Table of flat knot invariants
Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo
Glossary Reference List

Flat knot 5.48

Min(phi) over symmetries of the knot is: [-2,-2,1,1,2,-1,1,1,3,0,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['5.48', '7.32766']
Arrow polynomial of the knot is: 6*K1**2 + 4*K1*K2 - 2*K1 - 3*K2 - 2*K3 - 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['5.22', '5.48', '5.85', '5.96', '7.2267', '7.2278', '7.3359', '7.4682', '7.4684', '7.4701', '7.4863', '7.5192', '7.5252', '7.5422', '7.5520', '7.5604', '7.5955', '7.5956', '7.5959', '7.6468', '7.6488', '7.6515', '7.6560', '7.6909', '7.6967', '7.7037', '7.7181', '7.7988', '7.8046', '7.8072', '7.8119', '7.8399', '7.8471', '7.8538', '7.8822', '7.8825', '7.8950', '7.9078', '7.9769', '7.9770', '7.9915', '7.10248', '7.10261', '7.10266', '7.10278', '7.10763', '7.10768', '7.10770', '7.10924', '7.10962', '7.11304', '7.11823', '7.11981', '7.12042', '7.12102', '7.12181', '7.12189', '7.12315', '7.12325', '7.12375', '7.12428', '7.12542', '7.12551', '7.12738', '7.12859', '7.12894', '7.12943', '7.13610', '7.13694', '7.13737', '7.13813', '7.13950', '7.14057', '7.14303', '7.14369', '7.14550', '7.14925', '7.14934', '7.15058', '7.15084', '7.15127', '7.15146', '7.15454', '7.15480', '7.15515', '7.15643', '7.15666', '7.15674', '7.15978', '7.16030', '7.16144', '7.16508', '7.16513', '7.16667', '7.16684', '7.16913', '7.17008', '7.17012', '7.17016', '7.17236', '7.17441', '7.17451', '7.17669', '7.17692', '7.17723', '7.17811', '7.17834', '7.17996', '7.17998', '7.18034', '7.18047', '7.18155', '7.18474', '7.18674', '7.18692', '7.18706', '7.18922', '7.18937', '7.19014', '7.19351', '7.19365', '7.19373', '7.19376', '7.19415', '7.19590', '7.19815', '7.19841', '7.20248', '7.20273', '7.20367', '7.20441', '7.20447', '7.20891', '7.20982', '7.21294', '7.21312', '7.21333', '7.21341', '7.21388', '7.21412', '7.21416', '7.21478', '7.21492', '7.21645', '7.21685', '7.21705', '7.21738', '7.22574', '7.22596', '7.22669', '7.22771', '7.22780', '7.22803', '7.22828', '7.22845', '7.22861', '7.23368', '7.23438', '7.23462', '7.23464', '7.23467', '7.23500', '7.23551', '7.23560', '7.23900', '7.23909', '7.24153', '7.24270', '7.24287', '7.24348', '7.24484', '7.24693', '7.24813', '7.24826', '7.24858', '7.25050', '7.25061', '7.25068', '7.25327', '7.25464', '7.25530', '7.25653', '7.25680', '7.25803', '7.25805', '7.25810', '7.25833', '7.25973', '7.25977', '7.26023', '7.26103', '7.26169', '7.26171', '7.26176', '7.26194', '7.26250', '7.26254', '7.26356', '7.26445', '7.26476', '7.26528', '7.26553', '7.26572', '7.26638', '7.26652', '7.26679', '7.26759', '7.26800', '7.26831', '7.26944', '7.26953', '7.26994', '7.27027', '7.27125', '7.27146', '7.27150', '7.27172', '7.27221', '7.27341', '7.27348', '7.27356', '7.27508', '7.27580', '7.27581', '7.27639', '7.27656', '7.27663', '7.27710', '7.27848', '7.27948', '7.28161', '7.28189', '7.28229', '7.28261', '7.28366', '7.28372', '7.28376', '7.28456', '7.28515', '7.28543', '7.28548', '7.28605', '7.28662', '7.28713', '7.29034', '7.29036', '7.29040', '7.29068', '7.29078', '7.29086', '7.29466', '7.29592', '7.29670', '7.29672', '7.29796', '7.29809', '7.30443', '7.30490', '7.30510', '7.30550', '7.30552', '7.30558', '7.30817', '7.30901', '7.30961', '7.30975', '7.31017', '7.31062', '7.31095', '7.31123', '7.31194', '7.31248', '7.31266', '7.31617', '7.31644', '7.31797', '7.31987', '7.32060', '7.32126', '7.32171', '7.32191', '7.32574', '7.32593', '7.32744', '7.32746', '7.33418', '7.33449', '7.33943', '7.33986', '7.34058', '7.34165', '7.34198', '7.34306', '7.34314', '7.34333', '7.34605', '7.34621', '7.34687', '7.34688', '7.34774', '7.34869', '7.34890', '7.34975', '7.34995', '7.34998', '7.35088', '7.35095', '7.35143', '7.35168', '7.35994', '7.35997', '7.36030', '7.36032', '7.36117', '7.37151', '7.37153', '7.37303', '7.37353', '7.37367', '7.37390', '7.37398', '7.37456', '7.37457', '7.37617', '7.37631', '7.37632', '7.37637', '7.37652', '7.37672', '7.37673', '7.37682', '7.37683', '7.37703', '7.37726', '7.37727', '7.37739', '7.37740', '7.37843', '7.37844', '7.37891', '7.37895', '7.37914', '7.37915', '7.38030', '7.38108', '7.38139', '7.38152', '7.38244', '7.38327', '7.38371', '7.38381', '7.38464', '7.38487', '7.38499', '7.38537', '7.38553', '7.38576', '7.38627', '7.38680', '7.38692', '7.38702', '7.38704', '7.38736', '7.38772', '7.38786', '7.38816', '7.38826', '7.38861', '7.38869', '7.38870', '7.38878', '7.38880', '7.38891', '7.38932', '7.39488', '7.39516', '7.39517', '7.39581', '7.39608', '7.39626', '7.39634', '7.39667', '7.39706', '7.39724', '7.39844', '7.40133', '7.40224', '7.40263', '7.40281', '7.40348', '7.40369', '7.40400', '7.40407', '7.40441', '7.40476', '7.40491', '7.40499', '7.40516', '7.40537', '7.40593', '7.40595', '7.40631', '7.40700', '7.40701', '7.40702', '7.40709', '7.40722', '7.40723', '7.40728', '7.40744', '7.40745', '7.40893', '7.40894', '7.40914', '7.40915', '7.41082', '7.41138', '7.41164', '7.41182', '7.41228', '7.41232', '7.41303', '7.41309', '7.41321', '7.41323', '7.41353', '7.41454', '7.41509', '7.41527', '7.41602', '7.41701', '7.41774', '7.41980', '7.41991', '7.42045', '7.42046', '7.42047', '7.42061', '7.42328', '7.42332', '7.42333', '7.42334', '7.42348', '7.42353', '7.42544', '7.42576', '7.42643', '7.42660', '7.42930', '7.42956', '7.42986', '7.43002', '7.43175', '7.43211', '7.43213', '7.43228', '7.43232', '7.43249', '7.43251', '7.43319', '7.43342', '7.43397', '7.43555', '7.43616', '7.43817', '7.43825', '7.43937', '7.43944', '7.44030', '7.44042', '7.44090', '7.44099', '7.44122', '7.44199', '7.44713', '7.44717', '7.44757', '7.44761', '7.44894', '7.44895', '7.44911', '7.44912', '7.44960']
Outer characteristic polynomial of the knot is: t^6+31t^4+15t^2+1
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.48']
2-strand cable arrow polynomial of the knot is: 160*K1**4*K2 - 464*K1**4 + 32*K1**3*K2*K3 - 288*K1**3*K3 - 432*K1**2*K2**2 - 32*K1**2*K2*K4 + 1744*K1**2*K2 - 176*K1**2*K3**2 - 1512*K1**2 - 192*K1*K2**2*K3 - 96*K1*K2*K3*K4 + 1384*K1*K2*K3 + 384*K1*K3*K4 + 72*K1*K4*K5 - 24*K2**4 - 48*K2**2*K3**2 - 16*K2**2*K4**2 + 248*K2**2*K4 - 1180*K2**2 + 136*K2*K3*K5 + 32*K2*K4*K6 - 604*K3**2 - 234*K4**2 - 68*K5**2 - 12*K6**2 + 1184
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.48']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.141', 'vk5.152', 'vk5.158', 'vk5.259', 'vk5.269', 'vk5.276', 'vk5.286', 'vk5.567', 'vk5.577', 'vk5.1040', 'vk5.1055', 'vk5.1352', 'vk5.1363', 'vk5.1369', 'vk5.1584', 'vk5.1594']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3O4U2U4O5U1U5U3
R3 orbit {'O1O2O3O4U2U4O5U1U5U3'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U2U5U4O5U1U3
Gauss code of K* O1O2O3U1U4U3U5O4O5U2
Gauss code of -K* O1O2O3U2O4O5U4U1U5U3
Diagrammatic symmetry type c
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 -2 2 1 1],[ 2 0 -1 3 1 1],[ 2 1 0 2 1 0],[-2 -3 -2 0 0 0],[-1 -1 -1 0 0 0],[-1 -1 0 0 0 0]]
Primitive based matrix [[ 0 2 1 1 -2 -2],[-2 0 0 0 -2 -3],[-1 0 0 0 0 -1],[-1 0 0 0 -1 -1],[ 2 2 0 1 0 1],[ 2 3 1 1 -1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,2,2,0,0,2,3,0,0,1,1,1,-1]
Phi over symmetry [-2,-2,1,1,2,-1,1,1,3,0,1,2,0,0,0]
Phi of -K [-2,-2,1,1,2,-1,2,3,2,2,2,1,0,1,1]
Phi of K* [-2,-1,-1,2,2,1,1,1,2,0,2,2,2,3,-1]
Phi of -K* [-2,-2,1,1,2,-1,1,1,3,0,1,2,0,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial -11z-21
Enhanced Jones-Krushkal polynomial -11w^2z-21w
Inner characteristic polynomial t^5+17t^3+6t
Outer characteristic polynomial t^6+31t^4+15t^2+1
Flat arrow polynomial 6*K1**2 + 4*K1*K2 - 2*K1 - 3*K2 - 2*K3 - 2
2-strand cable arrow polynomial 160*K1**4*K2 - 464*K1**4 + 32*K1**3*K2*K3 - 288*K1**3*K3 - 432*K1**2*K2**2 - 32*K1**2*K2*K4 + 1744*K1**2*K2 - 176*K1**2*K3**2 - 1512*K1**2 - 192*K1*K2**2*K3 - 96*K1*K2*K3*K4 + 1384*K1*K2*K3 + 384*K1*K3*K4 + 72*K1*K4*K5 - 24*K2**4 - 48*K2**2*K3**2 - 16*K2**2*K4**2 + 248*K2**2*K4 - 1180*K2**2 + 136*K2*K3*K5 + 32*K2*K4*K6 - 604*K3**2 - 234*K4**2 - 68*K5**2 - 12*K6**2 + 1184
Genus of based matrix 1
Fillings of based matrix [[{3, 5}, {1, 4}, {2}], [{3, 5}, {2, 4}, {1}], [{3, 5}, {4}, {1, 2}]]
If K is slice False
Contact