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Input data and context
% (2,-3)-pretzel tangle
r1.x2.y0.x2.y0.x2.y0.x2.y0.x2.y0.x2.y0.x2.u1
,1,0
Bar-Natan multicurve \(\mathrm{\widetilde{BN}}(T)\)
options -c0
(diagram)
(metadata)
% file 'examples/pretzeltangles/PT6m7/cxBNr-c0', generated by kht, version v0.1, on Sun Mar 7 14:27:17 2021 GMT
1) h^-3 q^ -6 δ^ 0 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
2) h^-5 q^-10 δ^ 0 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬯<—
3) h^ 0 q^ -1 δ^-1/2 ⬮<—D——⬮<~⬮<—⬮<~⬮<—⬮<~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯
options -c2
(diagram)
(metadata)
% file 'examples/pretzeltangles/PT6m7/cxBNr-c2', generated by kht++ v0.1-110-gad0cb61-dirty on Mon Jun 1 16:47:17 2026 GMT
1) h^ 0 q^ -1 δ^-1/2 ⬮<—D——⬮<~⬮<—⬮<~⬮<—⬮<~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯
2) s8(0) q^-1 δ|^-1/2: h^-5 q^-10 δ^ 0 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬯<—
3) s4(0) q^-1 δ|^-1/2: h^-3 q^ -6 δ^ 0 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
options -c3
(diagram)
(metadata)
% file 'examples/pretzeltangles/PT6m7/cxBNr-c3', generated by kht++ v0.1-99-g54450e6-dirty on Mon Dec 29 17:40:02 2025 GMT
1) h^-3 q^-7 δ^-1/2 ⬮<~(-S)~~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—
2) h^-2 q^-5 δ^-1/2 ⬮<~(-S)~~⬯—>⬯~>⬮—>⬮~>⬮—>⬮~>⬯<—⬯<~⬮<—⬮<~⬮<—
3) h^ 0 q^-1 δ^-1/2 ⬮<—D——⬮<~⬮<—⬮<~⬮<—⬮<~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯
options -c5
(diagram)
(metadata)
% file 'examples/pretzeltangles/PT6m7/cxBNr-c5', generated by kht++ v0.1-99-g54450e6-dirty on Mon Dec 29 17:40:02 2025 GMT
1) h^-1 q^-3 δ^-1/2 ⬮<~(-S)~~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—
2) h^-1 q^-3 δ^-1/2 ⬮<~(-S^2)~~⬮<—⬮<~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—
3) h^ 0 q^-1 δ^-1/2 ⬮<—D——⬮<~⬮<—⬮<~⬮<—⬮<~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯
options -c7
(diagram)
(metadata)
% file 'examples/pretzeltangles/PT6m7/cxBNr-c7', generated by kht++ v0.1-99-g54450e6-dirty on Mon Dec 29 17:40:02 2025 GMT
1) h^-1 q^-3 δ^-1/2 ⬮<~(-S)~~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—
2) h^ 3 q^ 5 δ^-1/2 ⬮<~(-S^2)~~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—
3) h^ 0 q^-1 δ^-1/2 ⬮<—D——⬮<~⬮<—⬮<~⬮<—⬮<~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯
options -c11
(diagram)
(metadata)
% file 'examples/pretzeltangles/PT6m7/cxBNr-c11', generated by kht++ v0.1-99-g54450e6-dirty on Mon Dec 29 17:40:02 2025 GMT
1) h^-2 q^-5 δ^-1/2 ⬮<~(-S)~~⬯—>⬯~>⬮—>⬮~>⬮—>⬮~>⬯<—⬯<~⬮<—⬮<~⬮<—
2) h^-4 q^-9 δ^-1/2 ⬮<~(-S)~~⬯—>⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯<—⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—
3) h^ 0 q^-1 δ^-1/2 ⬮<—D——⬮<~⬮<—⬮<~⬮<—⬮<~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯
Khovanov multicurve \(\mathrm{\widetilde{Kh}}(T)\)
options -c0
(diagram)
(metadata)
% file 'examples/pretzeltangles/PT6m7/cxKhr-c0', generated by kht, version v0.1, on Sun Mar 7 14:27:17 2021 GMT
1) h^-4 q^ -7 δ^1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^-2 δ|^0
2) h^-3 q^ -5 δ^1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<— s2(0) q^0 δ|^0
3) h^-7 q^-13 δ^1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬯<— r(1/6) q^-48/7 δ|^0
4) h^-6 q^-11 δ^1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬯<— s4(0) q^-2 δ|^0
5) h^-5 q^ -9 δ^1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬯<— s4(0) q^0 δ|^0
6) h^-8 q^-15 δ^1/2 ⬯~~(-S)~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬯<— s6(0) q^-2 δ|^0
options -c2
(diagram)
(metadata)
% file 'examples/pretzeltangles/PT6m7/cxKhr-c2', generated by kht++ v0.1-110-gad0cb61-dirty on Mon Jun 1 16:47:17 2026 GMT
1) r(1/6) q^-48/7 δ|^0: h^-7 q^-13 δ^1/2 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬯<—
2) s12(0) q^-2 δ|^0: h^-8 q^-15 δ^1/2 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬯<—
3) s8(0) q^-2 δ|^0: h^-6 q^-11 δ^1/2 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬯<—
4) s8(0) q^0 δ|^0: h^-5 q^ -9 δ^1/2 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬯<—
5) s4(0) q^-2 δ|^0: h^-4 q^ -7 δ^1/2 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
6) s4(0) q^0 δ|^0: h^-3 q^ -5 δ^1/2 ⬯~~S~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬯<—
options -c3
(diagram)
(metadata)
% file 'examples/pretzeltangles/PT6m7/cxKhr-c3', generated by kht++ v0.1-99-g54450e6-dirty on Mon Dec 29 17:40:02 2025 GMT
1) h^-5 q^-10 δ^0 ⬮<~(-S)~~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮<—⬮<~⬮<—⬮<~⬮<— r(1/6) q^-48/7 δ|^0
2) h^-3 q^ -6 δ^0 ⬮<~(-S)~~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<— r2(-1/4) q^0 δ|^0
3) h^-6 q^-12 δ^0 ⬮<~(-S)~~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<— r2(-1/6) q^-2 δ|^0
4) h^-2 q^ -4 δ^0 ⬮<~(-S)~~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<— r2(-1/2) q^-2 δ|^0
5) h^ 1 q^ 2 δ^0 ⬮<~(-S^2)~~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬯—>⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯<—⬯<~⬮<— r2(-1/4) q^-2 δ|^0
6) h^ 1 q^ 2 δ^0 ⬮<~(-S^2)~~⬮<—⬮<~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<— r2(-1/2) q^0 δ|^0
options -c5
(diagram)
(metadata)
% file 'examples/pretzeltangles/PT6m7/cxKhr-c5', generated by kht++ v0.1-99-g54450e6-dirty on Mon Dec 29 17:40:02 2025 GMT
1) h^-1 q^ -2 δ^0 ⬮<~(-S)~~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<— r2(-1/2) q^0 δ|^0
2) h^-5 q^-10 δ^0 ⬮<~(-S)~~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮<—⬮<~⬮<—⬮<~⬮<— r(1/6) q^-48/7 δ|^0
3) h^-6 q^-12 δ^0 ⬮<~(-S)~~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<— r2(-1/6) q^-2 δ|^0
4) h^-2 q^ -4 δ^0 ⬮<~(-S)~~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<— r2(-1/2) q^-2 δ|^0
5) h^-5 q^-10 δ^0 ⬮<~(-S)~~⬯—>⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯<—⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<— r2(-1/4) q^-2 δ|^0
6) h^-2 q^ -4 δ^0 ⬮<~(-S^2)~~⬮<—⬮<~⬯—>⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯<—⬯<~⬮<—⬮<~⬮<—⬮<~⬮<— r2(-1/4) q^0 δ|^0
options -c7
(diagram)
(metadata)
% file 'examples/pretzeltangles/PT6m7/cxKhr-c7', generated by kht++ v0.1-99-g54450e6-dirty on Mon Dec 29 17:40:02 2025 GMT
1) h^-6 q^-12 δ^0 ⬮<~(-S)~~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<— r2(-1/6) q^-2 δ|^0
2) h^-4 q^ -8 δ^0 ⬮<~(-S)~~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<— r2(-1/4) q^-2 δ|^0
3) h^-1 q^ -2 δ^0 ⬮<~(-S)~~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<— r2(-1/2) q^0 δ|^0
4) h^-3 q^ -6 δ^0 ⬮<~(-S)~~⬯—>⬯~>⬮—>⬮~>⬮—>⬮~>⬯<—⬯<~⬮<—⬮<~⬮<— r2(-1/2) q^-2 δ|^0
5) h^-4 q^ -8 δ^0 ⬮<~(-S)~~⬯—>⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯<—⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<— r2(-1/4) q^0 δ|^0
6) h^-1 q^ -2 δ^0 ⬮<~(-S^2)~~⬮<—⬮<~⬮<—⬮<~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮<— r(1/6) q^-48/7 δ|^0
options -c11
(diagram)
(metadata)
% file 'examples/pretzeltangles/PT6m7/cxKhr-c11', generated by kht++ v0.1-99-g54450e6-dirty on Mon Dec 29 17:40:02 2025 GMT
1) h^-4 q^-8 δ^0 ⬮<~(-S)~~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬮<— r2(-1/4) q^-2 δ|^0
2) h^-2 q^-4 δ^0 ⬮<~(-S)~~⬯—>⬯~>⬮—>⬮~>⬮—>⬮~>⬯<—⬯<~⬮<—⬮<~⬮<— r2(-1/2) q^0 δ|^0
3) h^ 1 q^ 2 δ^0 ⬮<~(-S^2)~~⬮<—⬮<~⬮<—⬮<~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<—⬮<~⬮<— r2(-1/4) q^0 δ|^0
4) h^-1 q^-2 δ^0 ⬮<~(-S^2)~~⬮<—⬮<~⬮<—⬮<~⬮<—⬮<~⬯—>⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮~>⬯<—⬯<~⬮<—⬮<~⬮<—⬮<~⬮<— r2(-1/6) q^-2 δ|^0
5) h^-2 q^-4 δ^0 ⬮<~(-S^2)~~⬮<—⬮<~⬮<—⬮<~⬯—>⬯~>⬮—>⬮~>⬮—>⬮~>⬮—>⬮<~⬮<— r(1/6) q^-48/7 δ|^0
6) h^ 0 q^ 0 δ^0 ⬮<~(-S^2)~~⬮<—⬮<~⬯<—⬯~>⬮—>⬮~>⬮—>⬮~>⬯—>⬯<~⬮<— r2(-1/2) q^-2 δ|^0