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

Flat knot 5.99

Min(phi) over symmetries of the knot is: [-2,0,0,1,1,0,1,2,3,-1,1,0,1,0,0]
Flat knots (up to 7 crossings) with same phi are :['5.99', '7.33689', '7.39898']
Arrow polynomial of the knot is: 6*K1**2 - 3*K2 - 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['5.64', '5.71', '5.73', '5.76', '5.99', '5.100', '5.107', '7.12972', '7.14218', '7.17628', '7.18232', '7.18593', '7.18877', '7.19146', '7.19768', '7.19915', '7.19927', '7.19944', '7.20401', '7.20414', '7.20656', '7.20895', '7.20986', '7.21121', '7.21123', '7.21128', '7.21130', '7.21132', '7.21143', '7.21221', '7.21234', '7.21243', '7.21250', '7.21258', '7.21270', '7.21276', '7.21299', '7.21356', '7.21367', '7.21441', '7.21452', '7.21577', '7.21660', '7.21665', '7.21766', '7.22362', '7.23073', '7.23081', '7.23288', '7.23366', '7.24182', '7.24406', '7.24429', '7.24452', '7.24454', '7.24457', '7.24690', '7.24700', '7.24703', '7.24721', '7.24725', '7.24747', '7.24752', '7.24769', '7.24785', '7.24789', '7.24806', '7.24837', '7.24861', '7.24864', '7.24872', '7.24876', '7.24912', '7.24947', '7.24953', '7.24959', '7.24996', '7.24999', '7.25492', '7.25616', '7.25849', '7.25871', '7.25908', '7.25922', '7.25999', '7.26007', '7.26047', '7.26439', '7.26616', '7.26945', '7.26954', '7.26979', '7.26990', '7.27010', '7.27018', '7.27042', '7.27054', '7.27062', '7.27089', '7.27154', '7.27182', '7.27205', '7.27208', '7.27223', '7.27237', '7.27291', '7.27301', '7.27324', '7.27345', '7.27366', '7.27392', '7.27407', '7.27502', '7.27514', '7.27519', '7.27562', '7.27563', '7.27567', '7.27597', '7.27611', '7.27618', '7.27619', '7.27625', '7.27747', '7.27753', '7.27783', '7.27835', '7.27892', '7.27920', '7.28338', '7.28341', '7.28353', '7.28389', '7.28469', '7.28494', '7.28584', '7.28949', '7.28980', '7.29104', '7.29119', '7.29688', '7.29854', '7.29856', '7.29857', '7.30134', '7.30164', '7.30513', '7.30562', '7.30566', '7.30647', '7.30673', '7.30941', '7.30967', '7.31372', '7.31377', '7.31587', '7.31671', '7.31673', '7.31712', '7.31732', '7.31735', '7.31751', '7.31761', '7.31783', '7.31964', '7.31976', '7.32074', '7.32080', '7.32122', '7.32152', '7.32159', '7.32184', '7.32186', '7.32201', '7.32220', '7.32274', '7.32282', '7.32300', '7.32341', '7.32387', '7.32394', '7.32415', '7.32455', '7.32456', '7.32457', '7.32468', '7.32476', '7.32507', '7.32519', '7.32522', '7.32529', '7.32545', '7.32550', '7.32554', '7.32582', '7.32589', '7.32598', '7.32602', '7.32605', '7.32639', '7.32669', '7.32671', '7.32686', '7.32688', '7.32694', '7.32699', '7.32703', '7.32710', '7.32715', '7.32749', '7.32758', '7.32782', '7.32857', '7.32858', '7.32892', '7.32899', '7.32904', '7.32915', '7.32950', '7.33050', '7.33051', '7.33179', '7.33181', '7.33222', '7.33249', '7.33251', '7.33513', '7.33564', '7.33595', '7.33735', '7.33818', '7.33828', '7.33968', '7.34162', '7.34163', '7.34505', '7.34534', '7.34544', '7.34569', '7.34570', '7.34661', '7.34695', '7.34704', '7.34705', '7.34756', '7.34948', '7.34993', '7.35012', '7.35038', '7.35046', '7.35047', '7.35146', '7.35147', '7.35166', '7.35186', '7.35279', '7.35283', '7.35341', '7.35354', '7.35355', '7.35358', '7.35359', '7.35366', '7.35370', '7.35373', '7.35374', '7.35392', '7.35399', '7.35404', '7.35405', '7.35407', '7.35412', '7.35421', '7.35430', '7.35436', '7.35450', '7.35451', '7.35453', '7.35454', '7.35455', '7.35456', '7.35458', '7.35459', '7.35460', '7.35461', '7.35465', '7.35467', '7.35470', '7.35480', '7.35482', '7.35484', '7.35486', '7.35492', '7.35494', '7.35502', '7.35506', '7.35514', '7.35517', '7.35518', '7.35519', '7.35520', '7.35522', '7.35524', '7.35527', '7.35532', '7.35533', '7.35535', '7.35538', '7.35554', '7.35555', '7.35558', '7.35562', '7.35563', '7.35568', '7.35569', '7.35572', '7.35573', '7.35577', '7.35584', '7.35588', '7.35589', '7.35590', '7.35609', '7.35614', '7.35618', '7.35625', '7.35630', '7.35644', '7.35647', '7.35656', '7.35660', '7.35665', '7.35666', '7.35672', '7.35681', '7.35688', '7.35692', '7.35719', '7.35778', '7.35791', '7.35815', '7.35822', '7.35837', '7.35864', '7.35878', '7.35950', '7.36006', '7.36016', '7.36018', '7.36094', '7.36124', '7.36135', '7.36144', '7.36164', '7.36175', '7.36178', '7.36181', '7.36188', '7.36227', '7.36274', '7.36279', '7.36280', '7.36284', '7.36285', '7.36300', '7.36301', '7.36309', '7.36313', '7.36316', '7.36328', '7.36339', '7.36346', '7.36350', '7.36359', '7.36363', '7.36378', '7.36382', '7.36392', '7.36396', '7.36398', '7.36403', '7.36407', '7.36418', '7.36424', '7.36431', '7.36436', '7.36445', '7.36447', '7.36449', '7.36455', '7.36456', '7.36462', '7.36463', '7.36467', '7.36476', '7.36478', '7.36480', '7.36486', '7.36489', '7.36494', '7.36501', '7.36504', '7.36509', '7.36513', '7.36518', '7.36524', '7.36529', '7.36532', '7.36533', '7.36538', '7.36544', '7.36552', '7.36556', '7.36558', '7.36587', '7.36590', '7.36593', '7.36597', '7.36599', '7.36601', '7.36604', '7.36607', '7.36609', '7.36617', '7.36618', '7.36621', '7.36625', '7.36665', '7.36672', '7.36673', '7.36675', '7.36684', '7.36685', '7.36686', '7.36687', '7.36689', '7.36708', '7.36709', '7.36713', '7.36723', '7.36724', '7.36726', '7.36775', '7.36817', '7.36881', '7.36936', '7.36964', '7.36972', '7.37016', '7.37027', '7.37029', '7.37038', '7.37064', '7.37096', '7.37097', '7.37108', '7.37119', '7.37140', '7.37182', '7.37195', '7.37200', '7.37202', '7.37205', '7.37206', '7.37216', '7.37228', '7.37231', '7.37232', '7.37235', '7.37258', '7.37265', '7.37308', '7.37310', '7.37402', '7.37404', '7.37430', '7.37450', '7.37458', '7.37465', '7.37466', '7.37470', '7.37471', '7.37477', '7.37489', '7.37527', '7.37560', '7.37589', '7.37623', '7.37706', '7.37749', '7.37752', '7.37763', '7.37787', '7.37791', '7.37807', '7.37828', '7.37870', '7.37931', '7.37934', '7.37950', '7.37955', '7.38025', '7.38080', '7.38086', '7.38116', '7.38120', '7.38141', '7.38228', '7.38238', '7.38307', '7.38342', '7.38359', '7.38402', '7.38409', '7.38439', '7.38440', '7.38447', '7.38531', '7.38535', '7.38542', '7.38550', '7.38583', '7.38586', '7.38662', '7.38748', '7.38767', '7.38875', '7.38944', '7.38975', '7.38976', '7.38977', '7.38978', '7.38979', '7.38980', '7.38981', '7.38985', '7.38989', '7.38996', '7.38997', '7.38999', '7.39001', '7.39003', '7.39006', '7.39007', '7.39020', '7.39064', '7.39069', '7.39072', '7.39073', '7.39076', '7.39084', '7.39090', '7.39095', '7.39096', '7.39100', '7.39103', '7.39106', '7.39110', '7.39111', '7.39124', '7.39130', '7.39132', '7.39133', '7.39140', '7.39142', '7.39146', '7.39150', '7.39154', '7.39159', '7.39167', '7.39170', '7.39171', '7.39172', '7.39214', '7.39216', '7.39224', '7.39227', '7.39230', '7.39241', '7.39256', '7.39340', '7.39347', '7.39402', '7.39469', '7.39491', '7.39511', '7.39606', '7.39628', '7.39648', '7.39679', '7.39720', '7.39726', '7.39804', '7.39805', '7.40031', '7.40071', '7.40073', '7.40093', '7.40349', '7.40615', '7.40654', '7.40657', '7.40662', '7.40674', '7.40675', '7.40681', '7.40687', '7.40764', '7.40788', '7.40789', '7.40794', '7.40795', '7.40810', '7.40811', '7.40812', '7.40816', '7.40818', '7.40832', '7.40837', '7.40839', '7.40846', '7.40850', '7.40871', '7.40876', '7.40877', '7.40891', '7.40927', '7.40992', '7.40995', '7.40996', '7.41045', '7.41384', '7.41501', '7.41609', '7.41674', '7.41696', '7.41786', '7.41788', '7.41793', '7.41805', '7.41811', '7.41842', '7.41895', '7.41898', '7.41938', '7.41950', '7.41971', '7.41977', '7.42187', '7.42194', '7.42208', '7.42239', '7.42265', '7.42301', '7.42324', '7.42358', '7.42445', '7.42484', '7.42541', '7.42546', '7.42569', '7.42631', '7.42664', '7.42821', '7.42887', '7.42896', '7.42952', '7.43156', '7.43159', '7.43163', '7.43351', '7.43427', '7.43623', '7.43624', '7.43629', '7.43633', '7.43644', '7.43646', '7.43651', '7.43652', '7.43856', '7.44048', '7.44287', '7.44294', '7.44297', '7.44302', '7.44303', '7.44309', '7.44314', '7.44315', '7.44318', '7.44323', '7.44339', '7.44351', '7.44359', '7.44360', '7.44361', '7.44368', '7.44369', '7.44370', '7.44371', '7.44372', '7.44375', '7.44386', '7.44387', '7.44398', '7.44402', '7.44404', '7.44410', '7.44411', '7.44412', '7.44420', '7.44423', '7.44437', '7.44439', '7.44441', '7.44442', '7.44443', '7.44448', '7.44449', '7.44450', '7.44451', '7.44454', '7.44459', '7.44465', '7.44466', '7.44468', '7.44470', '7.44471', '7.44473', '7.44476', '7.44477', '7.44480', '7.44481', '7.44482', '7.44485', '7.44489', '7.44491', '7.44493', '7.44494', '7.44495', '7.44497', '7.44499', '7.44503', '7.44505', '7.44518', '7.44519', '7.44524', '7.44525', '7.44528', '7.44529', '7.44531', '7.44532', '7.44533', '7.44536', '7.44538', '7.44540', '7.44542', '7.44544', '7.44547', '7.44551', '7.44553', '7.44555', '7.44557', '7.44562', '7.44563', '7.44566', '7.44568', '7.44571', '7.44572', '7.44576', '7.44577', '7.44578', '7.44583', '7.44587', '7.44589', '7.44590', '7.44591', '7.44592', '7.44593', '7.44595', '7.44597', '7.44601', '7.44615', '7.44621', '7.44623', '7.44624', '7.44627', '7.44642', '7.44655', '7.44750', '7.44832', '7.44861', '7.44920', '7.44934', '7.45034', '7.45036', '7.45078', '7.45085', '7.45086', '7.45133', '7.45139', '7.45159', '7.45165', '7.45181']
Outer characteristic polynomial of the knot is: t^6+15t^4+19t^2
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.99', '7.33689', '7.39898', '7.44753']
2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 432*K1**4 + 32*K1**3*K2*K3 - 272*K1**2*K2**2 + 728*K1**2*K2 - 48*K1**2*K3**2 - 380*K1**2 + 392*K1*K2*K3 + 24*K1*K3*K4 - 24*K2**4 + 40*K2**2*K4 - 384*K2**2 - 140*K3**2 - 18*K4**2 + 384
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.99']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.177', 'vk5.181', 'vk5.226', 'vk5.230', 'vk5.320', 'vk5.324', 'vk5.368', 'vk5.671', 'vk5.673', 'vk5.822', 'vk5.1335', 'vk5.1422', 'vk5.1426', 'vk5.1453', 'vk5.1692', 'vk5.1694']
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 O1O2O3U1U3O4U2O5U4U5
R3 orbit {'O1O2O3U1U3O4U2O5U4U5'}
R3 orbit length 1
Gauss code of -K O1O2O3U4U5O4U2O5U1U3
Gauss code of K* O1O2U3O4U5O3O5U1U4U2
Gauss code of -K* O1O2U1O3U2O4O5U4U3U5
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -2 0 1 0 1],[ 2 0 2 1 1 0],[ 0 -2 0 0 1 1],[-1 -1 0 0 0 0],[ 0 -1 -1 0 0 1],[-1 0 -1 0 -1 0]]
Primitive based matrix [[ 0 1 1 0 0 -2],[-1 0 0 0 0 -1],[-1 0 0 -1 -1 0],[ 0 0 1 0 1 -2],[ 0 0 1 -1 0 -1],[ 2 1 0 2 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,0,0,2,0,0,0,1,1,1,0,-1,2,1]
Phi over symmetry [-2,0,0,1,1,0,1,2,3,-1,1,0,1,0,0]
Phi of -K [-2,0,0,1,1,0,1,2,3,-1,1,0,1,0,0]
Phi of K* [-1,-1,0,0,2,0,0,0,3,1,1,2,-1,1,0]
Phi of -K* [-2,0,0,1,1,1,2,0,1,-1,1,0,1,0,0]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial -7z-13
Enhanced Jones-Krushkal polynomial -7w^2z-13w
Inner characteristic polynomial t^5+9t^3+4t
Outer characteristic polynomial t^6+15t^4+19t^2
Flat arrow polynomial 6*K1**2 - 3*K2 - 2
2-strand cable arrow polynomial 96*K1**4*K2 - 432*K1**4 + 32*K1**3*K2*K3 - 272*K1**2*K2**2 + 728*K1**2*K2 - 48*K1**2*K3**2 - 380*K1**2 + 392*K1*K2*K3 + 24*K1*K3*K4 - 24*K2**4 + 40*K2**2*K4 - 384*K2**2 - 140*K3**2 - 18*K4**2 + 384
Genus of based matrix 1
Fillings of based matrix [[{1, 5}, {3, 4}, {2}], [{2, 5}, {3, 4}, {1}], [{3, 5}, {1, 4}, {2}], [{4, 5}, {1, 3}, {2}], [{5}, {3, 4}, {1, 2}], [{5}, {3, 4}, {2}, {1}]]
If K is slice False
Contact