The Alternating Knot 10₂₃

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_3,12,4,13 X_13,1,14,20 X_5,15,6,14 X_7,17,8,16 X_15,7,16,6 X_19,9,20,8 X_9,19,10,18 X_17,11,18,10 X_11,2,12,3

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

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

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

Reversible 1 3 2 / NotAvailable 1

Alexander Polynomial: 2t^-3 - 7t^-2 + 13t^-1 - 15 + 13t - 7t² + 2t³

Conway Polynomial: 1 + 3z² + 5z⁴ + 2z⁶

Other knots with the same Alexander/Conway Polynomial: {10₅₂, ...}

Determinant and Signature: {59, 2}

Jones Polynomial: - q^-2 + 3q^-1 - 5 + 8q - 9q² + 10q³ - 9q⁴ + 7q⁵ - 4q⁶ + 2q⁷ - q⁸

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

A2 (sl(3)) Invariant: - q^-6 + q^-4 + 2q² - 2q⁴ + 2q⁶ + q¹⁰ + 2q¹² - q¹⁴ + 2q¹⁶ - q¹⁸ - q²⁰ - q²⁴

HOMFLY-PT Polynomial: - 2a^-6 - 3a^-6z² - a^-6z⁴ + 3a^-4 + 6a^-4z² + 4a^-4z⁴ + a^-4z⁶ + 2a^-2z² + 3a^-2z⁴ + a^-2z⁶ - 2z² - z⁴

Kauffman Polynomial: 2a^-9z - 3a^-9z³ + a^-9z⁵ + 3a^-8z² - 5a^-8z⁴ + 2a^-8z⁶ + a^-7z - 2a^-7z³ - 2a^-7z⁵ + 2a^-7z⁷ + 2a^-6 - 6a^-6z² + 5a^-6z⁴ - 3a^-6z⁶ + 2a^-6z⁸ - 2a^-5z + 3a^-5z³ - 2a^-5z⁵ + a^-5z⁷ + a^-5z⁹ + 3a^-4 - 13a^-4z² + 20a^-4z⁴ - 13a^-4z⁶ + 5a^-4z⁸ - 2a^-3z + 9a^-3z³ - 9a^-3z⁵ + 3a^-3z⁷ + a^-3z⁹ - a^-2z² + 3a^-2z⁴ - 5a^-2z⁶ + 3a^-2z⁸ - a^-1z + 5a^-1z³ - 9a^-1z⁵ + 4a^-1z⁷ + 3z² - 7z⁴ + 3z⁶ - 2az³ + az⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {3, 5}

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=2 is the signature of 10₂₃. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7

j = 17 1

j = 15 1

j = 13 3 1

j = 11 4 1

j = 9 5 3

j = 7 5 4

j = 5 4 5

j = 3 4 5

j = 1 2 5

j = -1 1 3

j = -3 2

j = -5 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-7 - 3q^-6 + q^-5 + 8q^-4 - 14q^-3 - q^-2 + 28q^-1 - 28 - 14q + 57q² - 38q³ - 35q⁴ + 81q⁵ - 38q⁶ - 51q⁷ + 87q⁸ - 29q⁹ - 54q¹⁰ + 70q¹¹ - 14q¹² - 42q¹³ + 41q¹⁴ - 3q¹⁵ - 22q¹⁶ + 17q¹⁷ - 8q¹⁹ + 5q²⁰ - 2q²² + q²³

3 - q^-15 + 3q^-14 - q^-13 - 4q^-12 - q^-11 + 11q^-10 + 3q^-9 - 22q^-8 - 9q^-7 + 35q^-6 + 24q^-5 - 50q^-4 - 49q^-3 + 60q^-2 + 88q^-1 - 67 - 127q + 52q² + 185q³ - 40q⁴ - 224q⁵ + 2q⁶ + 274q⁷ + 26q⁸ - 297q⁹ - 70q¹⁰ + 323q¹¹ + 97q¹² - 322q¹³ - 131q¹⁴ + 316q¹⁵ + 148q¹⁶ - 286q¹⁷ - 167q¹⁸ + 252q¹⁹ + 169q²⁰ - 202q²¹ - 165q²² + 152q²³ + 146q²⁴ - 100q²⁵ - 126q²⁶ + 64q²⁷ + 93q²⁸ - 33q²⁹ - 64q³⁰ + 15q³¹ + 42q³² - 9q³³ - 22q³⁴ + 3q³⁵ + 14q³⁶ - 5q³⁷ - 5q³⁸ + q³⁹ + 5q⁴⁰ - 3q⁴¹ - q⁴² + 2q⁴⁴ - q⁴⁵

4 q^-26 - 3q^-25 + q^-24 + 4q^-23 - 3q^-22 + 4q^-21 - 14q^-20 + 5q^-19 + 19q^-18 - 10q^-17 + 12q^-16 - 51q^-15 + 4q^-14 + 63q^-13 - 3q^-12 + 39q^-11 - 139q^-10 - 41q^-9 + 118q^-8 + 55q^-7 + 157q^-6 - 259q^-5 - 192q^-4 + 87q^-3 + 145q^-2 + 451q^-1 - 291 - 425q - 145q² + 139q³ + 899q⁴ - 116q⁵ - 605q⁶ - 553q⁷ - 67q⁸ + 1335q⁹ + 238q¹⁰ - 612q¹¹ - 980q¹² - 425q¹³ + 1612q¹⁴ + 617q¹⁵ - 468q¹⁶ - 1276q¹⁷ - 785q¹⁸ + 1683q¹⁹ + 893q²⁰ - 248q²¹ - 1387q²² - 1055q²³ + 1562q²⁴ + 1024q²⁵ + 5q²⁶ - 1303q²⁷ - 1207q²⁸ + 1249q²⁹ + 999q³⁰ + 275q³¹ - 1024q³² - 1213q³³ + 793q³⁴ + 797q³⁵ + 480q³⁶ - 600q³⁷ - 1028q³⁸ + 335q³⁹ + 462q⁴⁰ + 514q⁴¹ - 196q⁴² - 687q⁴³ + 55q⁴⁴ + 140q⁴⁵ + 366q⁴⁶ + 32q⁴⁷ - 338q⁴⁸ - 16q⁴⁹ - 33q⁵⁰ + 173q⁵¹ + 71q⁵² - 122q⁵³ + 10q⁵⁴ - 59q⁵⁵ + 52q⁵⁶ + 35q⁵⁷ - 38q⁵⁸ + 26q⁵⁹ - 31q⁶⁰ + 10q⁶¹ + 8q⁶² - 15q⁶³ + 18q⁶⁴ - 9q⁶⁵ + 2q⁶⁶ + q⁶⁷ - 7q⁶⁸ + 6q⁶⁹ - q⁷⁰ + q⁷¹ - 2q⁷³ + q⁷⁴

5 - q^-40 + 3q^-39 - q^-38 - 4q^-37 + 3q^-36 - q^-34 + 6q^-33 - q^-32 - 14q^-31 + 2q^-30 + 12q^-29 + 5q^-28 + 8q^-27 - 9q^-26 - 35q^-25 - 18q^-24 + 32q^-23 + 56q^-22 + 35q^-21 - 19q^-20 - 102q^-19 - 103q^-18 + 5q^-17 + 152q^-16 + 194q^-15 + 70q^-14 - 162q^-13 - 335q^-12 - 242q^-11 + 118q^-10 + 477q^-9 + 502q^-8 + 80q^-7 - 563q^-6 - 874q^-5 - 440q^-4 + 501q^-3 + 1256q^-2 + 1052q^-1 - 248 - 1601q - 1767q² - 336q³ + 1758q⁴ + 2674q⁵ + 1136q⁶ - 1729q⁷ - 3433q⁸ - 2226q⁹ + 1345q¹⁰ + 4225q¹¹ + 3370q¹² - 795q¹³ - 4668q¹⁴ - 4570q¹⁵ - 44q¹⁶ + 5033q¹⁷ + 5623q¹⁸ + 876q¹⁹ - 5039q²⁰ - 6551q²¹ - 1794q²² + 4999q²³ + 7218q²⁴ + 2576q²⁵ - 4713q²⁶ - 7724q²⁷ - 3318q²⁸ + 4446q²⁹ + 7988q³⁰ + 3882q³¹ - 3992q³² - 8118q³³ - 4415q³⁴ + 3566q³⁵ + 8036q³⁶ + 4802q³⁷ - 2949q³⁸ - 7803q³⁹ - 5167q⁴⁰ + 2295q⁴¹ + 7358q⁴² + 5383q⁴³ - 1477q⁴⁴ - 6700q⁴⁵ - 5505q⁴⁶ + 633q⁴⁷ + 5807q⁴⁸ + 5422q⁴⁹ + 248q⁵⁰ - 4747q⁵¹ - 5126q⁵² - 997q⁵³ + 3543q⁵⁴ + 4573q⁵⁵ + 1622q⁵⁶ - 2381q⁵⁷ - 3843q⁵⁸ - 1912q⁵⁹ + 1289q⁶⁰ + 2960q⁶¹ + 1989q⁶² - 438q⁶³ - 2102q⁶⁴ - 1779q⁶⁵ - 135q⁶⁶ + 1296q⁶⁷ + 1435q⁶⁸ + 447q⁶⁹ - 677q⁷⁰ - 1043q⁷¹ - 518q⁷² + 264q⁷³ + 645q⁷⁴ + 470q⁷⁵ - 13q⁷⁶ - 373q⁷⁷ - 341q⁷⁸ - 73q⁷⁹ + 152q⁸⁰ + 225q⁸¹ + 108q⁸² - 66q⁸³ - 113q⁸⁴ - 78q⁸⁵ - 12q⁸⁶ + 62q⁸⁷ + 62q⁸⁸ + 3q⁸⁹ - 12q⁹⁰ - 24q⁹¹ - 31q⁹² + 7q⁹³ + 22q⁹⁴ + 2q⁹⁵ + 7q⁹⁶ + 2q⁹⁷ - 16q⁹⁸ - 3q⁹⁹ + 6q¹⁰⁰ - 2q¹⁰¹ + 3q¹⁰² + 5q¹⁰³ - 4q¹⁰⁴ - 2q¹⁰⁵ + q¹⁰⁶ - q¹⁰⁷ + 2q¹⁰⁹ - q¹¹⁰

6 q^-57 - 3q^-56 + q^-55 + 4q^-54 - 3q^-53 - 3q^-51 + 9q^-50 - 10q^-49 - 4q^-48 + 21q^-47 - 12q^-46 - 5q^-45 - 13q^-44 + 31q^-43 - 15q^-42 - 12q^-41 + 58q^-40 - 32q^-39 - 31q^-38 - 59q^-37 + 77q^-36 - 15q^-35 - q^-34 + 170q^-33 - 38q^-32 - 87q^-31 - 224q^-30 + 73q^-29 - 71q^-28 + 23q^-27 + 457q^-26 + 153q^-25 - 18q^-24 - 487q^-23 - 118q^-22 - 509q^-21 - 247q^-20 + 781q^-19 + 761q^-18 + 703q^-17 - 284q^-16 - 176q^-15 - 1599q^-14 - 1614q^-13 + 157q^-12 + 1220q^-11 + 2352q^-10 + 1551q^-9 + 1378q^-8 - 2388q^-7 - 4304q^-6 - 2972q^-5 - 523q^-4 + 3461q^-3 + 5171q^-2 + 6421q^-1 - 208 - 6308q - 8568q² - 6573q³ + 824q⁴ + 8058q⁵ + 14566q⁶ + 7118q⁷ - 4053q⁸ - 13598q⁹ - 16143q¹⁰ - 7574q¹¹ + 6499q¹² + 22281q¹³ + 18301q¹⁴ + 4128q¹⁵ - 14301q¹⁶ - 25357q¹⁷ - 19866q¹⁸ - 711q¹⁹ + 25815q²⁰ + 29167q²¹ + 15895q²² - 9790q²³ - 30622q²⁴ - 31701q²⁵ - 10960q²⁶ + 24452q²⁷ + 36274q²⁸ + 27008q²⁹ - 2562q³⁰ - 31363q³¹ - 39874q³² - 20434q³³ + 20320q³⁴ + 39083q³⁵ + 34768q³⁶ + 4319q³⁷ - 29319q³⁸ - 43968q³⁹ - 27138q⁴⁰ + 15662q⁴¹ + 38960q⁴² + 39080q⁴³ + 9668q⁴⁴ - 26050q⁴⁵ - 45125q⁴⁶ - 31401q⁴⁷ + 10954q⁴⁸ + 36931q⁴⁹ + 41035q⁵⁰ + 14213q⁵¹ - 21536q⁵² - 43984q⁵³ - 34283q⁵⁴ + 5219q⁵⁵ + 32583q⁵⁶ + 40962q⁵⁷ + 18893q⁵⁸ - 14694q⁵⁹ - 39791q⁶⁰ - 35722q⁶¹ - 2185q⁶² + 24760q⁶³ + 37723q⁶⁴ + 23073q⁶⁵ - 5273q⁶⁶ - 31338q⁶⁷ - 34056q⁶⁸ - 9826q⁶⁹ + 13646q⁷⁰ + 29944q⁷¹ + 24329q⁷² + 4499q⁷³ - 19214q⁷⁴ - 27630q⁷⁵ - 14418q⁷⁶ + 2210q⁷⁷ + 18465q⁷⁸ + 20597q⁷⁹ + 10716q⁸⁰ - 6965q⁸¹ - 17441q⁸² - 13580q⁸³ - 5282q⁸⁴ + 7100q⁸⁵ + 12954q⁸⁶ + 11113q⁸⁷ + 1084q⁸⁸ - 7429q⁸⁹ - 8586q⁹⁰ - 6893q⁹¹ - 102q⁹² + 5300q⁹³ + 7381q⁹⁴ + 3484q⁹⁵ - 1227q⁹⁶ - 3262q⁹⁷ - 4655q⁹⁸ - 2277q⁹⁹ + 771q¹⁰⁰ + 3250q¹⁰¹ + 2458q¹⁰² + 775q¹⁰³ - 260q¹⁰⁴ - 1970q¹⁰⁵ - 1684q¹⁰⁶ - 586q¹⁰⁷ + 899q¹⁰⁸ + 952q¹⁰⁹ + 653q¹¹⁰ + 527q¹¹¹ - 507q¹¹² - 715q¹¹³ - 498q¹¹⁴ + 134q¹¹⁵ + 184q¹¹⁶ + 220q¹¹⁷ + 409q¹¹⁸ - 55q¹¹⁹ - 208q¹²⁰ - 224q¹²¹ + 9q¹²² - 17q¹²³ + 17q¹²⁴ + 201q¹²⁵ + 18q¹²⁶ - 42q¹²⁷ - 81q¹²⁸ + 6q¹²⁹ - 30q¹³⁰ - 26q¹³¹ + 80q¹³² + 14q¹³³ - 27q¹³⁵ + 7q¹³⁶ - 14q¹³⁷ - 21q¹³⁸ + 26q¹³⁹ + 4q¹⁴⁰ + 5q¹⁴¹ - 7q¹⁴² + 4q¹⁴³ - 3q¹⁴⁴ - 9q¹⁴⁵ + 6q¹⁴⁶ + 2q¹⁴⁸ - q¹⁴⁹ + q¹⁵⁰ - 2q¹⁵² + q¹⁵³

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, 23]]

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

In[3]:=
GaussCode[Knot[10, 23]]

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

In[4]:=
DTCode[Knot[10, 23]]

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

In[5]:=
br = BR[Knot[10, 23]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 23]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 23]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 23]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 1, 3, 2, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 23]][t]

Out[10]=
2 7 13 2 3 -15 + -- - -- + -- + 13 t - 7 t + 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 23]][z]

Out[11]=
2 4 6 1 + 3 z + 5 z + 2 z

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

Out[12]=
{Knot[10, 23], Knot[10, 52]}

In[13]:=
{KnotDet[Knot[10, 23]], KnotSignature[Knot[10, 23]]}

Out[13]=
{59, 2}

In[14]:=
Jones[Knot[10, 23]][q]

Out[14]=
-2 3 2 3 4 5 6 7 8 -5 - q + - + 8 q - 9 q + 10 q - 9 q + 7 q - 4 q + 2 q - q q

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

Out[15]=
{Knot[10, 23]}

In[16]:=
A2Invariant[Knot[10, 23]][q]

Out[16]=
-6 -4 2 4 6 10 12 14 16 18 20 24 -q + q + 2 q - 2 q + 2 q + q + 2 q - q + 2 q - q - q - q

In[17]:=
HOMFLYPT[Knot[10, 23]][a, z]

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

In[18]:=
Kauffman[Knot[10, 23]][a, z]

Out[18]=
2 2 2 2 3 2 3 2 z z 2 z 2 z z 2 3 z 6 z 13 z z 3 z -- + -- + --- + -- - --- - --- - - + 3 z + ---- - ---- - ----- - -- - ---- - 6 4 9 7 5 3 a 8 6 4 2 9 a a a a a a a a a a a 3 3 3 3 4 4 4 4 2 z 3 z 9 z 5 z 3 4 5 z 5 z 20 z 3 z > ---- + ---- + ---- + ---- - 2 a z - 7 z - ---- + ---- + ----- + ---- + 7 5 3 a 8 6 4 2 a a a a a a a 5 5 5 5 5 6 6 6 6 z 2 z 2 z 9 z 9 z 5 6 2 z 3 z 13 z 5 z > -- - ---- - ---- - ---- - ---- + a z + 3 z + ---- - ---- - ----- - ---- + 9 7 5 3 a 8 6 4 2 a a a a a a a a 7 7 7 7 8 8 8 9 9 2 z z 3 z 4 z 2 z 5 z 3 z z z > ---- + -- + ---- + ---- + ---- + ---- + ---- + -- + -- 7 5 3 a 6 4 2 5 3 a a a a a a a a

In[19]:=
{Vassiliev[2][Knot[10, 23]], Vassiliev[3][Knot[10, 23]]}

Out[19]=
{3, 5}

In[20]:=
Kh[Knot[10, 23]][q, t]

Out[20]=
3 1 2 1 3 2 q 3 5 5 2 5 q + 4 q + ----- + ----- + ---- + --- + --- + 5 q t + 4 q t + 5 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 5 q t + 4 q t + 5 q t + 3 q t + 4 q t + q t + 3 q t + 13 6 15 6 17 7 > q t + q t + q t

In[21]:=
ColouredJones[Knot[10, 23], 2][q]

Out[21]=
-7 3 -5 8 14 -2 28 2 3 4 -28 + q - -- + q + -- - -- - q + -- - 14 q + 57 q - 38 q - 35 q + 6 4 3 q q q q 5 6 7 8 9 10 11 12 13 > 81 q - 38 q - 51 q + 87 q - 29 q - 54 q + 70 q - 14 q - 42 q + 14 15 16 17 19 20 22 23 > 41 q - 3 q - 22 q + 17 q - 8 q + 5 q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₂₃

10₂₂
10₂₄

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-7 - 3q^-6 + q^-5 + 8q^-4 - 14q^-3 - q^-2 + 28q^-1 - 28 - 14q + 57q² - 38q³ - 35q⁴ + 81q⁵ - 38q⁶ - 51q⁷ + 87q⁸ - 29q⁹ - 54q¹⁰ + 70q¹¹ - 14q¹² - 42q¹³ + 41q¹⁴ - 3q¹⁵ - 22q¹⁶ + 17q¹⁷ - 8q¹⁹ + 5q²⁰ - 2q²² + q²³
3	- q^-15 + 3q^-14 - q^-13 - 4q^-12 - q^-11 + 11q^-10 + 3q^-9 - 22q^-8 - 9q^-7 + 35q^-6 + 24q^-5 - 50q^-4 - 49q^-3 + 60q^-2 + 88q^-1 - 67 - 127q + 52q² + 185q³ - 40q⁴ - 224q⁵ + 2q⁶ + 274q⁷ + 26q⁸ - 297q⁹ - 70q¹⁰ + 323q¹¹ + 97q¹² - 322q¹³ - 131q¹⁴ + 316q¹⁵ + 148q¹⁶ - 286q¹⁷ - 167q¹⁸ + 252q¹⁹ + 169q²⁰ - 202q²¹ - 165q²² + 152q²³ + 146q²⁴ - 100q²⁵ - 126q²⁶ + 64q²⁷ + 93q²⁸ - 33q²⁹ - 64q³⁰ + 15q³¹ + 42q³² - 9q³³ - 22q³⁴ + 3q³⁵ + 14q³⁶ - 5q³⁷ - 5q³⁸ + q³⁹ + 5q⁴⁰ - 3q⁴¹ - q⁴² + 2q⁴⁴ - q⁴⁵
4	q^-26 - 3q^-25 + q^-24 + 4q^-23 - 3q^-22 + 4q^-21 - 14q^-20 + 5q^-19 + 19q^-18 - 10q^-17 + 12q^-16 - 51q^-15 + 4q^-14 + 63q^-13 - 3q^-12 + 39q^-11 - 139q^-10 - 41q^-9 + 118q^-8 + 55q^-7 + 157q^-6 - 259q^-5 - 192q^-4 + 87q^-3 + 145q^-2 + 451q^-1 - 291 - 425q - 145q² + 139q³ + 899q⁴ - 116q⁵ - 605q⁶ - 553q⁷ - 67q⁸ + 1335q⁹ + 238q¹⁰ - 612q¹¹ - 980q¹² - 425q¹³ + 1612q¹⁴ + 617q¹⁵ - 468q¹⁶ - 1276q¹⁷ - 785q¹⁸ + 1683q¹⁹ + 893q²⁰ - 248q²¹ - 1387q²² - 1055q²³ + 1562q²⁴ + 1024q²⁵ + 5q²⁶ - 1303q²⁷ - 1207q²⁸ + 1249q²⁹ + 999q³⁰ + 275q³¹ - 1024q³² - 1213q³³ + 793q³⁴ + 797q³⁵ + 480q³⁶ - 600q³⁷ - 1028q³⁸ + 335q³⁹ + 462q⁴⁰ + 514q⁴¹ - 196q⁴² - 687q⁴³ + 55q⁴⁴ + 140q⁴⁵ + 366q⁴⁶ + 32q⁴⁷ - 338q⁴⁸ - 16q⁴⁹ - 33q⁵⁰ + 173q⁵¹ + 71q⁵² - 122q⁵³ + 10q⁵⁴ - 59q⁵⁵ + 52q⁵⁶ + 35q⁵⁷ - 38q⁵⁸ + 26q⁵⁹ - 31q⁶⁰ + 10q⁶¹ + 8q⁶² - 15q⁶³ + 18q⁶⁴ - 9q⁶⁵ + 2q⁶⁶ + q⁶⁷ - 7q⁶⁸ + 6q⁶⁹ - q⁷⁰ + q⁷¹ - 2q⁷³ + q⁷⁴
5	- q^-40 + 3q^-39 - q^-38 - 4q^-37 + 3q^-36 - q^-34 + 6q^-33 - q^-32 - 14q^-31 + 2q^-30 + 12q^-29 + 5q^-28 + 8q^-27 - 9q^-26 - 35q^-25 - 18q^-24 + 32q^-23 + 56q^-22 + 35q^-21 - 19q^-20 - 102q^-19 - 103q^-18 + 5q^-17 + 152q^-16 + 194q^-15 + 70q^-14 - 162q^-13 - 335q^-12 - 242q^-11 + 118q^-10 + 477q^-9 + 502q^-8 + 80q^-7 - 563q^-6 - 874q^-5 - 440q^-4 + 501q^-3 + 1256q^-2 + 1052q^-1 - 248 - 1601q - 1767q² - 336q³ + 1758q⁴ + 2674q⁵ + 1136q⁶ - 1729q⁷ - 3433q⁸ - 2226q⁹ + 1345q¹⁰ + 4225q¹¹ + 3370q¹² - 795q¹³ - 4668q¹⁴ - 4570q¹⁵ - 44q¹⁶ + 5033q¹⁷ + 5623q¹⁸ + 876q¹⁹ - 5039q²⁰ - 6551q²¹ - 1794q²² + 4999q²³ + 7218q²⁴ + 2576q²⁵ - 4713q²⁶ - 7724q²⁷ - 3318q²⁸ + 4446q²⁹ + 7988q³⁰ + 3882q³¹ - 3992q³² - 8118q³³ - 4415q³⁴ + 3566q³⁵ + 8036q³⁶ + 4802q³⁷ - 2949q³⁸ - 7803q³⁹ - 5167q⁴⁰ + 2295q⁴¹ + 7358q⁴² + 5383q⁴³ - 1477q⁴⁴ - 6700q⁴⁵ - 5505q⁴⁶ + 633q⁴⁷ + 5807q⁴⁸ + 5422q⁴⁹ + 248q⁵⁰ - 4747q⁵¹ - 5126q⁵² - 997q⁵³ + 3543q⁵⁴ + 4573q⁵⁵ + 1622q⁵⁶ - 2381q⁵⁷ - 3843q⁵⁸ - 1912q⁵⁹ + 1289q⁶⁰ + 2960q⁶¹ + 1989q⁶² - 438q⁶³ - 2102q⁶⁴ - 1779q⁶⁵ - 135q⁶⁶ + 1296q⁶⁷ + 1435q⁶⁸ + 447q⁶⁹ - 677q⁷⁰ - 1043q⁷¹ - 518q⁷² + 264q⁷³ + 645q⁷⁴ + 470q⁷⁵ - 13q⁷⁶ - 373q⁷⁷ - 341q⁷⁸ - 73q⁷⁹ + 152q⁸⁰ + 225q⁸¹ + 108q⁸² - 66q⁸³ - 113q⁸⁴ - 78q⁸⁵ - 12q⁸⁶ + 62q⁸⁷ + 62q⁸⁸ + 3q⁸⁹ - 12q⁹⁰ - 24q⁹¹ - 31q⁹² + 7q⁹³ + 22q⁹⁴ + 2q⁹⁵ + 7q⁹⁶ + 2q⁹⁷ - 16q⁹⁸ - 3q⁹⁹ + 6q¹⁰⁰ - 2q¹⁰¹ + 3q¹⁰² + 5q¹⁰³ - 4q¹⁰⁴ - 2q¹⁰⁵ + q¹⁰⁶ - q¹⁰⁷ + 2q¹⁰⁹ - q¹¹⁰
6	q^-57 - 3q^-56 + q^-55 + 4q^-54 - 3q^-53 - 3q^-51 + 9q^-50 - 10q^-49 - 4q^-48 + 21q^-47 - 12q^-46 - 5q^-45 - 13q^-44 + 31q^-43 - 15q^-42 - 12q^-41 + 58q^-40 - 32q^-39 - 31q^-38 - 59q^-37 + 77q^-36 - 15q^-35 - q^-34 + 170q^-33 - 38q^-32 - 87q^-31 - 224q^-30 + 73q^-29 - 71q^-28 + 23q^-27 + 457q^-26 + 153q^-25 - 18q^-24 - 487q^-23 - 118q^-22 - 509q^-21 - 247q^-20 + 781q^-19 + 761q^-18 + 703q^-17 - 284q^-16 - 176q^-15 - 1599q^-14 - 1614q^-13 + 157q^-12 + 1220q^-11 + 2352q^-10 + 1551q^-9 + 1378q^-8 - 2388q^-7 - 4304q^-6 - 2972q^-5 - 523q^-4 + 3461q^-3 + 5171q^-2 + 6421q^-1 - 208 - 6308q - 8568q² - 6573q³ + 824q⁴ + 8058q⁵ + 14566q⁶ + 7118q⁷ - 4053q⁸ - 13598q⁹ - 16143q¹⁰ - 7574q¹¹ + 6499q¹² + 22281q¹³ + 18301q¹⁴ + 4128q¹⁵ - 14301q¹⁶ - 25357q¹⁷ - 19866q¹⁸ - 711q¹⁹ + 25815q²⁰ + 29167q²¹ + 15895q²² - 9790q²³ - 30622q²⁴ - 31701q²⁵ - 10960q²⁶ + 24452q²⁷ + 36274q²⁸ + 27008q²⁹ - 2562q³⁰ - 31363q³¹ - 39874q³² - 20434q³³ + 20320q³⁴ + 39083q³⁵ + 34768q³⁶ + 4319q³⁷ - 29319q³⁸ - 43968q³⁹ - 27138q⁴⁰ + 15662q⁴¹ + 38960q⁴² + 39080q⁴³ + 9668q⁴⁴ - 26050q⁴⁵ - 45125q⁴⁶ - 31401q⁴⁷ + 10954q⁴⁸ + 36931q⁴⁹ + 41035q⁵⁰ + 14213q⁵¹ - 21536q⁵² - 43984q⁵³ - 34283q⁵⁴ + 5219q⁵⁵ + 32583q⁵⁶ + 40962q⁵⁷ + 18893q⁵⁸ - 14694q⁵⁹ - 39791q⁶⁰ - 35722q⁶¹ - 2185q⁶² + 24760q⁶³ + 37723q⁶⁴ + 23073q⁶⁵ - 5273q⁶⁶ - 31338q⁶⁷ - 34056q⁶⁸ - 9826q⁶⁹ + 13646q⁷⁰ + 29944q⁷¹ + 24329q⁷² + 4499q⁷³ - 19214q⁷⁴ - 27630q⁷⁵ - 14418q⁷⁶ + 2210q⁷⁷ + 18465q⁷⁸ + 20597q⁷⁹ + 10716q⁸⁰ - 6965q⁸¹ - 17441q⁸² - 13580q⁸³ - 5282q⁸⁴ + 7100q⁸⁵ + 12954q⁸⁶ + 11113q⁸⁷ + 1084q⁸⁸ - 7429q⁸⁹ - 8586q⁹⁰ - 6893q⁹¹ - 102q⁹² + 5300q⁹³ + 7381q⁹⁴ + 3484q⁹⁵ - 1227q⁹⁶ - 3262q⁹⁷ - 4655q⁹⁸ - 2277q⁹⁹ + 771q¹⁰⁰ + 3250q¹⁰¹ + 2458q¹⁰² + 775q¹⁰³ - 260q¹⁰⁴ - 1970q¹⁰⁵ - 1684q¹⁰⁶ - 586q¹⁰⁷ + 899q¹⁰⁸ + 952q¹⁰⁹ + 653q¹¹⁰ + 527q¹¹¹ - 507q¹¹² - 715q¹¹³ - 498q¹¹⁴ + 134q¹¹⁵ + 184q¹¹⁶ + 220q¹¹⁷ + 409q¹¹⁸ - 55q¹¹⁹ - 208q¹²⁰ - 224q¹²¹ + 9q¹²² - 17q¹²³ + 17q¹²⁴ + 201q¹²⁵ + 18q¹²⁶ - 42q¹²⁷ - 81q¹²⁸ + 6q¹²⁹ - 30q¹³⁰ - 26q¹³¹ + 80q¹³² + 14q¹³³ - 27q¹³⁵ + 7q¹³⁶ - 14q¹³⁷ - 21q¹³⁸ + 26q¹³⁹ + 4q¹⁴⁰ + 5q¹⁴¹ - 7q¹⁴² + 4q¹⁴³ - 3q¹⁴⁴ - 9q¹⁴⁵ + 6q¹⁴⁶ + 2q¹⁴⁸ - q¹⁴⁹ + q¹⁵⁰ - 2q¹⁵² + q¹⁵³

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 23]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[13, 1, 14, 20], X[5, 15, 6, 14], > X[7, 17, 8, 16], X[15, 7, 16, 6], X[19, 9, 20, 8], X[9, 19, 10, 18], > X[17, 11, 18, 10], X[11, 2, 12, 3]]
In[3]:=	GaussCode[Knot[10, 23]]
Out[3]=	GaussCode[-1, 10, -2, 1, -4, 6, -5, 7, -8, 9, -10, 2, -3, 4, -6, 5, -9, 8, -7, > 3]
In[4]:=	DTCode[Knot[10, 23]]
Out[4]=	DTCode[4, 12, 14, 16, 18, 2, 20, 6, 10, 8]
In[5]:=	br = BR[Knot[10, 23]]
Out[5]=	BR[4, {-1, -1, 2, -1, 2, 2, 2, 2, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 23]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 23]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 23]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 23]][t]
Out[10]=	2 7 13 2 3 -15 + -- - -- + -- + 13 t - 7 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 23]][z]
Out[11]=	2 4 6 1 + 3 z + 5 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 23], Knot[10, 52]}
In[13]:=	{KnotDet[Knot[10, 23]], KnotSignature[Knot[10, 23]]}
Out[13]=	{59, 2}
In[14]:=	Jones[Knot[10, 23]][q]
Out[14]=	-2 3 2 3 4 5 6 7 8 -5 - q + - + 8 q - 9 q + 10 q - 9 q + 7 q - 4 q + 2 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 23]}
In[16]:=	A2Invariant[Knot[10, 23]][q]
Out[16]=	-6 -4 2 4 6 10 12 14 16 18 20 24 -q + q + 2 q - 2 q + 2 q + q + 2 q - q + 2 q - q - q - q
In[17]:=	HOMFLYPT[Knot[10, 23]][a, z]
Out[17]=	2 2 2 4 4 4 6 6 -2 3 2 3 z 6 z 2 z 4 z 4 z 3 z z z -- + -- - 2 z - ---- + ---- + ---- - z - -- + ---- + ---- + -- + -- 6 4 6 4 2 6 4 2 4 2 a a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 23]][a, z]
Out[18]=	2 2 2 2 3 2 3 2 z z 2 z 2 z z 2 3 z 6 z 13 z z 3 z -- + -- + --- + -- - --- - --- - - + 3 z + ---- - ---- - ----- - -- - ---- - 6 4 9 7 5 3 a 8 6 4 2 9 a a a a a a a a a a a 3 3 3 3 4 4 4 4 2 z 3 z 9 z 5 z 3 4 5 z 5 z 20 z 3 z > ---- + ---- + ---- + ---- - 2 a z - 7 z - ---- + ---- + ----- + ---- + 7 5 3 a 8 6 4 2 a a a a a a a 5 5 5 5 5 6 6 6 6 z 2 z 2 z 9 z 9 z 5 6 2 z 3 z 13 z 5 z > -- - ---- - ---- - ---- - ---- + a z + 3 z + ---- - ---- - ----- - ---- + 9 7 5 3 a 8 6 4 2 a a a a a a a a 7 7 7 7 8 8 8 9 9 2 z z 3 z 4 z 2 z 5 z 3 z z z > ---- + -- + ---- + ---- + ---- + ---- + ---- + -- + -- 7 5 3 a 6 4 2 5 3 a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 23]], Vassiliev[3][Knot[10, 23]]}
Out[19]=	{3, 5}
In[20]:=	Kh[Knot[10, 23]][q, t]
Out[20]=	3 1 2 1 3 2 q 3 5 5 2 5 q + 4 q + ----- + ----- + ---- + --- + --- + 5 q t + 4 q t + 5 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 5 q t + 4 q t + 5 q t + 3 q t + 4 q t + q t + 3 q t + 13 6 15 6 17 7 > q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 23], 2][q]
Out[21]=	-7 3 -5 8 14 -2 28 2 3 4 -28 + q - -- + q + -- - -- - q + -- - 14 q + 57 q - 38 q - 35 q + 6 4 3 q q q q 5 6 7 8 9 10 11 12 13 > 81 q - 38 q - 51 q + 87 q - 29 q - 54 q + 70 q - 14 q - 42 q + 14 15 16 17 19 20 22 23 > 41 q - 3 q - 22 q + 17 q - 8 q + 5 q - 2 q + q

The Alternating Knot 1023

The Alternating Knot 10₂₃