10 123

10_122

10_124

Visit 10 123's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 123's page at Knotilus! Visit 10 123's page at the original Knot Atlas! 10_123 can be depicted with five-fold rotational symmetry (like 5 1).

Quasi-floral decorative knot.

Decorative pentagonal representation.

Symmetrical "flower".

Cylindrical depiction

Knot presentations

Planar diagram presentation	X8291 X10,3,11,4 X12,6,13,5 X4,18,5,17 X18,11,19,12 X2,15,3,16 X16,10,17,9 X20,14,1,13 X14,7,15,8 X6,19,7,20
Gauss code	1, -6, 2, -4, 3, -10, 9, -1, 7, -2, 5, -3, 8, -9, 6, -7, 4, -5, 10, -8
Dowker-Thistlethwaite code	8 10 12 14 16 18 20 2 4 6
Conway Notation	[10*]

Minimum Braid Representative: Length is 10, width is 3. Braid index is 3.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Fully amphicheiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 17.0857 A-Polynomial See Data:10 123/A-polynomial

[edit Notes for 10 123's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $0$ Topological 4 genus $0$ Concordance genus $0$ Rasmussen s-Invariant 0

[edit Notes for 10 123's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+15t^{2}-24t+29-24t^{-1}+15t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}-z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{t^{4}-3t^{3}+3t^{2}-3t+1\right\}$
Determinant and Signature	{ 121, 0 }
Jones polynomial	$-q^{5}+5q^{4}-10q^{3}+15q^{2}-19q+21-19q^{-1}+15q^{-2}-10q^{-3}+5q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+4z^{6}-2a^{2}z^{4}-2z^{4}a^{-2}+3z^{4}+a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+2a^{2}+2a^{-2}-3$
Kauffman polynomial (db, data sources)	$4az^{9}+4z^{9}a^{-1}+10a^{2}z^{8}+10z^{8}a^{-2}+20z^{8}+10a^{3}z^{7}+14az^{7}+14z^{7}a^{-1}+10z^{7}a^{-3}+5a^{4}z^{6}-11a^{2}z^{6}-11z^{6}a^{-2}+5z^{6}a^{-4}-32z^{6}+a^{5}z^{5}-15a^{3}z^{5}-38az^{5}-38z^{5}a^{-1}-15z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}-3a^{2}z^{4}-3z^{4}a^{-2}-5z^{4}a^{-4}+4z^{4}+5a^{3}z^{3}+21az^{3}+21z^{3}a^{-1}+5z^{3}a^{-3}+6a^{2}z^{2}+6z^{2}a^{-2}+12z^{2}-2az-2za^{-1}-2a^{2}-2a^{-2}-3$
The A2 invariant	$-q^{14}+3q^{12}-2q^{10}+3q^{8}-3q^{4}+4q^{2}-5+4q^{-2}-3q^{-4}+3q^{-8}-2q^{-10}+3q^{-12}-q^{-14}$
The G2 invariant	$q^{80}-4q^{78}+10q^{76}-20q^{74}+26q^{72}-25q^{70}+10q^{68}+30q^{66}-80q^{64}+140q^{62}-180q^{60}+158q^{58}-71q^{56}-100q^{54}+308q^{52}-473q^{50}+528q^{48}-391q^{46}+69q^{44}+343q^{42}-693q^{40}+822q^{38}-656q^{36}+239q^{34}+267q^{32}-647q^{30}+750q^{28}-495q^{26}+29q^{24}+435q^{22}-675q^{20}+551q^{18}-133q^{16}-414q^{14}+836q^{12}-944q^{10}+710q^{8}-173q^{6}-472q^{4}+970q^{2}-1165+970q^{-2}-472q^{-4}-173q^{-6}+710q^{-8}-944q^{-10}+836q^{-12}-414q^{-14}-133q^{-16}+551q^{-18}-675q^{-20}+435q^{-22}+29q^{-24}-495q^{-26}+750q^{-28}-647q^{-30}+267q^{-32}+239q^{-34}-656q^{-36}+822q^{-38}-693q^{-40}+343q^{-42}+69q^{-44}-391q^{-46}+528q^{-48}-473q^{-50}+308q^{-52}-100q^{-54}-71q^{-56}+158q^{-58}-180q^{-60}+140q^{-62}-80q^{-64}+30q^{-66}+10q^{-68}-25q^{-70}+26q^{-72}-20q^{-74}+10q^{-76}-4q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_123.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+4q^{9}-5q^{7}+5q^{5}-4q^{3}+2q+2q^{-1}-4q^{-3}+5q^{-5}-5q^{-7}+4q^{-9}-q^{-11}$
2	$q^{32}-4q^{30}+q^{28}+15q^{26}-21q^{24}-12q^{22}+53q^{20}-27q^{18}-51q^{16}+75q^{14}-q^{12}-76q^{10}+49q^{8}+31q^{6}-53q^{4}-q^{2}+45-q^{-2}-53q^{-4}+31q^{-6}+49q^{-8}-76q^{-10}-q^{-12}+75q^{-14}-51q^{-16}-27q^{-18}+53q^{-20}-12q^{-22}-21q^{-24}+15q^{-26}+q^{-28}-4q^{-30}+q^{-32}$
3	$-q^{63}+4q^{61}-q^{59}-11q^{57}+q^{55}+27q^{53}+12q^{51}-73q^{49}-38q^{47}+131q^{45}+126q^{43}-184q^{41}-289q^{39}+185q^{37}+493q^{35}-82q^{33}-682q^{31}-127q^{29}+783q^{27}+387q^{25}-754q^{23}-619q^{21}+601q^{19}+772q^{17}-376q^{15}-810q^{13}+130q^{11}+751q^{9}+102q^{7}-636q^{5}-298q^{3}+478q+478q^{-1}-298q^{-3}-636q^{-5}+102q^{-7}+751q^{-9}+130q^{-11}-810q^{-13}-376q^{-15}+772q^{-17}+601q^{-19}-619q^{-21}-754q^{-23}+387q^{-25}+783q^{-27}-127q^{-29}-682q^{-31}-82q^{-33}+493q^{-35}+185q^{-37}-289q^{-39}-184q^{-41}+126q^{-43}+131q^{-45}-38q^{-47}-73q^{-49}+12q^{-51}+27q^{-53}+q^{-55}-11q^{-57}-q^{-59}+4q^{-61}-q^{-63}$
5	$-q^{155}+4q^{153}-q^{151}-11q^{149}+5q^{147}+11q^{145}+7q^{143}-3q^{141}-28q^{139}-43q^{137}+23q^{135}+137q^{133}+116q^{131}-95q^{129}-381q^{127}-416q^{125}+53q^{123}+958q^{121}+1448q^{119}+427q^{117}-1828q^{115}-3593q^{113}-2582q^{111}+1979q^{109}+7323q^{107}+7942q^{105}+557q^{103}-11096q^{101}-17370q^{99}-9355q^{97}+11333q^{95}+29603q^{93}+26624q^{91}-2590q^{89}-39209q^{87}-51313q^{85}-19986q^{83}+38304q^{81}+77229q^{79}+56261q^{77}-19581q^{75}-93770q^{73}-99666q^{71}-19217q^{69}+90984q^{67}+138225q^{65}+72047q^{63}-64007q^{61}-159134q^{59}-126635q^{57}+16445q^{55}+154719q^{53}+168882q^{51}+40575q^{49}-125523q^{47}-188457q^{45}-93249q^{43}+79592q^{41}+182862q^{39}+130390q^{37}-29132q^{35}-156911q^{33}-146829q^{31}-15052q^{29}+119894q^{27}+144524q^{25}+46592q^{23}-81740q^{21}-129908q^{19}-64506q^{17}+49163q^{15}+110867q^{13}+72745q^{11}-24867q^{9}-94167q^{7}-76937q^{5}+7459q^{3}+82883q+82883q^{-1}+7459q^{-3}-76937q^{-5}-94167q^{-7}-24867q^{-9}+72745q^{-11}+110867q^{-13}+49163q^{-15}-64506q^{-17}-129908q^{-19}-81740q^{-21}+46592q^{-23}+144524q^{-25}+119894q^{-27}-15052q^{-29}-146829q^{-31}-156911q^{-33}-29132q^{-35}+130390q^{-37}+182862q^{-39}+79592q^{-41}-93249q^{-43}-188457q^{-45}-125523q^{-47}+40575q^{-49}+168882q^{-51}+154719q^{-53}+16445q^{-55}-126635q^{-57}-159134q^{-59}-64007q^{-61}+72047q^{-63}+138225q^{-65}+90984q^{-67}-19217q^{-69}-99666q^{-71}-93770q^{-73}-19581q^{-75}+56261q^{-77}+77229q^{-79}+38304q^{-81}-19986q^{-83}-51313q^{-85}-39209q^{-87}-2590q^{-89}+26624q^{-91}+29603q^{-93}+11333q^{-95}-9355q^{-97}-17370q^{-99}-11096q^{-101}+557q^{-103}+7942q^{-105}+7323q^{-107}+1979q^{-109}-2582q^{-111}-3593q^{-113}-1828q^{-115}+427q^{-117}+1448q^{-119}+958q^{-121}+53q^{-123}-416q^{-125}-381q^{-127}-95q^{-129}+116q^{-131}+137q^{-133}+23q^{-135}-43q^{-137}-28q^{-139}-3q^{-141}+7q^{-143}+11q^{-145}+5q^{-147}-11q^{-149}-q^{-151}+4q^{-153}-q^{-155}$
6	$q^{216}-4q^{214}+q^{212}+11q^{210}-5q^{208}-11q^{206}-11q^{204}+23q^{202}+13q^{200}-12q^{198}+32q^{196}-63q^{194}-102q^{192}-35q^{190}+216q^{188}+331q^{186}+151q^{184}-63q^{182}-828q^{180}-1302q^{178}-822q^{176}+1158q^{174}+3193q^{172}+3762q^{170}+2167q^{168}-3467q^{166}-9707q^{164}-11888q^{162}-4605q^{160}+9930q^{158}+24650q^{156}+29895q^{154}+12734q^{152}-22818q^{150}-59258q^{148}-66630q^{146}-29644q^{144}+45161q^{142}+122100q^{140}+139844q^{138}+65184q^{136}-85492q^{134}-227185q^{132}-263331q^{130}-129523q^{128}+140894q^{126}+394984q^{124}+456871q^{122}+224564q^{120}-215686q^{118}-631328q^{116}-729568q^{114}-362254q^{112}+323275q^{110}+944757q^{108}+1069701q^{106}+528876q^{104}-463516q^{102}-1326402q^{100}-1467670q^{98}-686860q^{96}+650850q^{94}+1746741q^{92}+1872976q^{90}+805959q^{88}-889284q^{86}-2184458q^{84}-2211519q^{82}-835611q^{80}+1167684q^{78}+2577899q^{76}+2429580q^{74}+743587q^{72}-1482269q^{70}-2853591q^{68}-2467337q^{66}-527285q^{64}+1780716q^{62}+2971580q^{60}+2302095q^{58}+195777q^{56}-1998465q^{54}-2893466q^{52}-1955828q^{50}+181758q^{48}+2103658q^{46}+2622082q^{44}+1470680q^{42}-527658q^{40}-2065674q^{38}-2201421q^{36}-938315q^{34}+801680q^{32}+1892462q^{30}+1687640q^{28}+437651q^{26}-973947q^{24}-1624896q^{22}-1162065q^{20}+1047q^{18}+1054281q^{16}+1308526q^{14}+672400q^{12}-368280q^{10}-1077408q^{8}-993482q^{6}-218554q^{4}+688126q^{2}+1077985+688126q^{-2}-218554q^{-4}-993482q^{-6}-1077408q^{-8}-368280q^{-10}+672400q^{-12}+1308526q^{-14}+1054281q^{-16}+1047q^{-18}-1162065q^{-20}-1624896q^{-22}-973947q^{-24}+437651q^{-26}+1687640q^{-28}+1892462q^{-30}+801680q^{-32}-938315q^{-34}-2201421q^{-36}-2065674q^{-38}-527658q^{-40}+1470680q^{-42}+2622082q^{-44}+2103658q^{-46}+181758q^{-48}-1955828q^{-50}-2893466q^{-52}-1998465q^{-54}+195777q^{-56}+2302095q^{-58}+2971580q^{-60}+1780716q^{-62}-527285q^{-64}-2467337q^{-66}-2853591q^{-68}-1482269q^{-70}+743587q^{-72}+2429580q^{-74}+2577899q^{-76}+1167684q^{-78}-835611q^{-80}-2211519q^{-82}-2184458q^{-84}-889284q^{-86}+805959q^{-88}+1872976q^{-90}+1746741q^{-92}+650850q^{-94}-686860q^{-96}-1467670q^{-98}-1326402q^{-100}-463516q^{-102}+528876q^{-104}+1069701q^{-106}+944757q^{-108}+323275q^{-110}-362254q^{-112}-729568q^{-114}-631328q^{-116}-215686q^{-118}+224564q^{-120}+456871q^{-122}+394984q^{-124}+140894q^{-126}-129523q^{-128}-263331q^{-130}-227185q^{-132}-85492q^{-134}+65184q^{-136}+139844q^{-138}+122100q^{-140}+45161q^{-142}-29644q^{-144}-66630q^{-146}-59258q^{-148}-22818q^{-150}+12734q^{-152}+29895q^{-154}+24650q^{-156}+9930q^{-158}-4605q^{-160}-11888q^{-162}-9707q^{-164}-3467q^{-166}+2167q^{-168}+3762q^{-170}+3193q^{-172}+1158q^{-174}-822q^{-176}-1302q^{-178}-828q^{-180}-63q^{-182}+151q^{-184}+331q^{-186}+216q^{-188}-35q^{-190}-102q^{-192}-63q^{-194}+32q^{-196}-12q^{-198}+13q^{-200}+23q^{-202}-11q^{-204}-11q^{-206}-5q^{-208}+11q^{-210}+q^{-212}-4q^{-214}+q^{-216}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{14}+3q^{12}-2q^{10}+3q^{8}-3q^{4}+4q^{2}-5+4q^{-2}-3q^{-4}+3q^{-8}-2q^{-10}+3q^{-12}-q^{-14}$
1,1	$q^{44}-8q^{42}+32q^{40}-88q^{38}+196q^{36}-392q^{34}+716q^{32}-1194q^{30}+1831q^{28}-2600q^{26}+3434q^{24}-4172q^{22}+4631q^{20}-4638q^{18}+4072q^{16}-2860q^{14}+1036q^{12}+1192q^{10}-3588q^{8}+5840q^{6}-7694q^{4}+8922q^{2}-9330+8922q^{-2}-7694q^{-4}+5840q^{-6}-3588q^{-8}+1192q^{-10}+1036q^{-12}-2860q^{-14}+4072q^{-16}-4638q^{-18}+4631q^{-20}-4172q^{-22}+3434q^{-24}-2600q^{-26}+1831q^{-28}-1194q^{-30}+716q^{-32}-392q^{-34}+196q^{-36}-88q^{-38}+32q^{-40}-8q^{-42}+q^{-44}$
2,0	$q^{38}-3q^{36}-q^{34}+9q^{32}-6q^{30}-9q^{28}+14q^{26}+9q^{24}-13q^{22}-10q^{20}+20q^{18}+10q^{16}-35q^{14}+2q^{12}+24q^{10}-20q^{8}-14q^{6}+21q^{4}+11q^{2}-14+11q^{-2}+21q^{-4}-14q^{-6}-20q^{-8}+24q^{-10}+2q^{-12}-35q^{-14}+10q^{-16}+20q^{-18}-10q^{-20}-13q^{-22}+9q^{-24}+14q^{-26}-9q^{-28}-6q^{-30}+9q^{-32}-q^{-34}-3q^{-36}+q^{-38}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-4q^{32}+2q^{30}+11q^{28}-19q^{26}+4q^{24}+29q^{22}-43q^{20}+6q^{18}+49q^{16}-54q^{14}+5q^{12}+50q^{10}-40q^{8}-8q^{6}+28q^{4}-6q^{2}-16-6q^{-2}+28q^{-4}-8q^{-6}-40q^{-8}+50q^{-10}+5q^{-12}-54q^{-14}+49q^{-16}+6q^{-18}-43q^{-20}+29q^{-22}+4q^{-24}-19q^{-26}+11q^{-28}+2q^{-30}-4q^{-32}+q^{-34}$
1,0,0	$-q^{17}+3q^{15}-3q^{13}+6q^{11}-3q^{9}+4q^{7}-3q^{5}+q^{3}-2q-2q^{-1}+q^{-3}-3q^{-5}+4q^{-7}-3q^{-9}+6q^{-11}-3q^{-13}+3q^{-15}-q^{-17}$
1,0,1	$q^{56}-8q^{54}+28q^{52}-52q^{50}+38q^{48}+65q^{46}-253q^{44}+409q^{42}-326q^{40}-151q^{38}+912q^{36}-1524q^{34}+1436q^{32}-334q^{30}-1493q^{28}+3199q^{26}-3741q^{24}+2539q^{22}+148q^{20}-3147q^{18}+5022q^{16}-4816q^{14}+2595q^{12}+368q^{10}-2626q^{8}+3081q^{6}-1935q^{4}+375q^{2}+395+375q^{-2}-1935q^{-4}+3081q^{-6}-2626q^{-8}+368q^{-10}+2595q^{-12}-4816q^{-14}+5022q^{-16}-3147q^{-18}+148q^{-20}+2539q^{-22}-3741q^{-24}+3199q^{-26}-1493q^{-28}-334q^{-30}+1436q^{-32}-1524q^{-34}+912q^{-36}-151q^{-38}-326q^{-40}+409q^{-42}-253q^{-44}+65q^{-46}+38q^{-48}-52q^{-50}+28q^{-52}-8q^{-54}+q^{-56}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{40}-3q^{38}-q^{36}+10q^{34}-8q^{32}-9q^{30}+25q^{28}-5q^{26}-30q^{24}+23q^{22}+26q^{20}-36q^{18}-11q^{16}+43q^{14}+q^{12}-48q^{10}+18q^{8}+46q^{6}-50q^{4}-18q^{2}+62-18q^{-2}-50q^{-4}+46q^{-6}+18q^{-8}-48q^{-10}+q^{-12}+43q^{-14}-11q^{-16}-36q^{-18}+26q^{-20}+23q^{-22}-30q^{-24}-5q^{-26}+25q^{-28}-9q^{-30}-8q^{-32}+10q^{-34}-q^{-36}-3q^{-38}+q^{-40}$
1,0,0,0	$-q^{20}+3q^{18}-3q^{16}+5q^{14}+q^{10}+3q^{8}-3q^{6}+2q^{4}-5q^{2}+1-5q^{-2}+2q^{-4}-3q^{-6}+3q^{-8}+q^{-10}+5q^{-14}-3q^{-16}+3q^{-18}-q^{-20}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+4q^{32}-10q^{30}+19q^{28}-31q^{26}+46q^{24}-59q^{22}+69q^{20}-70q^{18}+65q^{16}-48q^{14}+23q^{12}+10q^{10}-46q^{8}+80q^{6}-110q^{4}+130q^{2}-138+130q^{-2}-110q^{-4}+80q^{-6}-46q^{-8}+10q^{-10}+23q^{-12}-48q^{-14}+65q^{-16}-70q^{-18}+69q^{-20}-59q^{-22}+46q^{-24}-31q^{-26}+19q^{-28}-10q^{-30}+4q^{-32}-q^{-34}$
1,0	$q^{56}-4q^{52}-4q^{50}+6q^{48}+15q^{46}+q^{44}-25q^{42}-20q^{40}+25q^{38}+44q^{36}-6q^{34}-62q^{32}-28q^{30}+57q^{28}+61q^{26}-29q^{24}-75q^{22}-4q^{20}+71q^{18}+32q^{16}-52q^{14}-45q^{12}+32q^{10}+48q^{8}-17q^{6}-49q^{4}+5q^{2}+49+5q^{-2}-49q^{-4}-17q^{-6}+48q^{-8}+32q^{-10}-45q^{-12}-52q^{-14}+32q^{-16}+71q^{-18}-4q^{-20}-75q^{-22}-29q^{-24}+61q^{-26}+57q^{-28}-28q^{-30}-62q^{-32}-6q^{-34}+44q^{-36}+25q^{-38}-20q^{-40}-25q^{-42}+q^{-44}+15q^{-46}+6q^{-48}-4q^{-50}-4q^{-52}+q^{-56}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{46}-4q^{44}+6q^{42}-8q^{40}+16q^{38}-25q^{36}+30q^{34}-36q^{32}+48q^{30}-56q^{28}+53q^{26}-53q^{24}+56q^{22}-42q^{20}+26q^{18}-12q^{16}+2q^{14}+30q^{12}-51q^{10}+61q^{8}-79q^{6}+98q^{4}-106q^{2}+98-106q^{-2}+98q^{-4}-79q^{-6}+61q^{-8}-51q^{-10}+30q^{-12}+2q^{-14}-12q^{-16}+26q^{-18}-42q^{-20}+56q^{-22}-53q^{-24}+53q^{-26}-56q^{-28}+48q^{-30}-36q^{-32}+30q^{-34}-25q^{-36}+16q^{-38}-8q^{-40}+6q^{-42}-4q^{-44}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-4q^{78}+10q^{76}-20q^{74}+26q^{72}-25q^{70}+10q^{68}+30q^{66}-80q^{64}+140q^{62}-180q^{60}+158q^{58}-71q^{56}-100q^{54}+308q^{52}-473q^{50}+528q^{48}-391q^{46}+69q^{44}+343q^{42}-693q^{40}+822q^{38}-656q^{36}+239q^{34}+267q^{32}-647q^{30}+750q^{28}-495q^{26}+29q^{24}+435q^{22}-675q^{20}+551q^{18}-133q^{16}-414q^{14}+836q^{12}-944q^{10}+710q^{8}-173q^{6}-472q^{4}+970q^{2}-1165+970q^{-2}-472q^{-4}-173q^{-6}+710q^{-8}-944q^{-10}+836q^{-12}-414q^{-14}-133q^{-16}+551q^{-18}-675q^{-20}+435q^{-22}+29q^{-24}-495q^{-26}+750q^{-28}-647q^{-30}+267q^{-32}+239q^{-34}-656q^{-36}+822q^{-38}-693q^{-40}+343q^{-42}+69q^{-44}-391q^{-46}+528q^{-48}-473q^{-50}+308q^{-52}-100q^{-54}-71q^{-56}+158q^{-58}-180q^{-60}+140q^{-62}-80q^{-64}+30q^{-66}+10q^{-68}-25q^{-70}+26q^{-72}-20q^{-74}+10q^{-76}-4q^{-78}+q^{-80}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 123"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$t^{4}-6t^{3}+15t^{2}-24t+29-24t^{-1}+15t^{-2}-6t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+2z^{6}-z^{4}-2z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\left\{t^{4}-3t^{3}+3t^{2}-3t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 121, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q^{5}+5q^{4}-10q^{3}+15q^{2}-19q+21-19q^{-1}+15q^{-2}-10q^{-3}+5q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+4z^{6}-2a^{2}z^{4}-2z^{4}a^{-2}+3z^{4}+a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+2a^{2}+2a^{-2}-3$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$4az^{9}+4z^{9}a^{-1}+10a^{2}z^{8}+10z^{8}a^{-2}+20z^{8}+10a^{3}z^{7}+14az^{7}+14z^{7}a^{-1}+10z^{7}a^{-3}+5a^{4}z^{6}-11a^{2}z^{6}-11z^{6}a^{-2}+5z^{6}a^{-4}-32z^{6}+a^{5}z^{5}-15a^{3}z^{5}-38az^{5}-38z^{5}a^{-1}-15z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}-3a^{2}z^{4}-3z^{4}a^{-2}-5z^{4}a^{-4}+4z^{4}+5a^{3}z^{3}+21az^{3}+21z^{3}a^{-1}+5z^{3}a^{-3}+6a^{2}z^{2}+6z^{2}a^{-2}+12z^{2}-2az-2za^{-1}-2a^{2}-2a^{-2}-3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a28, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {...}

Vassiliev invariants

V2 and V3:

(-2, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-8$ $0$ $32$ ${\frac {164}{3}}$ ${\frac {76}{3}}$ $0$ $0$ $0$ $0$ $-{\frac {256}{3}}$ $0$ $-{\frac {1312}{3}}$ $-{\frac {608}{3}}$ $-{\frac {6271}{15}}$ ${\frac {1484}{15}}$ $-{\frac {16684}{45}}$ ${\frac {31}{9}}$ $-{\frac {1471}{15}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of 10 123. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                     1 -1 9                   4   4 7                 6 1   -5 5               9 4     5 3             10 6       -4 1           11 9         2 -1         9 11           2 -3       6 10             -4 -5     4 9               5 -7   1 6                 -5 -9   4                   4 -11 1                     -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{15}-5q^{14}+5q^{13}+15q^{12}-41q^{11}+14q^{10}+80q^{9}-121q^{8}-10q^{7}+206q^{6}-197q^{5}-85q^{4}+331q^{3}-215q^{2}-169q+383-169q^{-1}-215q^{-2}+331q^{-3}-85q^{-4}-197q^{-5}+206q^{-6}-10q^{-7}-121q^{-8}+80q^{-9}+14q^{-10}-41q^{-11}+15q^{-12}+5q^{-13}-5q^{-14}+q^{-15}$
3	$-q^{30}+5q^{29}-5q^{28}-10q^{27}+11q^{26}+31q^{25}-20q^{24}-95q^{23}+46q^{22}+200q^{21}-25q^{20}-405q^{19}-59q^{18}+674q^{17}+283q^{16}-980q^{15}-659q^{14}+1229q^{13}+1193q^{12}-1376q^{11}-1800q^{10}+1364q^{9}+2413q^{8}-1205q^{7}-2948q^{6}+930q^{5}+3353q^{4}-584q^{3}-3597q^{2}+192q+3691+192q^{-1}-3597q^{-2}-584q^{-3}+3353q^{-4}+930q^{-5}-2948q^{-6}-1205q^{-7}+2413q^{-8}+1364q^{-9}-1800q^{-10}-1376q^{-11}+1193q^{-12}+1229q^{-13}-659q^{-14}-980q^{-15}+283q^{-16}+674q^{-17}-59q^{-18}-405q^{-19}-25q^{-20}+200q^{-21}+46q^{-22}-95q^{-23}-20q^{-24}+31q^{-25}+11q^{-26}-10q^{-27}-5q^{-28}+5q^{-29}-q^{-30}$
4	$q^{50}-5q^{49}+5q^{48}+10q^{47}-16q^{46}-q^{45}-25q^{44}+50q^{43}+75q^{42}-111q^{41}-76q^{40}-144q^{39}+300q^{38}+521q^{37}-300q^{36}-645q^{35}-1029q^{34}+795q^{33}+2396q^{32}+536q^{31}-1785q^{30}-4555q^{29}-371q^{28}+5976q^{27}+5172q^{26}-707q^{25}-11055q^{24}-6857q^{23}+7535q^{22}+13845q^{21}+6944q^{20}-15956q^{19}-18552q^{18}+2209q^{17}+21368q^{16}+20543q^{15}-14191q^{14}-29641q^{13}-9205q^{12}+22728q^{11}+33968q^{10}-6460q^{9}-35045q^{8}-21175q^{7}+18340q^{6}+42330q^{5}+2887q^{4}-34554q^{3}-29818q^{2}+11268q+44955+11268q^{-1}-29818q^{-2}-34554q^{-3}+2887q^{-4}+42330q^{-5}+18340q^{-6}-21175q^{-7}-35045q^{-8}-6460q^{-9}+33968q^{-10}+22728q^{-11}-9205q^{-12}-29641q^{-13}-14191q^{-14}+20543q^{-15}+21368q^{-16}+2209q^{-17}-18552q^{-18}-15956q^{-19}+6944q^{-20}+13845q^{-21}+7535q^{-22}-6857q^{-23}-11055q^{-24}-707q^{-25}+5172q^{-26}+5976q^{-27}-371q^{-28}-4555q^{-29}-1785q^{-30}+536q^{-31}+2396q^{-32}+795q^{-33}-1029q^{-34}-645q^{-35}-300q^{-36}+521q^{-37}+300q^{-38}-144q^{-39}-76q^{-40}-111q^{-41}+75q^{-42}+50q^{-43}-25q^{-44}-q^{-45}-16q^{-46}+10q^{-47}+5q^{-48}-5q^{-49}+q^{-50}$
5	$-q^{75}+5q^{74}-5q^{73}-10q^{72}+16q^{71}+6q^{70}-5q^{69}-5q^{68}-30q^{67}-25q^{66}+82q^{65}+120q^{64}-26q^{63}-216q^{62}-316q^{61}-60q^{60}+551q^{59}+1025q^{58}+464q^{57}-1237q^{56}-2571q^{55}-1825q^{54}+1562q^{53}+5586q^{52}+5808q^{51}-618q^{50}-9956q^{49}-13478q^{48}-4712q^{47}+13601q^{46}+26496q^{45}+17652q^{44}-12935q^{43}-42692q^{42}-41331q^{41}+1497q^{40}+57823q^{39}+75942q^{38}+25990q^{37}-63660q^{36}-117173q^{35}-72692q^{34}+51927q^{33}+156391q^{32}+136191q^{31}-16419q^{30}-183351q^{29}-208746q^{28}-43200q^{27}+188890q^{26}+279271q^{25}+121855q^{24}-169188q^{23}-337053q^{22}-209298q^{21}+125956q^{20}+374479q^{19}+294696q^{18}-65918q^{17}-389525q^{16}-368820q^{15}-1823q^{14}+384561q^{13}+426473q^{12}+69028q^{11}-364895q^{10}-466752q^{9}-130155q^{8}+336393q^{7}+491875q^{6}+182697q^{5}-303191q^{4}-504874q^{3}-227767q^{2}+267093q+509105+267093q^{-1}-227767q^{-2}-504874q^{-3}-303191q^{-4}+182697q^{-5}+491875q^{-6}+336393q^{-7}-130155q^{-8}-466752q^{-9}-364895q^{-10}+69028q^{-11}+426473q^{-12}+384561q^{-13}-1823q^{-14}-368820q^{-15}-389525q^{-16}-65918q^{-17}+294696q^{-18}+374479q^{-19}+125956q^{-20}-209298q^{-21}-337053q^{-22}-169188q^{-23}+121855q^{-24}+279271q^{-25}+188890q^{-26}-43200q^{-27}-208746q^{-28}-183351q^{-29}-16419q^{-30}+136191q^{-31}+156391q^{-32}+51927q^{-33}-72692q^{-34}-117173q^{-35}-63660q^{-36}+25990q^{-37}+75942q^{-38}+57823q^{-39}+1497q^{-40}-41331q^{-41}-42692q^{-42}-12935q^{-43}+17652q^{-44}+26496q^{-45}+13601q^{-46}-4712q^{-47}-13478q^{-48}-9956q^{-49}-618q^{-50}+5808q^{-51}+5586q^{-52}+1562q^{-53}-1825q^{-54}-2571q^{-55}-1237q^{-56}+464q^{-57}+1025q^{-58}+551q^{-59}-60q^{-60}-316q^{-61}-216q^{-62}-26q^{-63}+120q^{-64}+82q^{-65}-25q^{-66}-30q^{-67}-5q^{-68}-5q^{-69}+6q^{-70}+16q^{-71}-10q^{-72}-5q^{-73}+5q^{-74}-q^{-75}$
6	$q^{105}-5q^{104}+5q^{103}+10q^{102}-16q^{101}-6q^{100}+35q^{98}-15q^{97}-20q^{96}+54q^{95}-111q^{94}-45q^{93}+67q^{92}+286q^{91}+100q^{90}-200q^{89}-160q^{88}-876q^{87}-519q^{86}+547q^{85}+2266q^{84}+2135q^{83}+369q^{82}-1755q^{81}-6510q^{80}-6759q^{79}-1634q^{78}+9549q^{77}+16670q^{76}+15089q^{75}+3490q^{74}-23671q^{73}-42311q^{72}-38074q^{71}+2177q^{70}+53656q^{69}+89894q^{68}+80429q^{67}-5927q^{66}-116971q^{65}-188750q^{64}-139516q^{63}+17510q^{62}+223702q^{61}+350846q^{60}+248163q^{59}-55084q^{58}-421057q^{57}-579766q^{56}-398132q^{55}+125462q^{54}+718160q^{53}+933692q^{52}+566398q^{51}-296113q^{50}-1120591q^{49}-1390524q^{48}-737424q^{47}+576892q^{46}+1714502q^{45}+1904108q^{44}+799778q^{43}-994356q^{42}-2457541q^{41}-2432667q^{40}-718282q^{39}+1687441q^{38}+3280016q^{37}+2803073q^{36}+415859q^{35}-2605860q^{34}-4118660q^{33}-2944138q^{32}+316119q^{31}+3666270q^{30}+4743125q^{29}+2723860q^{28}-1414996q^{27}-4788145q^{26}-5050456q^{25}-1878123q^{24}+2771269q^{23}+5680763q^{22}+4861446q^{21}+506904q^{20}-4269721q^{19}-6201858q^{18}-3876461q^{17}+1233253q^{16}+5545016q^{15}+6124552q^{14}+2246899q^{13}-3178939q^{12}-6406680q^{11}-5126450q^{10}-178345q^{9}+4894067q^{8}+6587383q^{7}+3410011q^{6}-2125705q^{5}-6152435q^{4}-5762576q^{3}-1219025q^{2}+4184939q+6671309+4184939q^{-1}-1219025q^{-2}-5762576q^{-3}-6152435q^{-4}-2125705q^{-5}+3410011q^{-6}+6587383q^{-7}+4894067q^{-8}-178345q^{-9}-5126450q^{-10}-6406680q^{-11}-3178939q^{-12}+2246899q^{-13}+6124552q^{-14}+5545016q^{-15}+1233253q^{-16}-3876461q^{-17}-6201858q^{-18}-4269721q^{-19}+506904q^{-20}+4861446q^{-21}+5680763q^{-22}+2771269q^{-23}-1878123q^{-24}-5050456q^{-25}-4788145q^{-26}-1414996q^{-27}+2723860q^{-28}+4743125q^{-29}+3666270q^{-30}+316119q^{-31}-2944138q^{-32}-4118660q^{-33}-2605860q^{-34}+415859q^{-35}+2803073q^{-36}+3280016q^{-37}+1687441q^{-38}-718282q^{-39}-2432667q^{-40}-2457541q^{-41}-994356q^{-42}+799778q^{-43}+1904108q^{-44}+1714502q^{-45}+576892q^{-46}-737424q^{-47}-1390524q^{-48}-1120591q^{-49}-296113q^{-50}+566398q^{-51}+933692q^{-52}+718160q^{-53}+125462q^{-54}-398132q^{-55}-579766q^{-56}-421057q^{-57}-55084q^{-58}+248163q^{-59}+350846q^{-60}+223702q^{-61}+17510q^{-62}-139516q^{-63}-188750q^{-64}-116971q^{-65}-5927q^{-66}+80429q^{-67}+89894q^{-68}+53656q^{-69}+2177q^{-70}-38074q^{-71}-42311q^{-72}-23671q^{-73}+3490q^{-74}+15089q^{-75}+16670q^{-76}+9549q^{-77}-1634q^{-78}-6759q^{-79}-6510q^{-80}-1755q^{-81}+369q^{-82}+2135q^{-83}+2266q^{-84}+547q^{-85}-519q^{-86}-876q^{-87}-160q^{-88}-200q^{-89}+100q^{-90}+286q^{-91}+67q^{-92}-45q^{-93}-111q^{-94}+54q^{-95}-20q^{-96}-15q^{-97}+35q^{-98}-6q^{-100}-16q^{-101}+10q^{-102}+5q^{-103}-5q^{-104}+q^{-105}$
7	$-q^{140}+5q^{139}-5q^{138}-10q^{137}+16q^{136}+6q^{135}-30q^{133}-15q^{132}+65q^{131}-9q^{130}-25q^{129}+36q^{128}-11q^{127}-42q^{126}-195q^{125}-115q^{124}+385q^{123}+395q^{122}+319q^{121}+136q^{120}-646q^{119}-1137q^{118}-1890q^{117}-1295q^{116}+1720q^{115}+4250q^{114}+5950q^{113}+4577q^{112}-1660q^{111}-9312q^{110}-17220q^{109}-18195q^{108}-4883q^{107}+17046q^{106}+41839q^{105}+52772q^{104}+33340q^{103}-13156q^{102}-78969q^{101}-129393q^{100}-120485q^{99}-37709q^{98}+110481q^{97}+258889q^{96}+312390q^{95}+212678q^{94}-58409q^{93}-408671q^{92}-655018q^{91}-629387q^{90}-225618q^{89}+455543q^{88}+1111993q^{87}+1386952q^{86}+974072q^{85}-120295q^{84}-1490425q^{83}-2486737q^{82}-2411772q^{81}-992130q^{80}+1359420q^{79}+3658056q^{78}+4600767q^{77}+3312410q^{76}-69631q^{75}-4295721q^{74}-7250018q^{73}-7038624q^{72}-3064728q^{71}+3458772q^{70}+9564329q^{69}+11904596q^{68}+8481876q^{67}-127110q^{66}-10320484q^{65}-16996224q^{64}-16013247q^{63}-6427765q^{62}+8135986q^{61}+20830152q^{60}+24710309q^{59}+16239649q^{58}-1944871q^{57}-21701623q^{56}-32924207q^{55}-28421781q^{54}-8547351q^{53}+18203416q^{52}+38688427q^{51}+41268842q^{50}+22629174q^{49}-9727922q^{48}-40298558q^{47}-52668841q^{46}-38661673q^{45}-3265789q^{44}+36818791q^{43}+60675114q^{42}+54508287q^{41}+19350236q^{40}-28369637q^{39}-64052090q^{38}-68100082q^{37}-36505322q^{36}+16059829q^{35}+62539208q^{34}+77949831q^{33}+52680978q^{32}-1622508q^{31}-56831245q^{30}-83452910q^{29}-66273208q^{28}-13063190q^{27}+48269134q^{26}+84870725q^{25}+76415546q^{24}+26435776q^{23}-38413430q^{22}-83109137q^{21}-83022604q^{20}-37518962q^{19}+28658568q^{18}+79366271q^{17}+86611850q^{16}+45992202q^{15}-19960604q^{14}-74793555q^{13}-88046411q^{12}-52101041q^{11}+12750874q^{10}+70271955q^{9}+88260356q^{8}+56446409q^{7}-6976270q^{6}-66276594q^{5}-88039714q^{4}-59795350q^{3}+2203806q^{2}+62882041q+87910159+62882041q^{-1}+2203806q^{-2}-59795350q^{-3}-88039714q^{-4}-66276594q^{-5}-6976270q^{-6}+56446409q^{-7}+88260356q^{-8}+70271955q^{-9}+12750874q^{-10}-52101041q^{-11}-88046411q^{-12}-74793555q^{-13}-19960604q^{-14}+45992202q^{-15}+86611850q^{-16}+79366271q^{-17}+28658568q^{-18}-37518962q^{-19}-83022604q^{-20}-83109137q^{-21}-38413430q^{-22}+26435776q^{-23}+76415546q^{-24}+84870725q^{-25}+48269134q^{-26}-13063190q^{-27}-66273208q^{-28}-83452910q^{-29}-56831245q^{-30}-1622508q^{-31}+52680978q^{-32}+77949831q^{-33}+62539208q^{-34}+16059829q^{-35}-36505322q^{-36}-68100082q^{-37}-64052090q^{-38}-28369637q^{-39}+19350236q^{-40}+54508287q^{-41}+60675114q^{-42}+36818791q^{-43}-3265789q^{-44}-38661673q^{-45}-52668841q^{-46}-40298558q^{-47}-9727922q^{-48}+22629174q^{-49}+41268842q^{-50}+38688427q^{-51}+18203416q^{-52}-8547351q^{-53}-28421781q^{-54}-32924207q^{-55}-21701623q^{-56}-1944871q^{-57}+16239649q^{-58}+24710309q^{-59}+20830152q^{-60}+8135986q^{-61}-6427765q^{-62}-16013247q^{-63}-16996224q^{-64}-10320484q^{-65}-127110q^{-66}+8481876q^{-67}+11904596q^{-68}+9564329q^{-69}+3458772q^{-70}-3064728q^{-71}-7038624q^{-72}-7250018q^{-73}-4295721q^{-74}-69631q^{-75}+3312410q^{-76}+4600767q^{-77}+3658056q^{-78}+1359420q^{-79}-992130q^{-80}-2411772q^{-81}-2486737q^{-82}-1490425q^{-83}-120295q^{-84}+974072q^{-85}+1386952q^{-86}+1111993q^{-87}+455543q^{-88}-225618q^{-89}-629387q^{-90}-655018q^{-91}-408671q^{-92}-58409q^{-93}+212678q^{-94}+312390q^{-95}+258889q^{-96}+110481q^{-97}-37709q^{-98}-120485q^{-99}-129393q^{-100}-78969q^{-101}-13156q^{-102}+33340q^{-103}+52772q^{-104}+41839q^{-105}+17046q^{-106}-4883q^{-107}-18195q^{-108}-17220q^{-109}-9312q^{-110}-1660q^{-111}+4577q^{-112}+5950q^{-113}+4250q^{-114}+1720q^{-115}-1295q^{-116}-1890q^{-117}-1137q^{-118}-646q^{-119}+136q^{-120}+319q^{-121}+395q^{-122}+385q^{-123}-115q^{-124}-195q^{-125}-42q^{-126}-11q^{-127}+36q^{-128}-25q^{-129}-9q^{-130}+65q^{-131}-15q^{-132}-30q^{-133}+6q^{-135}+16q^{-136}-10q^{-137}-5q^{-138}+5q^{-139}-q^{-140}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[10, 123]]
Out[2]=	PD[X[8, 2, 9, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[4, 18, 5, 17], X[18, 11, 19, 12], X[2, 15, 3, 16], X[16, 10, 17, 9], X[20, 14, 1, 13], X[14, 7, 15, 8], X[6, 19, 7, 20]]
In[3]:=	GaussCode[Knot[10, 123]]
Out[3]=	GaussCode[1, -6, 2, -4, 3, -10, 9, -1, 7, -2, 5, -3, 8, -9, 6, -7, 4, -5, 10, -8]
In[4]:=	DTCode[Knot[10, 123]]
Out[4]=	DTCode[8, 10, 12, 14, 16, 18, 20, 2, 4, 6]
In[5]:=	br = BR[Knot[10, 123]]
Out[5]=	BR[3, {-1, 2, -1, 2, -1, 2, -1, 2, -1, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 123]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 123]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[10, 123]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{FullyAmphicheiral, 2, 4, 3, NotAvailable, 2}
In[10]:=	alex = Alexander[Knot[10, 123]][t]
Out[10]=	-4 6 15 24 2 3 4 29 + t - -- + -- - -- - 24 t + 15 t - 6 t + t 3 2 t t t
In[11]:=	Conway[Knot[10, 123]][z]
Out[11]=	2 4 6 8 1 - 2 z - z + 2 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 123], Knot[11, Alternating, 28]}
In[13]:=	{KnotDet[Knot[10, 123]], KnotSignature[Knot[10, 123]]}
Out[13]=	{121, 0}
In[14]:=	Jones[Knot[10, 123]][q]
Out[14]=	-5 5 10 15 19 2 3 4 5 21 - q + -- - -- + -- - -- - 19 q + 15 q - 10 q + 5 q - q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 123]}
In[16]:=	A2Invariant[Knot[10, 123]][q]
Out[16]=	-14 3 2 3 3 4 2 4 8 10 -5 - q + --- - --- + -- - -- + -- + 4 q - 3 q + 3 q - 2 q + 12 10 8 4 2 q q q q q 12 14 3 q - q
In[17]:=	HOMFLYPT[Knot[10, 123]][a, z]
Out[17]=	2 4 2 2 2 z 2 2 4 2 z 2 4 6 -3 + -- + 2 a - 4 z + -- + a z + 3 z - ---- - 2 a z + 4 z - 2 2 2 a a a 6 z 2 6 8 -- - a z + z 2 a
In[18]:=	Kauffman[Knot[10, 123]][a, z]
Out[18]=	2 3 3 2 2 2 z 2 6 z 2 2 5 z 21 z -3 - -- - 2 a - --- - 2 a z + 12 z + ---- + 6 a z + ---- + ----- + 2 a 2 3 a a a a 4 4 5 3 3 3 4 5 z 3 z 2 4 4 4 z 21 a z + 5 a z + 4 z - ---- - ---- - 3 a z - 5 a z + -- - 4 2 5 a a a 5 5 6 6 15 z 38 z 5 3 5 5 5 6 5 z 11 z ----- - ----- - 38 a z - 15 a z + a z - 32 z + ---- - ----- - 3 a 4 2 a a a 7 7 2 6 4 6 10 z 14 z 7 3 7 8 11 a z + 5 a z + ----- + ----- + 14 a z + 10 a z + 20 z + 3 a a 8 9 10 z 2 8 4 z 9 ----- + 10 a z + ---- + 4 a z 2 a a
In[19]:=	{Vassiliev[2][Knot[10, 123]], Vassiliev[3][Knot[10, 123]]}
Out[19]=	{-2, 0}
In[20]:=	Kh[Knot[10, 123]][q, t]
Out[20]=	11 1 4 1 6 4 9 6 -- + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 10 9 3 3 2 5 2 5 3 ---- + --- + 9 q t + 10 q t + 6 q t + 9 q t + 4 q t + 3 q t q t 7 3 7 4 9 4 11 5 6 q t + q t + 4 q t + q t
In[21]:=	ColouredJones[Knot[10, 123], 2][q]
Out[21]=	-15 5 5 15 41 14 80 121 10 206 197 383 + q - --- + --- + --- - --- + --- + -- - --- - -- + --- - --- - 14 13 12 11 10 9 8 7 6 5 q q q q q q q q q q 85 331 215 169 2 3 4 5 -- + --- - --- - --- - 169 q - 215 q + 331 q - 85 q - 197 q + 4 3 2 q q q q 6 7 8 9 10 11 12 13 206 q - 10 q - 121 q + 80 q + 14 q - 41 q + 15 q + 5 q - 14 15 5 q + q

See/edit the Rolfsen_Splice_Template.