1 q^ 2 δ^0 ⬮——H—>⬮
8) h^ 1 q^ 2 δ^0 ⬮——H—>⬮
9) h^ 3 q^ 6 δ^0 ⬮——H—>⬮
options -c11
(diagram)
(metadata)
% file 'examples/strong_inversions/2-bridge_knots/8_3-2/cxCKh-c11', generated by kht, version v0.1, on Sun Mar 7 15:58:53 2021 GMT
1) h^ 0 q^ 0 δ^0 ⬮
2) h^-4 q^-8 δ^0 ⬮——H—>⬮
3) h^-2 q^-4 δ^0 ⬮——H—>⬮
4) h^-2 q^-4 δ^0 ⬮——H—>⬮
5) h^-1 q^-2 δ^0 ⬮——H—>⬮
6) h^ 0 q^ 0 δ^0 ⬮——H—>⬮
7) h^ 1 q^ 2 δ^0 ⬮——H—>⬮
8) h^ 1 q^ 2 δ^0 ⬮——H—>⬮
9) h^ 3 q^ 6 δ^0 ⬮——H—>⬮