The Alternating Knot 10₁₀₂

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X_16,10,17,9 X_10,3,11,4 X_2,15,3,16 X_14,5,15,6 X_18,8,19,7 X_4,11,5,12 X_8,18,9,17 X_20,14,1,13 X_12,20,13,19

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

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

Minimum Braid Representative:

Length is 11, width is 4
Braid index is 4

A Morse Link Presentation:

3D Invariants:

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

Chiral 1 3 3 / NotAvailable 1

Alexander Polynomial: - 2t^-3 + 8t^-2 - 16t^-1 + 21 - 16t + 8t² - 2t³

Conway Polynomial: 1 - 2z² - 4z⁴ - 2z⁶

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

Determinant and Signature: {73, 0}

Jones Polynomial: q^-4 - 3q^-3 + 6q^-2 - 9q^-1 + 12 - 12q + 11q² - 9q³ + 6q⁴ - 3q⁵ + q⁶

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

A2 (sl(3)) Invariant: q^-12 - q^-10 + q^-8 + q^-6 - 2q^-4 + 3q^-2 - 1 + q² - 2q⁶ + 2q⁸ - 2q¹⁰ + q¹² + q¹⁴ - q¹⁶ + q¹⁸

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

Kauffman Polynomial: 2a^-6z² - 3a^-6z⁴ + a^-6z⁶ - 2a^-5z + 7a^-5z³ - 9a^-5z⁵ + 3a^-5z⁷ + a^-4 - 4a^-4z² + 8a^-4z⁴ - 11a^-4z⁶ + 4a^-4z⁸ - 4a^-3z + 13a^-3z³ - 14a^-3z⁵ + a^-3z⁷ + 2a^-3z⁹ + a^-2 - 8a^-2z² + 21a^-2z⁴ - 24a^-2z⁶ + 9a^-2z⁸ - 4a^-1z + 16a^-1z³ - 17a^-1z⁵ + 4a^-1z⁷ + 2a^-1z⁹ + 2z² + 3z⁴ - 7z⁶ + 5z⁸ - 2az + 7az³ - 9az⁵ + 6az⁷ - a² + 3a²z² - 6a²z⁴ + 5a²z⁶ - 3a³z³ + 3a³z⁵ - a⁴z² + a⁴z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-2, -1}

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 10₁₀₂. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6

j = 13 1

j = 11 2

j = 9 4 1

j = 7 5 2

j = 5 6 4

j = 3 6 5

j = 1 6 6

j = -1 4 7

j = -3 2 5

j = -5 1 4

j = -7 2

j = -9 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-12 - 3q^-11 + 2q^-10 + 6q^-9 - 15q^-8 + 10q^-7 + 18q^-6 - 45q^-5 + 24q^-4 + 46q^-3 - 87q^-2 + 28q^-1 + 83 - 113q + 15q² + 104q³ - 106q⁴ - 7q⁵ + 101q⁶ - 74q⁷ - 26q⁸ + 76q⁹ - 34q¹⁰ - 29q¹¹ + 39q¹² - 6q¹³ - 16q¹⁴ + 10q¹⁵ + q¹⁶ - 3q¹⁷ + q¹⁸

3 q^-24 - 3q^-23 + 2q^-22 + 2q^-21 - 7q^-19 + 4q^-18 + 9q^-17 - 11q^-16 - 12q^-15 + 28q^-14 + 19q^-13 - 58q^-12 - 42q^-11 + 109q^-10 + 82q^-9 - 162q^-8 - 158q^-7 + 214q^-6 + 254q^-5 - 240q^-4 - 368q^-3 + 247q^-2 + 460q^-1 - 205 - 548q + 162q² + 585q³ - 88q⁴ - 602q⁵ + 18q⁶ + 582q⁷ + 60q⁸ - 542q⁹ - 135q¹⁰ + 481q¹¹ + 201q¹² - 399q¹³ - 254q¹⁴ + 302q¹⁵ + 284q¹⁶ - 197q¹⁷ - 284q¹⁸ + 99q¹⁹ + 249q²⁰ - 17q²¹ - 191q²² - 37q²³ + 129q²⁴ + 52q²⁵ - 67q²⁶ - 49q²⁷ + 28q²⁸ + 32q²⁹ - 8q³⁰ - 16q³¹ + 2q³² + 5q³³ + q³⁴ - 3q³⁵ + q³⁶

4 q^-40 - 3q^-39 + 2q^-38 + 2q^-37 - 4q^-36 + 8q^-35 - 13q^-34 + 6q^-33 + 4q^-32 - 16q^-31 + 40q^-30 - 29q^-29 + 6q^-28 - 19q^-27 - 65q^-26 + 132q^-25 + 16q^-24 + 40q^-23 - 140q^-22 - 298q^-21 + 243q^-20 + 273q^-19 + 348q^-18 - 305q^-17 - 977q^-16 + 25q^-15 + 686q^-14 + 1279q^-13 - 68q^-12 - 1992q^-11 - 922q^-10 + 693q^-9 + 2632q^-8 + 975q^-7 - 2639q^-6 - 2287q^-5 - 109q^-4 + 3579q^-3 + 2403q^-2 - 2436q^-1 - 3215 - 1318q + 3638q² + 3418q³ - 1678q⁴ - 3336q⁵ - 2276q⁶ + 3059q⁷ + 3738q⁸ - 816q⁹ - 2887q¹⁰ - 2832q¹¹ + 2177q¹² + 3579q¹³ + 65q¹⁴ - 2115q¹⁵ - 3118q¹⁶ + 1059q¹⁷ + 3045q¹⁸ + 958q¹⁹ - 1021q²⁰ - 3039q²¹ - 186q²² + 2032q²³ + 1529q²⁴ + 255q²⁵ - 2312q²⁶ - 1081q²⁷ + 675q²⁸ + 1348q²⁹ + 1159q³⁰ - 1079q³¹ - 1125q³² - 372q³³ + 549q³⁴ + 1182q³⁵ - 60q³⁶ - 510q³⁷ - 595q³⁸ - 122q³⁹ + 603q⁴⁰ + 234q⁴¹ + 9q⁴² - 283q⁴³ - 239q⁴⁴ + 135q⁴⁵ + 101q⁴⁶ + 103q⁴⁷ - 42q⁴⁸ - 98q⁴⁹ + 8q⁵⁰ + 4q⁵¹ + 35q⁵² + 4q⁵³ - 19q⁵⁴ + 2q⁵⁵ - 3q⁵⁶ + 5q⁵⁷ + q⁵⁸ - 3q⁵⁹ + q⁶⁰

5 q^-60 - 3q^-59 + 2q^-58 + 2q^-57 - 4q^-56 + 4q^-55 + 2q^-54 - 11q^-53 + q^-52 + 10q^-51 - 3q^-50 + 13q^-49 + 8q^-48 - 40q^-47 - 32q^-46 + 13q^-45 + 50q^-44 + 84q^-43 + 41q^-42 - 124q^-41 - 225q^-40 - 109q^-39 + 187q^-38 + 459q^-37 + 370q^-36 - 207q^-35 - 875q^-34 - 923q^-33 + 42q^-32 + 1431q^-31 + 1934q^-30 + 586q^-29 - 1961q^-28 - 3534q^-27 - 2047q^-26 + 2170q^-25 + 5657q^-24 + 4561q^-23 - 1488q^-22 - 7902q^-21 - 8329q^-20 - 549q^-19 + 9782q^-18 + 12919q^-17 + 4184q^-16 - 10441q^-15 - 17801q^-14 - 9293q^-13 + 9535q^-12 + 22028q^-11 + 15224q^-10 - 6866q^-9 - 24932q^-8 - 21127q^-7 + 2937q^-6 + 25991q^-5 + 26177q^-4 + 1714q^-3 - 25525q^-2 - 29716q^-1 - 6124 + 23621q + 31695q² + 10011q³ - 21220q⁴ - 32257q⁵ - 12795q⁶ + 18420q⁷ + 31811q⁸ + 14891q⁹ - 15790q¹⁰ - 30781q¹¹ - 16275q¹² + 13141q¹³ + 29344q¹⁴ + 17472q¹⁵ - 10427q¹⁶ - 27651q¹⁷ - 18533q¹⁸ + 7395q¹⁹ + 25525q²⁰ + 19550q²¹ - 3923q²² - 22779q²³ - 20330q²⁴ + 44q²⁵ + 19230q²⁶ + 20514q²⁷ + 4003q²⁸ - 14815q²⁹ - 19725q³⁰ - 7753q³¹ + 9687q³² + 17699q³³ + 10632q³⁴ - 4326q³⁵ - 14368q³⁶ - 12077q³⁷ - 659q³⁸ + 10028q³⁹ + 11851q⁴⁰ + 4510q⁴¹ - 5332q⁴² - 9976q⁴³ - 6689q⁴⁴ + 1015q⁴⁵ + 6977q⁴⁶ + 7099q⁴⁷ + 2126q⁴⁸ - 3646q⁴⁹ - 5956q⁵⁰ - 3730q⁵¹ + 714q⁵² + 3977q⁵³ + 3922q⁵⁴ + 1146q⁵⁵ - 1897q⁵⁶ - 3045q⁵⁷ - 1920q⁵⁸ + 283q⁵⁹ + 1853q⁶⁰ + 1791q⁶¹ + 550q⁶² - 757q⁶³ - 1232q⁶⁴ - 758q⁶⁵ + 82q⁶⁶ + 637q⁶⁷ + 605q⁶⁸ + 179q⁶⁹ - 235q⁷⁰ - 335q⁷¹ - 192q⁷² + 17q⁷³ + 157q⁷⁴ + 128q⁷⁵ + 16q⁷⁶ - 50q⁷⁷ - 45q⁷⁸ - 30q⁷⁹ + 8q⁸⁰ + 30q⁸¹ + 6q⁸² - 7q⁸³ - q⁸⁴ - 3q⁸⁵ - 3q⁸⁶ + 5q⁸⁷ + q⁸⁸ - 3q⁸⁹ + q⁹⁰

6 q^-84 - 3q^-83 + 2q^-82 + 2q^-81 - 4q^-80 + 4q^-79 - 2q^-78 + 4q^-77 - 16q^-76 + 7q^-75 + 23q^-74 - 19q^-73 + 11q^-72 - 8q^-71 - 4q^-70 - 60q^-69 + 22q^-68 + 110q^-67 - 11q^-66 + 27q^-65 - 52q^-64 - 102q^-63 - 239q^-62 + 51q^-61 + 411q^-60 + 215q^-59 + 202q^-58 - 191q^-57 - 619q^-56 - 1038q^-55 - 185q^-54 + 1212q^-53 + 1457q^-52 + 1504q^-51 + 28q^-50 - 2246q^-49 - 4221q^-48 - 2636q^-47 + 1824q^-46 + 5249q^-45 + 7427q^-44 + 4221q^-43 - 3840q^-42 - 12778q^-41 - 13618q^-40 - 3983q^-39 + 9688q^-38 + 22982q^-37 + 22655q^-36 + 4683q^-35 - 23473q^-34 - 40384q^-33 - 31353q^-32 - 315q^-31 + 42586q^-30 + 64186q^-29 + 43494q^-28 - 15256q^-27 - 73297q^-26 - 89136q^-25 - 49053q^-24 + 38645q^-23 + 113070q^-22 + 118814q^-21 + 38491q^-20 - 78103q^-19 - 153672q^-18 - 137253q^-17 - 15560q^-16 + 129097q^-15 + 198022q^-14 + 131162q^-13 - 30045q^-12 - 180644q^-11 - 224132q^-10 - 105680q^-9 + 91357q^-8 + 235317q^-7 + 216685q^-6 + 49941q^-5 - 154497q^-4 - 265980q^-3 - 184231q^-2 + 25412q^-1 + 221314 + 257165q + 116818q² - 103337q³ - 260122q⁴ - 221404q⁵ - 28946q⁶ + 184503q⁷ + 256463q⁸ + 149756q⁹ - 60229q¹⁰ - 234155q¹¹ - 226308q¹² - 59247q¹³ + 150885q¹⁴ + 240301q¹⁵ + 161460q¹⁶ - 30735q¹⁷ - 207773q¹⁸ - 221925q¹⁹ - 80040q²⁰ + 120894q²¹ + 222886q²² + 171609q²³ + 117q²⁴ - 177844q²⁵ - 217422q²⁶ - 107233q²⁷ + 80015q²⁸ + 197880q²⁹ + 184290q³⁰ + 44450q³¹ - 130123q³² - 202709q³³ - 139593q³⁴ + 19514q³⁵ + 150009q³⁶ + 184571q³⁷ + 95729q³⁸ - 58160q³⁹ - 160004q⁴⁰ - 156602q⁴¹ - 48988q⁴² + 73937q⁴³ + 151189q⁴⁴ + 127695q⁴⁵ + 21804q⁴⁶ - 84650q⁴⁷ - 133650q⁴⁸ - 93783q⁴⁹ - 9705q⁵⁰ + 80747q⁵¹ + 113490q⁵² + 73356q⁵³ - 2028q⁵⁴ - 70063q⁵⁵ - 87335q⁵⁶ - 60852q⁵⁷ + 3884q⁵⁸ + 56932q⁵⁹ + 70330q⁶⁰ + 45337q⁶¹ - 1927q⁶² - 39274q⁶³ - 56952q⁶⁴ - 36130q⁶⁵ - 1052q⁶⁶ + 28989q⁶⁷ + 40296q⁶⁸ + 29024q⁶⁹ + 6473q⁷⁰ - 20834q⁷¹ - 28992q⁷² - 22897q⁷³ - 6186q⁷⁴ + 10828q⁷⁵ + 20077q⁷⁶ + 19052q⁷⁷ + 5394q⁷⁸ - 5607q⁷⁹ - 13210q⁸⁰ - 12597q⁸¹ - 6422q⁸² + 2364q⁸³ + 9048q⁸⁴ + 7925q⁸⁵ + 4951q⁸⁶ - 654q⁸⁷ - 4408q⁸⁸ - 5774q⁸⁹ - 3476q⁹⁰ + 358q⁹¹ + 2014q⁹² + 3217q⁹³ + 2130q⁹⁴ + 541q⁹⁵ - 1349q⁹⁶ - 1696q⁹⁷ - 950q⁹⁸ - 475q⁹⁹ + 506q¹⁰⁰ + 764q¹⁰¹ + 714q¹⁰² + 66q¹⁰³ - 224q¹⁰⁴ - 221q¹⁰⁵ - 318q¹⁰⁶ - 79q¹⁰⁷ + 62q¹⁰⁸ + 184q¹⁰⁹ + 50q¹¹⁰ + 7q¹¹¹ + 12q¹¹² - 59q¹¹³ - 30q¹¹⁴ - 11q¹¹⁵ + 33q¹¹⁶ + q¹¹⁷ - 5q¹¹⁸ + 11q¹¹⁹ - 6q¹²⁰ - 3q¹²¹ - 3q¹²² + 5q¹²³ + q¹²⁴ - 3q¹²⁵ + q¹²⁶

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, 102]]

Out[2]=
PD[X[6, 2, 7, 1], X[16, 10, 17, 9], X[10, 3, 11, 4], X[2, 15, 3, 16], > X[14, 5, 15, 6], X[18, 8, 19, 7], X[4, 11, 5, 12], X[8, 18, 9, 17], > X[20, 14, 1, 13], X[12, 20, 13, 19]]

In[3]:=
GaussCode[Knot[10, 102]]

Out[3]=
GaussCode[1, -4, 3, -7, 5, -1, 6, -8, 2, -3, 7, -10, 9, -5, 4, -2, 8, -6, 10, > -9]

In[4]:=
DTCode[Knot[10, 102]]

Out[4]=
DTCode[6, 10, 14, 18, 16, 4, 20, 2, 8, 12]

In[5]:=
br = BR[Knot[10, 102]]

Out[5]=
BR[4, {-1, -1, 2, -1, -3, 2, -1, 2, 2, 3, 3}]

In[6]:=
{First[br], Crossings[br]}

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 102]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 102]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 102]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Chiral, 1, 3, 3, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 102]][t]

Out[10]=
2 8 16 2 3 21 - -- + -- - -- - 16 t + 8 t - 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 102]][z]

Out[11]=
2 4 6 1 - 2 z - 4 z - 2 z

In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]

Out[12]=
{Knot[10, 102]}

In[13]:=
{KnotDet[Knot[10, 102]], KnotSignature[Knot[10, 102]]}

Out[13]=
{73, 0}

In[14]:=
Jones[Knot[10, 102]][q]

Out[14]=
-4 3 6 9 2 3 4 5 6 12 + q - -- + -- - - - 12 q + 11 q - 9 q + 6 q - 3 q + q 3 2 q q q

In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]

Out[15]=
{Knot[10, 102]}

In[16]:=
A2Invariant[Knot[10, 102]][q]

Out[16]=
-12 -10 -8 -6 2 3 2 6 8 10 12 14 -1 + q - q + q + q - -- + -- + q - 2 q + 2 q - 2 q + q + q - 4 2 q q 16 18 > q + q

In[17]:=
HOMFLYPT[Knot[10, 102]][a, z]

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

In[18]:=
Kauffman[Knot[10, 102]][a, z]

Out[18]=
2 2 2 -4 -2 2 2 z 4 z 4 z 2 2 z 4 z 8 z a + a - a - --- - --- - --- - 2 a z + 2 z + ---- - ---- - ---- + 5 3 a 6 4 2 a a a a a 3 3 3 4 2 2 4 2 7 z 13 z 16 z 3 3 3 4 3 z > 3 a z - a z + ---- + ----- + ----- + 7 a z - 3 a z + 3 z - ---- + 5 3 a 6 a a a 4 4 5 5 5 8 z 21 z 2 4 4 4 9 z 14 z 17 z 5 3 5 > ---- + ----- - 6 a z + a z - ---- - ----- - ----- - 9 a z + 3 a z - 4 2 5 3 a a a a a 6 6 6 7 7 7 6 z 11 z 24 z 2 6 3 z z 4 z 7 8 > 7 z + -- - ----- - ----- + 5 a z + ---- + -- + ---- + 6 a z + 5 z + 6 4 2 5 3 a a a a a a 8 8 9 9 4 z 9 z 2 z 2 z > ---- + ---- + ---- + ---- 4 2 3 a a a a

In[19]:=
{Vassiliev[2][Knot[10, 102]], Vassiliev[3][Knot[10, 102]]}

Out[19]=
{-2, -1}

In[20]:=
Kh[Knot[10, 102]][q, t]

Out[20]=
7 1 2 1 4 2 5 4 3 - + 6 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 6 q t + 6 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 5 q t + 6 q t + 4 q t + 5 q t + 2 q t + 4 q t + q t + 11 5 13 6 > 2 q t + q t

In[21]:=
ColouredJones[Knot[10, 102], 2][q]

Out[21]=
-12 3 2 6 15 10 18 45 24 46 87 28 83 + q - --- + --- + -- - -- + -- + -- - -- + -- + -- - -- + -- - 113 q + 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 2 3 4 5 6 7 8 9 10 > 15 q + 104 q - 106 q - 7 q + 101 q - 74 q - 26 q + 76 q - 34 q - 11 12 13 14 15 16 17 18 > 29 q + 39 q - 6 q - 16 q + 10 q + q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₀₂

10₁₀₁
10₁₀₃

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-12 - 3q^-11 + 2q^-10 + 6q^-9 - 15q^-8 + 10q^-7 + 18q^-6 - 45q^-5 + 24q^-4 + 46q^-3 - 87q^-2 + 28q^-1 + 83 - 113q + 15q² + 104q³ - 106q⁴ - 7q⁵ + 101q⁶ - 74q⁷ - 26q⁸ + 76q⁹ - 34q¹⁰ - 29q¹¹ + 39q¹² - 6q¹³ - 16q¹⁴ + 10q¹⁵ + q¹⁶ - 3q¹⁷ + q¹⁸
3	q^-24 - 3q^-23 + 2q^-22 + 2q^-21 - 7q^-19 + 4q^-18 + 9q^-17 - 11q^-16 - 12q^-15 + 28q^-14 + 19q^-13 - 58q^-12 - 42q^-11 + 109q^-10 + 82q^-9 - 162q^-8 - 158q^-7 + 214q^-6 + 254q^-5 - 240q^-4 - 368q^-3 + 247q^-2 + 460q^-1 - 205 - 548q + 162q² + 585q³ - 88q⁴ - 602q⁵ + 18q⁶ + 582q⁷ + 60q⁸ - 542q⁹ - 135q¹⁰ + 481q¹¹ + 201q¹² - 399q¹³ - 254q¹⁴ + 302q¹⁵ + 284q¹⁶ - 197q¹⁷ - 284q¹⁸ + 99q¹⁹ + 249q²⁰ - 17q²¹ - 191q²² - 37q²³ + 129q²⁴ + 52q²⁵ - 67q²⁶ - 49q²⁷ + 28q²⁸ + 32q²⁹ - 8q³⁰ - 16q³¹ + 2q³² + 5q³³ + q³⁴ - 3q³⁵ + q³⁶
4	q^-40 - 3q^-39 + 2q^-38 + 2q^-37 - 4q^-36 + 8q^-35 - 13q^-34 + 6q^-33 + 4q^-32 - 16q^-31 + 40q^-30 - 29q^-29 + 6q^-28 - 19q^-27 - 65q^-26 + 132q^-25 + 16q^-24 + 40q^-23 - 140q^-22 - 298q^-21 + 243q^-20 + 273q^-19 + 348q^-18 - 305q^-17 - 977q^-16 + 25q^-15 + 686q^-14 + 1279q^-13 - 68q^-12 - 1992q^-11 - 922q^-10 + 693q^-9 + 2632q^-8 + 975q^-7 - 2639q^-6 - 2287q^-5 - 109q^-4 + 3579q^-3 + 2403q^-2 - 2436q^-1 - 3215 - 1318q + 3638q² + 3418q³ - 1678q⁴ - 3336q⁵ - 2276q⁶ + 3059q⁷ + 3738q⁸ - 816q⁹ - 2887q¹⁰ - 2832q¹¹ + 2177q¹² + 3579q¹³ + 65q¹⁴ - 2115q¹⁵ - 3118q¹⁶ + 1059q¹⁷ + 3045q¹⁸ + 958q¹⁹ - 1021q²⁰ - 3039q²¹ - 186q²² + 2032q²³ + 1529q²⁴ + 255q²⁵ - 2312q²⁶ - 1081q²⁷ + 675q²⁸ + 1348q²⁹ + 1159q³⁰ - 1079q³¹ - 1125q³² - 372q³³ + 549q³⁴ + 1182q³⁵ - 60q³⁶ - 510q³⁷ - 595q³⁸ - 122q³⁹ + 603q⁴⁰ + 234q⁴¹ + 9q⁴² - 283q⁴³ - 239q⁴⁴ + 135q⁴⁵ + 101q⁴⁶ + 103q⁴⁷ - 42q⁴⁸ - 98q⁴⁹ + 8q⁵⁰ + 4q⁵¹ + 35q⁵² + 4q⁵³ - 19q⁵⁴ + 2q⁵⁵ - 3q⁵⁶ + 5q⁵⁷ + q⁵⁸ - 3q⁵⁹ + q⁶⁰
5	q^-60 - 3q^-59 + 2q^-58 + 2q^-57 - 4q^-56 + 4q^-55 + 2q^-54 - 11q^-53 + q^-52 + 10q^-51 - 3q^-50 + 13q^-49 + 8q^-48 - 40q^-47 - 32q^-46 + 13q^-45 + 50q^-44 + 84q^-43 + 41q^-42 - 124q^-41 - 225q^-40 - 109q^-39 + 187q^-38 + 459q^-37 + 370q^-36 - 207q^-35 - 875q^-34 - 923q^-33 + 42q^-32 + 1431q^-31 + 1934q^-30 + 586q^-29 - 1961q^-28 - 3534q^-27 - 2047q^-26 + 2170q^-25 + 5657q^-24 + 4561q^-23 - 1488q^-22 - 7902q^-21 - 8329q^-20 - 549q^-19 + 9782q^-18 + 12919q^-17 + 4184q^-16 - 10441q^-15 - 17801q^-14 - 9293q^-13 + 9535q^-12 + 22028q^-11 + 15224q^-10 - 6866q^-9 - 24932q^-8 - 21127q^-7 + 2937q^-6 + 25991q^-5 + 26177q^-4 + 1714q^-3 - 25525q^-2 - 29716q^-1 - 6124 + 23621q + 31695q² + 10011q³ - 21220q⁴ - 32257q⁵ - 12795q⁶ + 18420q⁷ + 31811q⁸ + 14891q⁹ - 15790q¹⁰ - 30781q¹¹ - 16275q¹² + 13141q¹³ + 29344q¹⁴ + 17472q¹⁵ - 10427q¹⁶ - 27651q¹⁷ - 18533q¹⁸ + 7395q¹⁹ + 25525q²⁰ + 19550q²¹ - 3923q²² - 22779q²³ - 20330q²⁴ + 44q²⁵ + 19230q²⁶ + 20514q²⁷ + 4003q²⁸ - 14815q²⁹ - 19725q³⁰ - 7753q³¹ + 9687q³² + 17699q³³ + 10632q³⁴ - 4326q³⁵ - 14368q³⁶ - 12077q³⁷ - 659q³⁸ + 10028q³⁹ + 11851q⁴⁰ + 4510q⁴¹ - 5332q⁴² - 9976q⁴³ - 6689q⁴⁴ + 1015q⁴⁵ + 6977q⁴⁶ + 7099q⁴⁷ + 2126q⁴⁸ - 3646q⁴⁹ - 5956q⁵⁰ - 3730q⁵¹ + 714q⁵² + 3977q⁵³ + 3922q⁵⁴ + 1146q⁵⁵ - 1897q⁵⁶ - 3045q⁵⁷ - 1920q⁵⁸ + 283q⁵⁹ + 1853q⁶⁰ + 1791q⁶¹ + 550q⁶² - 757q⁶³ - 1232q⁶⁴ - 758q⁶⁵ + 82q⁶⁶ + 637q⁶⁷ + 605q⁶⁸ + 179q⁶⁹ - 235q⁷⁰ - 335q⁷¹ - 192q⁷² + 17q⁷³ + 157q⁷⁴ + 128q⁷⁵ + 16q⁷⁶ - 50q⁷⁷ - 45q⁷⁸ - 30q⁷⁹ + 8q⁸⁰ + 30q⁸¹ + 6q⁸² - 7q⁸³ - q⁸⁴ - 3q⁸⁵ - 3q⁸⁶ + 5q⁸⁷ + q⁸⁸ - 3q⁸⁹ + q⁹⁰
6	q^-84 - 3q^-83 + 2q^-82 + 2q^-81 - 4q^-80 + 4q^-79 - 2q^-78 + 4q^-77 - 16q^-76 + 7q^-75 + 23q^-74 - 19q^-73 + 11q^-72 - 8q^-71 - 4q^-70 - 60q^-69 + 22q^-68 + 110q^-67 - 11q^-66 + 27q^-65 - 52q^-64 - 102q^-63 - 239q^-62 + 51q^-61 + 411q^-60 + 215q^-59 + 202q^-58 - 191q^-57 - 619q^-56 - 1038q^-55 - 185q^-54 + 1212q^-53 + 1457q^-52 + 1504q^-51 + 28q^-50 - 2246q^-49 - 4221q^-48 - 2636q^-47 + 1824q^-46 + 5249q^-45 + 7427q^-44 + 4221q^-43 - 3840q^-42 - 12778q^-41 - 13618q^-40 - 3983q^-39 + 9688q^-38 + 22982q^-37 + 22655q^-36 + 4683q^-35 - 23473q^-34 - 40384q^-33 - 31353q^-32 - 315q^-31 + 42586q^-30 + 64186q^-29 + 43494q^-28 - 15256q^-27 - 73297q^-26 - 89136q^-25 - 49053q^-24 + 38645q^-23 + 113070q^-22 + 118814q^-21 + 38491q^-20 - 78103q^-19 - 153672q^-18 - 137253q^-17 - 15560q^-16 + 129097q^-15 + 198022q^-14 + 131162q^-13 - 30045q^-12 - 180644q^-11 - 224132q^-10 - 105680q^-9 + 91357q^-8 + 235317q^-7 + 216685q^-6 + 49941q^-5 - 154497q^-4 - 265980q^-3 - 184231q^-2 + 25412q^-1 + 221314 + 257165q + 116818q² - 103337q³ - 260122q⁴ - 221404q⁵ - 28946q⁶ + 184503q⁷ + 256463q⁸ + 149756q⁹ - 60229q¹⁰ - 234155q¹¹ - 226308q¹² - 59247q¹³ + 150885q¹⁴ + 240301q¹⁵ + 161460q¹⁶ - 30735q¹⁷ - 207773q¹⁸ - 221925q¹⁹ - 80040q²⁰ + 120894q²¹ + 222886q²² + 171609q²³ + 117q²⁴ - 177844q²⁵ - 217422q²⁶ - 107233q²⁷ + 80015q²⁸ + 197880q²⁹ + 184290q³⁰ + 44450q³¹ - 130123q³² - 202709q³³ - 139593q³⁴ + 19514q³⁵ + 150009q³⁶ + 184571q³⁷ + 95729q³⁸ - 58160q³⁹ - 160004q⁴⁰ - 156602q⁴¹ - 48988q⁴² + 73937q⁴³ + 151189q⁴⁴ + 127695q⁴⁵ + 21804q⁴⁶ - 84650q⁴⁷ - 133650q⁴⁸ - 93783q⁴⁹ - 9705q⁵⁰ + 80747q⁵¹ + 113490q⁵² + 73356q⁵³ - 2028q⁵⁴ - 70063q⁵⁵ - 87335q⁵⁶ - 60852q⁵⁷ + 3884q⁵⁸ + 56932q⁵⁹ + 70330q⁶⁰ + 45337q⁶¹ - 1927q⁶² - 39274q⁶³ - 56952q⁶⁴ - 36130q⁶⁵ - 1052q⁶⁶ + 28989q⁶⁷ + 40296q⁶⁸ + 29024q⁶⁹ + 6473q⁷⁰ - 20834q⁷¹ - 28992q⁷² - 22897q⁷³ - 6186q⁷⁴ + 10828q⁷⁵ + 20077q⁷⁶ + 19052q⁷⁷ + 5394q⁷⁸ - 5607q⁷⁹ - 13210q⁸⁰ - 12597q⁸¹ - 6422q⁸² + 2364q⁸³ + 9048q⁸⁴ + 7925q⁸⁵ + 4951q⁸⁶ - 654q⁸⁷ - 4408q⁸⁸ - 5774q⁸⁹ - 3476q⁹⁰ + 358q⁹¹ + 2014q⁹² + 3217q⁹³ + 2130q⁹⁴ + 541q⁹⁵ - 1349q⁹⁶ - 1696q⁹⁷ - 950q⁹⁸ - 475q⁹⁹ + 506q¹⁰⁰ + 764q¹⁰¹ + 714q¹⁰² + 66q¹⁰³ - 224q¹⁰⁴ - 221q¹⁰⁵ - 318q¹⁰⁶ - 79q¹⁰⁷ + 62q¹⁰⁸ + 184q¹⁰⁹ + 50q¹¹⁰ + 7q¹¹¹ + 12q¹¹² - 59q¹¹³ - 30q¹¹⁴ - 11q¹¹⁵ + 33q¹¹⁶ + q¹¹⁷ - 5q¹¹⁸ + 11q¹¹⁹ - 6q¹²⁰ - 3q¹²¹ - 3q¹²² + 5q¹²³ + q¹²⁴ - 3q¹²⁵ + q¹²⁶

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 102]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 10, 17, 9], X[10, 3, 11, 4], X[2, 15, 3, 16], > X[14, 5, 15, 6], X[18, 8, 19, 7], X[4, 11, 5, 12], X[8, 18, 9, 17], > X[20, 14, 1, 13], X[12, 20, 13, 19]]
In[3]:=	GaussCode[Knot[10, 102]]
Out[3]=	GaussCode[1, -4, 3, -7, 5, -1, 6, -8, 2, -3, 7, -10, 9, -5, 4, -2, 8, -6, 10, > -9]
In[4]:=	DTCode[Knot[10, 102]]
Out[4]=	DTCode[6, 10, 14, 18, 16, 4, 20, 2, 8, 12]
In[5]:=	br = BR[Knot[10, 102]]
Out[5]=	BR[4, {-1, -1, 2, -1, -3, 2, -1, 2, 2, 3, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 102]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 102]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 102]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 102]][t]
Out[10]=	2 8 16 2 3 21 - -- + -- - -- - 16 t + 8 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 102]][z]
Out[11]=	2 4 6 1 - 2 z - 4 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 102]}
In[13]:=	{KnotDet[Knot[10, 102]], KnotSignature[Knot[10, 102]]}
Out[13]=	{73, 0}
In[14]:=	Jones[Knot[10, 102]][q]
Out[14]=	-4 3 6 9 2 3 4 5 6 12 + q - -- + -- - - - 12 q + 11 q - 9 q + 6 q - 3 q + q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 102]}
In[16]:=	A2Invariant[Knot[10, 102]][q]
Out[16]=	-12 -10 -8 -6 2 3 2 6 8 10 12 14 -1 + q - q + q + q - -- + -- + q - 2 q + 2 q - 2 q + q + q - 4 2 q q 16 18 > q + q
In[17]:=	HOMFLYPT[Knot[10, 102]][a, z]
Out[17]=	2 2 4 4 -4 -2 2 2 2 z 3 z 2 2 4 z 3 z 2 4 6 a - a + a - 3 z + ---- - ---- + 2 a z - 3 z + -- - ---- + a z - z - 4 2 4 2 a a a a 6 z > -- 2 a
In[18]:=	Kauffman[Knot[10, 102]][a, z]
Out[18]=	2 2 2 -4 -2 2 2 z 4 z 4 z 2 2 z 4 z 8 z a + a - a - --- - --- - --- - 2 a z + 2 z + ---- - ---- - ---- + 5 3 a 6 4 2 a a a a a 3 3 3 4 2 2 4 2 7 z 13 z 16 z 3 3 3 4 3 z > 3 a z - a z + ---- + ----- + ----- + 7 a z - 3 a z + 3 z - ---- + 5 3 a 6 a a a 4 4 5 5 5 8 z 21 z 2 4 4 4 9 z 14 z 17 z 5 3 5 > ---- + ----- - 6 a z + a z - ---- - ----- - ----- - 9 a z + 3 a z - 4 2 5 3 a a a a a 6 6 6 7 7 7 6 z 11 z 24 z 2 6 3 z z 4 z 7 8 > 7 z + -- - ----- - ----- + 5 a z + ---- + -- + ---- + 6 a z + 5 z + 6 4 2 5 3 a a a a a a 8 8 9 9 4 z 9 z 2 z 2 z > ---- + ---- + ---- + ---- 4 2 3 a a a a
In[19]:=	{Vassiliev[2][Knot[10, 102]], Vassiliev[3][Knot[10, 102]]}
Out[19]=	{-2, -1}
In[20]:=	Kh[Knot[10, 102]][q, t]
Out[20]=	7 1 2 1 4 2 5 4 3 - + 6 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 6 q t + 6 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 5 q t + 6 q t + 4 q t + 5 q t + 2 q t + 4 q t + q t + 11 5 13 6 > 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 102], 2][q]
Out[21]=	-12 3 2 6 15 10 18 45 24 46 87 28 83 + q - --- + --- + -- - -- + -- + -- - -- + -- + -- - -- + -- - 113 q + 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 2 3 4 5 6 7 8 9 10 > 15 q + 104 q - 106 q - 7 q + 101 q - 74 q - 26 q + 76 q - 34 q - 11 12 13 14 15 16 17 18 > 29 q + 39 q - 6 q - 16 q + 10 q + q - 3 q + q

The Alternating Knot 10102

The Alternating Knot 10₁₀₂