The Alternating Knot 10₁₂₃

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Acknowledgement

KnotPlot

PD Presentation: X₈₂₉₁ 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

Gauss Code: {1, -6, 2, -4, 3, -10, 9, -1, 7, -2, 5, -3, 8, -9, 6, -7, 4, -5, 10, -8}

DT (Dowker-Thistlethwaite) Code: 8 10 12 14 16 18 20 2 4 6

Minimum Braid Representative:

Length is 10, width is 3
Braid index is 3

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index

FullyAmphicheiral 2 4 3 / NotAvailable 2

Alexander Polynomial: t^-4 - 6t^-3 + 15t^-2 - 24t^-1 + 29 - 24t + 15t² - 6t³ + t⁴

Conway Polynomial: 1 - 2z² - z⁴ + 2z⁶ + z⁸

Other knots with the same Alexander/Conway Polynomial: {K11a28, ...}

Determinant and Signature: {121, 0}

Jones Polynomial: - q^-5 + 5q^-4 - 10q^-3 + 15q^-2 - 19q^-1 + 21 - 19q + 15q² - 10q³ + 5q⁴ - q⁵

Other knots (up to mirrors) with the same Jones Polynomial: {...}

A2 (sl(3)) Invariant: - q^-14 + 3q^-12 - 2q^-10 + 3q^-8 - 3q^-4 + 4q^-2 - 5 + 4q² - 3q⁴ + 3q⁸ - 2q¹⁰ + 3q¹² - q¹⁴

HOMFLY-PT Polynomial: 2a^-2 + a^-2z² - 2a^-2z⁴ - a^-2z⁶ - 3 - 4z² + 3z⁴ + 4z⁶ + z⁸ + 2a² + a²z² - 2a²z⁴ - a²z⁶

Kauffman Polynomial: a^-5z⁵ - 5a^-4z⁴ + 5a^-4z⁶ + 5a^-3z³ - 15a^-3z⁵ + 10a^-3z⁷ - 2a^-2 + 6a^-2z² - 3a^-2z⁴ - 11a^-2z⁶ + 10a^-2z⁸ - 2a^-1z + 21a^-1z³ - 38a^-1z⁵ + 14a^-1z⁷ + 4a^-1z⁹ - 3 + 12z² + 4z⁴ - 32z⁶ + 20z⁸ - 2az + 21az³ - 38az⁵ + 14az⁷ + 4az⁹ - 2a² + 6a²z² - 3a²z⁴ - 11a²z⁶ + 10a²z⁸ + 5a³z³ - 15a³z⁵ + 10a³z⁷ - 5a⁴z⁴ + 5a⁴z⁶ + a⁵z⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-2, 0}

Khovanov Homology:
(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 10₁₂₃. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5

j = 11 1

j = 9 4

j = 7 6 1

j = 5 9 4

j = 3 10 6

j = 1 11 9

j = -1 9 11

j = -3 6 10

j = -5 4 9

j = -7 1 6

j = -9 4

j = -11 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
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^-1 + 383 - 169q - 215q² + 331q³ - 85q⁴ - 197q⁵ + 206q⁶ - 10q⁷ - 121q⁸ + 80q⁹ + 14q¹⁰ - 41q¹¹ + 15q¹² + 5q¹³ - 5q¹⁴ + q¹⁵

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^-1 + 3691 + 192q - 3597q² - 584q³ + 3353q⁴ + 930q⁵ - 2948q⁶ - 1205q⁷ + 2413q⁸ + 1364q⁹ - 1800q¹⁰ - 1376q¹¹ + 1193q¹² + 1229q¹³ - 659q¹⁴ - 980q¹⁵ + 283q¹⁶ + 674q¹⁷ - 59q¹⁸ - 405q¹⁹ - 25q²⁰ + 200q²¹ + 46q²² - 95q²³ - 20q²⁴ + 31q²⁵ + 11q²⁶ - 10q²⁷ - 5q²⁸ + 5q²⁹ - q³⁰

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^-1 + 44955 + 11268q - 29818q² - 34554q³ + 2887q⁴ + 42330q⁵ + 18340q⁶ - 21175q⁷ - 35045q⁸ - 6460q⁹ + 33968q¹⁰ + 22728q¹¹ - 9205q¹² - 29641q¹³ - 14191q¹⁴ + 20543q¹⁵ + 21368q¹⁶ + 2209q¹⁷ - 18552q¹⁸ - 15956q¹⁹ + 6944q²⁰ + 13845q²¹ + 7535q²² - 6857q²³ - 11055q²⁴ - 707q²⁵ + 5172q²⁶ + 5976q²⁷ - 371q²⁸ - 4555q²⁹ - 1785q³⁰ + 536q³¹ + 2396q³² + 795q³³ - 1029q³⁴ - 645q³⁵ - 300q³⁶ + 521q³⁷ + 300q³⁸ - 144q³⁹ - 76q⁴⁰ - 111q⁴¹ + 75q⁴² + 50q⁴³ - 25q⁴⁴ - q⁴⁵ - 16q⁴⁶ + 10q⁴⁷ + 5q⁴⁸ - 5q⁴⁹ + q⁵⁰

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^-1 + 509105 + 267093q - 227767q² - 504874q³ - 303191q⁴ + 182697q⁵ + 491875q⁶ + 336393q⁷ - 130155q⁸ - 466752q⁹ - 364895q¹⁰ + 69028q¹¹ + 426473q¹² + 384561q¹³ - 1823q¹⁴ - 368820q¹⁵ - 389525q¹⁶ - 65918q¹⁷ + 294696q¹⁸ + 374479q¹⁹ + 125956q²⁰ - 209298q²¹ - 337053q²² - 169188q²³ + 121855q²⁴ + 279271q²⁵ + 188890q²⁶ - 43200q²⁷ - 208746q²⁸ - 183351q²⁹ - 16419q³⁰ + 136191q³¹ + 156391q³² + 51927q³³ - 72692q³⁴ - 117173q³⁵ - 63660q³⁶ + 25990q³⁷ + 75942q³⁸ + 57823q³⁹ + 1497q⁴⁰ - 41331q⁴¹ - 42692q⁴² - 12935q⁴³ + 17652q⁴⁴ + 26496q⁴⁵ + 13601q⁴⁶ - 4712q⁴⁷ - 13478q⁴⁸ - 9956q⁴⁹ - 618q⁵⁰ + 5808q⁵¹ + 5586q⁵² + 1562q⁵³ - 1825q⁵⁴ - 2571q⁵⁵ - 1237q⁵⁶ + 464q⁵⁷ + 1025q⁵⁸ + 551q⁵⁹ - 60q⁶⁰ - 316q⁶¹ - 216q⁶² - 26q⁶³ + 120q⁶⁴ + 82q⁶⁵ - 25q⁶⁶ - 30q⁶⁷ - 5q⁶⁸ - 5q⁶⁹ + 6q⁷⁰ + 16q⁷¹ - 10q⁷² - 5q⁷³ + 5q⁷⁴ - q⁷⁵

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^-1 + 6671309 + 4184939q - 1219025q² - 5762576q³ - 6152435q⁴ - 2125705q⁵ + 3410011q⁶ + 6587383q⁷ + 4894067q⁸ - 178345q⁹ - 5126450q¹⁰ - 6406680q¹¹ - 3178939q¹² + 2246899q¹³ + 6124552q¹⁴ + 5545016q¹⁵ + 1233253q¹⁶ - 3876461q¹⁷ - 6201858q¹⁸ - 4269721q¹⁹ + 506904q²⁰ + 4861446q²¹ + 5680763q²² + 2771269q²³ - 1878123q²⁴ - 5050456q²⁵ - 4788145q²⁶ - 1414996q²⁷ + 2723860q²⁸ + 4743125q²⁹ + 3666270q³⁰ + 316119q³¹ - 2944138q³² - 4118660q³³ - 2605860q³⁴ + 415859q³⁵ + 2803073q³⁶ + 3280016q³⁷ + 1687441q³⁸ - 718282q³⁹ - 2432667q⁴⁰ - 2457541q⁴¹ - 994356q⁴² + 799778q⁴³ + 1904108q⁴⁴ + 1714502q⁴⁵ + 576892q⁴⁶ - 737424q⁴⁷ - 1390524q⁴⁸ - 1120591q⁴⁹ - 296113q⁵⁰ + 566398q⁵¹ + 933692q⁵² + 718160q⁵³ + 125462q⁵⁴ - 398132q⁵⁵ - 579766q⁵⁶ - 421057q⁵⁷ - 55084q⁵⁸ + 248163q⁵⁹ + 350846q⁶⁰ + 223702q⁶¹ + 17510q⁶² - 139516q⁶³ - 188750q⁶⁴ - 116971q⁶⁵ - 5927q⁶⁶ + 80429q⁶⁷ + 89894q⁶⁸ + 53656q⁶⁹ + 2177q⁷⁰ - 38074q⁷¹ - 42311q⁷² - 23671q⁷³ + 3490q⁷⁴ + 15089q⁷⁵ + 16670q⁷⁶ + 9549q⁷⁷ - 1634q⁷⁸ - 6759q⁷⁹ - 6510q⁸⁰ - 1755q⁸¹ + 369q⁸² + 2135q⁸³ + 2266q⁸⁴ + 547q⁸⁵ - 519q⁸⁶ - 876q⁸⁷ - 160q⁸⁸ - 200q⁸⁹ + 100q⁹⁰ + 286q⁹¹ + 67q⁹² - 45q⁹³ - 111q⁹⁴ + 54q⁹⁵ - 20q⁹⁶ - 15q⁹⁷ + 35q⁹⁸ - 6q¹⁰⁰ - 16q¹⁰¹ + 10q¹⁰² + 5q¹⁰³ - 5q¹⁰⁴ + q¹⁰⁵

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^-1 + 87910159 + 62882041q + 2203806q² - 59795350q³ - 88039714q⁴ - 66276594q⁵ - 6976270q⁶ + 56446409q⁷ + 88260356q⁸ + 70271955q⁹ + 12750874q¹⁰ - 52101041q¹¹ - 88046411q¹² - 74793555q¹³ - 19960604q¹⁴ + 45992202q¹⁵ + 86611850q¹⁶ + 79366271q¹⁷ + 28658568q¹⁸ - 37518962q¹⁹ - 83022604q²⁰ - 83109137q²¹ - 38413430q²² + 26435776q²³ + 76415546q²⁴ + 84870725q²⁵ + 48269134q²⁶ - 13063190q²⁷ - 66273208q²⁸ - 83452910q²⁹ - 56831245q³⁰ - 1622508q³¹ + 52680978q³² + 77949831q³³ + 62539208q³⁴ + 16059829q³⁵ - 36505322q³⁶ - 68100082q³⁷ - 64052090q³⁸ - 28369637q³⁹ + 19350236q⁴⁰ + 54508287q⁴¹ + 60675114q⁴² + 36818791q⁴³ - 3265789q⁴⁴ - 38661673q⁴⁵ - 52668841q⁴⁶ - 40298558q⁴⁷ - 9727922q⁴⁸ + 22629174q⁴⁹ + 41268842q⁵⁰ + 38688427q⁵¹ + 18203416q⁵² - 8547351q⁵³ - 28421781q⁵⁴ - 32924207q⁵⁵ - 21701623q⁵⁶ - 1944871q⁵⁷ + 16239649q⁵⁸ + 24710309q⁵⁹ + 20830152q⁶⁰ + 8135986q⁶¹ - 6427765q⁶² - 16013247q⁶³ - 16996224q⁶⁴ - 10320484q⁶⁵ - 127110q⁶⁶ + 8481876q⁶⁷ + 11904596q⁶⁸ + 9564329q⁶⁹ + 3458772q⁷⁰ - 3064728q⁷¹ - 7038624q⁷² - 7250018q⁷³ - 4295721q⁷⁴ - 69631q⁷⁵ + 3312410q⁷⁶ + 4600767q⁷⁷ + 3658056q⁷⁸ + 1359420q⁷⁹ - 992130q⁸⁰ - 2411772q⁸¹ - 2486737q⁸² - 1490425q⁸³ - 120295q⁸⁴ + 974072q⁸⁵ + 1386952q⁸⁶ + 1111993q⁸⁷ + 455543q⁸⁸ - 225618q⁸⁹ - 629387q⁹⁰ - 655018q⁹¹ - 408671q⁹² - 58409q⁹³ + 212678q⁹⁴ + 312390q⁹⁵ + 258889q⁹⁶ + 110481q⁹⁷ - 37709q⁹⁸ - 120485q⁹⁹ - 129393q¹⁰⁰ - 78969q¹⁰¹ - 13156q¹⁰² + 33340q¹⁰³ + 52772q¹⁰⁴ + 41839q¹⁰⁵ + 17046q¹⁰⁶ - 4883q¹⁰⁷ - 18195q¹⁰⁸ - 17220q¹⁰⁹ - 9312q¹¹⁰ - 1660q¹¹¹ + 4577q¹¹² + 5950q¹¹³ + 4250q¹¹⁴ + 1720q¹¹⁵ - 1295q¹¹⁶ - 1890q¹¹⁷ - 1137q¹¹⁸ - 646q¹¹⁹ + 136q¹²⁰ + 319q¹²¹ + 395q¹²² + 385q¹²³ - 115q¹²⁴ - 195q¹²⁵ - 42q¹²⁶ - 11q¹²⁷ + 36q¹²⁸ - 25q¹²⁹ - 9q¹³⁰ + 65q¹³¹ - 15q¹³² - 30q¹³³ + 6q¹³⁵ + 16q¹³⁶ - 10q¹³⁷ - 5q¹³⁸ + 5q¹³⁹ - q¹⁴⁰

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=

<< KnotTheory`

Loading KnotTheory` (version of August 30, 2005, 10:15:35)...

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 12 14 -5 - q + --- - --- + -- - -- + -- + 4 q - 3 q + 3 q - 2 q + 3 q - q 12 10 8 4 2 q q q q q

In[17]:=
HOMFLYPT[Knot[10, 123]][a, z]

Out[17]=
2 4 6 2 2 2 z 2 2 4 2 z 2 4 6 z -3 + -- + 2 a - 4 z + -- + a z + 3 z - ---- - 2 a z + 4 z - -- - 2 2 2 2 a a a a 2 6 8 > a z + z

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 5 3 3 3 4 5 z 3 z 2 4 4 4 z 15 z > 21 a z + 5 a z + 4 z - ---- - ---- - 3 a z - 5 a z + -- - ----- - 4 2 5 3 a a a a 5 6 6 38 z 5 3 5 5 5 6 5 z 11 z 2 6 > ----- - 38 a z - 15 a z + a z - 32 z + ---- - ----- - 11 a z + a 4 2 a a 7 7 8 4 6 10 z 14 z 7 3 7 8 10 z 2 8 > 5 a z + ----- + ----- + 14 a z + 10 a z + 20 z + ----- + 10 a z + 3 a 2 a a 9 4 z 9 > ---- + 4 a z 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 10 -- + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 9 3 3 2 5 2 5 3 7 3 7 4 > --- + 9 q t + 10 q t + 6 q t + 9 q t + 4 q t + 6 q t + q t + q t 9 4 11 5 > 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 85 383 + q - --- + --- + --- - --- + --- + -- - --- - -- + --- - --- - -- + 14 13 12 11 10 9 8 7 6 5 4 q q q q q q q q q q q 331 215 169 2 3 4 5 6 > --- - --- - --- - 169 q - 215 q + 331 q - 85 q - 197 q + 206 q - 3 2 q q q 7 8 9 10 11 12 13 14 15 > 10 q - 121 q + 80 q + 14 q - 41 q + 15 q + 5 q - 5 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₂₃

10₁₂₂
10₁₂₄

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
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^-1 + 383 - 169q - 215q² + 331q³ - 85q⁴ - 197q⁵ + 206q⁶ - 10q⁷ - 121q⁸ + 80q⁹ + 14q¹⁰ - 41q¹¹ + 15q¹² + 5q¹³ - 5q¹⁴ + q¹⁵
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^-1 + 3691 + 192q - 3597q² - 584q³ + 3353q⁴ + 930q⁵ - 2948q⁶ - 1205q⁷ + 2413q⁸ + 1364q⁹ - 1800q¹⁰ - 1376q¹¹ + 1193q¹² + 1229q¹³ - 659q¹⁴ - 980q¹⁵ + 283q¹⁶ + 674q¹⁷ - 59q¹⁸ - 405q¹⁹ - 25q²⁰ + 200q²¹ + 46q²² - 95q²³ - 20q²⁴ + 31q²⁵ + 11q²⁶ - 10q²⁷ - 5q²⁸ + 5q²⁹ - q³⁰
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^-1 + 44955 + 11268q - 29818q² - 34554q³ + 2887q⁴ + 42330q⁵ + 18340q⁶ - 21175q⁷ - 35045q⁸ - 6460q⁹ + 33968q¹⁰ + 22728q¹¹ - 9205q¹² - 29641q¹³ - 14191q¹⁴ + 20543q¹⁵ + 21368q¹⁶ + 2209q¹⁷ - 18552q¹⁸ - 15956q¹⁹ + 6944q²⁰ + 13845q²¹ + 7535q²² - 6857q²³ - 11055q²⁴ - 707q²⁵ + 5172q²⁶ + 5976q²⁷ - 371q²⁸ - 4555q²⁹ - 1785q³⁰ + 536q³¹ + 2396q³² + 795q³³ - 1029q³⁴ - 645q³⁵ - 300q³⁶ + 521q³⁷ + 300q³⁸ - 144q³⁹ - 76q⁴⁰ - 111q⁴¹ + 75q⁴² + 50q⁴³ - 25q⁴⁴ - q⁴⁵ - 16q⁴⁶ + 10q⁴⁷ + 5q⁴⁸ - 5q⁴⁹ + q⁵⁰
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^-1 + 509105 + 267093q - 227767q² - 504874q³ - 303191q⁴ + 182697q⁵ + 491875q⁶ + 336393q⁷ - 130155q⁸ - 466752q⁹ - 364895q¹⁰ + 69028q¹¹ + 426473q¹² + 384561q¹³ - 1823q¹⁴ - 368820q¹⁵ - 389525q¹⁶ - 65918q¹⁷ + 294696q¹⁸ + 374479q¹⁹ + 125956q²⁰ - 209298q²¹ - 337053q²² - 169188q²³ + 121855q²⁴ + 279271q²⁵ + 188890q²⁶ - 43200q²⁷ - 208746q²⁸ - 183351q²⁹ - 16419q³⁰ + 136191q³¹ + 156391q³² + 51927q³³ - 72692q³⁴ - 117173q³⁵ - 63660q³⁶ + 25990q³⁷ + 75942q³⁸ + 57823q³⁹ + 1497q⁴⁰ - 41331q⁴¹ - 42692q⁴² - 12935q⁴³ + 17652q⁴⁴ + 26496q⁴⁵ + 13601q⁴⁶ - 4712q⁴⁷ - 13478q⁴⁸ - 9956q⁴⁹ - 618q⁵⁰ + 5808q⁵¹ + 5586q⁵² + 1562q⁵³ - 1825q⁵⁴ - 2571q⁵⁵ - 1237q⁵⁶ + 464q⁵⁷ + 1025q⁵⁸ + 551q⁵⁹ - 60q⁶⁰ - 316q⁶¹ - 216q⁶² - 26q⁶³ + 120q⁶⁴ + 82q⁶⁵ - 25q⁶⁶ - 30q⁶⁷ - 5q⁶⁸ - 5q⁶⁹ + 6q⁷⁰ + 16q⁷¹ - 10q⁷² - 5q⁷³ + 5q⁷⁴ - q⁷⁵
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^-1 + 6671309 + 4184939q - 1219025q² - 5762576q³ - 6152435q⁴ - 2125705q⁵ + 3410011q⁶ + 6587383q⁷ + 4894067q⁸ - 178345q⁹ - 5126450q¹⁰ - 6406680q¹¹ - 3178939q¹² + 2246899q¹³ + 6124552q¹⁴ + 5545016q¹⁵ + 1233253q¹⁶ - 3876461q¹⁷ - 6201858q¹⁸ - 4269721q¹⁹ + 506904q²⁰ + 4861446q²¹ + 5680763q²² + 2771269q²³ - 1878123q²⁴ - 5050456q²⁵ - 4788145q²⁶ - 1414996q²⁷ + 2723860q²⁸ + 4743125q²⁹ + 3666270q³⁰ + 316119q³¹ - 2944138q³² - 4118660q³³ - 2605860q³⁴ + 415859q³⁵ + 2803073q³⁶ + 3280016q³⁷ + 1687441q³⁸ - 718282q³⁹ - 2432667q⁴⁰ - 2457541q⁴¹ - 994356q⁴² + 799778q⁴³ + 1904108q⁴⁴ + 1714502q⁴⁵ + 576892q⁴⁶ - 737424q⁴⁷ - 1390524q⁴⁸ - 1120591q⁴⁹ - 296113q⁵⁰ + 566398q⁵¹ + 933692q⁵² + 718160q⁵³ + 125462q⁵⁴ - 398132q⁵⁵ - 579766q⁵⁶ - 421057q⁵⁷ - 55084q⁵⁸ + 248163q⁵⁹ + 350846q⁶⁰ + 223702q⁶¹ + 17510q⁶² - 139516q⁶³ - 188750q⁶⁴ - 116971q⁶⁵ - 5927q⁶⁶ + 80429q⁶⁷ + 89894q⁶⁸ + 53656q⁶⁹ + 2177q⁷⁰ - 38074q⁷¹ - 42311q⁷² - 23671q⁷³ + 3490q⁷⁴ + 15089q⁷⁵ + 16670q⁷⁶ + 9549q⁷⁷ - 1634q⁷⁸ - 6759q⁷⁹ - 6510q⁸⁰ - 1755q⁸¹ + 369q⁸² + 2135q⁸³ + 2266q⁸⁴ + 547q⁸⁵ - 519q⁸⁶ - 876q⁸⁷ - 160q⁸⁸ - 200q⁸⁹ + 100q⁹⁰ + 286q⁹¹ + 67q⁹² - 45q⁹³ - 111q⁹⁴ + 54q⁹⁵ - 20q⁹⁶ - 15q⁹⁷ + 35q⁹⁸ - 6q¹⁰⁰ - 16q¹⁰¹ + 10q¹⁰² + 5q¹⁰³ - 5q¹⁰⁴ + q¹⁰⁵
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^-1 + 87910159 + 62882041q + 2203806q² - 59795350q³ - 88039714q⁴ - 66276594q⁵ - 6976270q⁶ + 56446409q⁷ + 88260356q⁸ + 70271955q⁹ + 12750874q¹⁰ - 52101041q¹¹ - 88046411q¹² - 74793555q¹³ - 19960604q¹⁴ + 45992202q¹⁵ + 86611850q¹⁶ + 79366271q¹⁷ + 28658568q¹⁸ - 37518962q¹⁹ - 83022604q²⁰ - 83109137q²¹ - 38413430q²² + 26435776q²³ + 76415546q²⁴ + 84870725q²⁵ + 48269134q²⁶ - 13063190q²⁷ - 66273208q²⁸ - 83452910q²⁹ - 56831245q³⁰ - 1622508q³¹ + 52680978q³² + 77949831q³³ + 62539208q³⁴ + 16059829q³⁵ - 36505322q³⁶ - 68100082q³⁷ - 64052090q³⁸ - 28369637q³⁹ + 19350236q⁴⁰ + 54508287q⁴¹ + 60675114q⁴² + 36818791q⁴³ - 3265789q⁴⁴ - 38661673q⁴⁵ - 52668841q⁴⁶ - 40298558q⁴⁷ - 9727922q⁴⁸ + 22629174q⁴⁹ + 41268842q⁵⁰ + 38688427q⁵¹ + 18203416q⁵² - 8547351q⁵³ - 28421781q⁵⁴ - 32924207q⁵⁵ - 21701623q⁵⁶ - 1944871q⁵⁷ + 16239649q⁵⁸ + 24710309q⁵⁹ + 20830152q⁶⁰ + 8135986q⁶¹ - 6427765q⁶² - 16013247q⁶³ - 16996224q⁶⁴ - 10320484q⁶⁵ - 127110q⁶⁶ + 8481876q⁶⁷ + 11904596q⁶⁸ + 9564329q⁶⁹ + 3458772q⁷⁰ - 3064728q⁷¹ - 7038624q⁷² - 7250018q⁷³ - 4295721q⁷⁴ - 69631q⁷⁵ + 3312410q⁷⁶ + 4600767q⁷⁷ + 3658056q⁷⁸ + 1359420q⁷⁹ - 992130q⁸⁰ - 2411772q⁸¹ - 2486737q⁸² - 1490425q⁸³ - 120295q⁸⁴ + 974072q⁸⁵ + 1386952q⁸⁶ + 1111993q⁸⁷ + 455543q⁸⁸ - 225618q⁸⁹ - 629387q⁹⁰ - 655018q⁹¹ - 408671q⁹² - 58409q⁹³ + 212678q⁹⁴ + 312390q⁹⁵ + 258889q⁹⁶ + 110481q⁹⁷ - 37709q⁹⁸ - 120485q⁹⁹ - 129393q¹⁰⁰ - 78969q¹⁰¹ - 13156q¹⁰² + 33340q¹⁰³ + 52772q¹⁰⁴ + 41839q¹⁰⁵ + 17046q¹⁰⁶ - 4883q¹⁰⁷ - 18195q¹⁰⁸ - 17220q¹⁰⁹ - 9312q¹¹⁰ - 1660q¹¹¹ + 4577q¹¹² + 5950q¹¹³ + 4250q¹¹⁴ + 1720q¹¹⁵ - 1295q¹¹⁶ - 1890q¹¹⁷ - 1137q¹¹⁸ - 646q¹¹⁹ + 136q¹²⁰ + 319q¹²¹ + 395q¹²² + 385q¹²³ - 115q¹²⁴ - 195q¹²⁵ - 42q¹²⁶ - 11q¹²⁷ + 36q¹²⁸ - 25q¹²⁹ - 9q¹³⁰ + 65q¹³¹ - 15q¹³² - 30q¹³³ + 6q¹³⁵ + 16q¹³⁶ - 10q¹³⁷ - 5q¹³⁸ + 5q¹³⁹ - q¹⁴⁰

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
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 12 14 -5 - q + --- - --- + -- - -- + -- + 4 q - 3 q + 3 q - 2 q + 3 q - q 12 10 8 4 2 q q q q q
In[17]:=	HOMFLYPT[Knot[10, 123]][a, z]
Out[17]=	2 4 6 2 2 2 z 2 2 4 2 z 2 4 6 z -3 + -- + 2 a - 4 z + -- + a z + 3 z - ---- - 2 a z + 4 z - -- - 2 2 2 2 a a a a 2 6 8 > a z + z
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 5 3 3 3 4 5 z 3 z 2 4 4 4 z 15 z > 21 a z + 5 a z + 4 z - ---- - ---- - 3 a z - 5 a z + -- - ----- - 4 2 5 3 a a a a 5 6 6 38 z 5 3 5 5 5 6 5 z 11 z 2 6 > ----- - 38 a z - 15 a z + a z - 32 z + ---- - ----- - 11 a z + a 4 2 a a 7 7 8 4 6 10 z 14 z 7 3 7 8 10 z 2 8 > 5 a z + ----- + ----- + 14 a z + 10 a z + 20 z + ----- + 10 a z + 3 a 2 a a 9 4 z 9 > ---- + 4 a z 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 10 -- + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 9 3 3 2 5 2 5 3 7 3 7 4 > --- + 9 q t + 10 q t + 6 q t + 9 q t + 4 q t + 6 q t + q t + q t 9 4 11 5 > 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 85 383 + q - --- + --- + --- - --- + --- + -- - --- - -- + --- - --- - -- + 14 13 12 11 10 9 8 7 6 5 4 q q q q q q q q q q q 331 215 169 2 3 4 5 6 > --- - --- - --- - 169 q - 215 q + 331 q - 85 q - 197 q + 206 q - 3 2 q q q 7 8 9 10 11 12 13 14 15 > 10 q - 121 q + 80 q + 14 q - 41 q + 15 q + 5 q - 5 q + q

The Alternating Knot 10123

The Alternating Knot 10₁₂₃