The Alternating Knot 8₁₇

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Acknowledgement

KnotPlot

Further views: A knot in Brian Sanderson's Garden
A knot in Brian Sanderson's Garden

PD Presentation: X₆₂₇₁ X_14,8,15,7 X₈₃₉₄ X_2,13,3,14 X_12,5,13,6 X_4,9,5,10 X_16,12,1,11 X_10,16,11,15

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

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

Minimum Braid Representative:

Length is 8, 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

NegativeAmphicheiral 1 3 3 / 4 1

Alexander Polynomial: - t^-3 + 4t^-2 - 8t^-1 + 11 - 8t + 4t² - t³

Conway Polynomial: 1 - z² - 2z⁴ - z⁶

Other knots with the same Alexander/Conway Polynomial: {K11n53, ...}

Determinant and Signature: {37, 0}

Jones Polynomial: q^-4 - 3q^-3 + 5q^-2 - 6q^-1 + 7 - 6q + 5q² - 3q³ + q⁴

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

A2 (sl(3)) Invariant: q^-12 - q^-10 + q^-8 - q^-4 + 2q^-2 - 1 + 2q² - q⁴ + q⁸ - q¹⁰ + q¹²

HOMFLY-PT Polynomial: a^-2 + 2a^-2z² + a^-2z⁴ - 1 - 5z² - 4z⁴ - z⁶ + a² + 2a²z² + a²z⁴

Kauffman Polynomial: - a^-4z² + a^-4z⁴ + a^-3z - 4a^-3z³ + 3a^-3z⁵ - a^-2 + 3a^-2z² - 6a^-2z⁴ + 4a^-2z⁶ + 2a^-1z - 6a^-1z³ + 2a^-1z⁵ + 2a^-1z⁷ - 1 + 8z² - 14z⁴ + 8z⁶ + 2az - 6az³ + 2az⁵ + 2az⁷ - a² + 3a²z² - 6a²z⁴ + 4a²z⁶ + a³z - 4a³z³ + 3a³z⁵ - a⁴z² + a⁴z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-1, 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 8₁₇. 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

j = 9 1

j = 7 2

j = 5 3 1

j = 3 3 2

j = 1 4 3

j = -1 3 4

j = -3 2 3

j = -5 1 3

j = -7 2

j = -9 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-12 - 3q^-11 + q^-10 + 9q^-9 - 14q^-8 - 3q^-7 + 28q^-6 - 25q^-5 - 14q^-4 + 47q^-3 - 29q^-2 - 25q^-1 + 55 - 25q - 29q² + 47q³ - 14q⁴ - 25q⁵ + 28q⁶ - 3q⁷ - 14q⁸ + 9q⁹ + q¹⁰ - 3q¹¹ + q¹²

3 q^-24 - 3q^-23 + q^-22 + 5q^-21 + q^-20 - 14q^-19 - 6q^-18 + 29q^-17 + 17q^-16 - 43q^-15 - 40q^-14 + 55q^-13 + 73q^-12 - 64q^-11 - 108q^-10 + 61q^-9 + 146q^-8 - 53q^-7 - 177q^-6 + 38q^-5 + 205q^-4 - 26q^-3 - 216q^-2 + 6q^-1 + 225 + 6q - 216q² - 26q³ + 205q⁴ + 38q⁵ - 177q⁶ - 53q⁷ + 146q⁸ + 61q⁹ - 108q¹⁰ - 64q¹¹ + 73q¹² + 55q¹³ - 40q¹⁴ - 43q¹⁵ + 17q¹⁶ + 29q¹⁷ - 6q¹⁸ - 14q¹⁹ + q²⁰ + 5q²¹ + q²² - 3q²³ + q²⁴

4 q^-40 - 3q^-39 + q^-38 + 5q^-37 - 3q^-36 + q^-35 - 17q^-34 + 6q^-33 + 31q^-32 + q^-31 - 82q^-29 - 16q^-28 + 96q^-27 + 69q^-26 + 52q^-25 - 216q^-24 - 146q^-23 + 120q^-22 + 216q^-21 + 260q^-20 - 323q^-19 - 393q^-18 - 7q^-17 + 340q^-16 + 605q^-15 - 292q^-14 - 631q^-13 - 265q^-12 + 347q^-11 + 945q^-10 - 149q^-9 - 759q^-8 - 522q^-7 + 261q^-6 + 1161q^-5 + 11q^-4 - 771q^-3 - 694q^-2 + 144q^-1 + 1233 + 144q - 694q² - 771q³ + 11q⁴ + 1161q⁵ + 261q⁶ - 522q⁷ - 759q⁸ - 149q⁹ + 945q¹⁰ + 347q¹¹ - 265q¹² - 631q¹³ - 292q¹⁴ + 605q¹⁵ + 340q¹⁶ - 7q¹⁷ - 393q¹⁸ - 323q¹⁹ + 260q²⁰ + 216q²¹ + 120q²² - 146q²³ - 216q²⁴ + 52q²⁵ + 69q²⁶ + 96q²⁷ - 16q²⁸ - 82q²⁹ + q³¹ + 31q³² + 6q³³ - 17q³⁴ + q³⁵ - 3q³⁶ + 5q³⁷ + q³⁸ - 3q³⁹ + q⁴⁰

5 q^-60 - 3q^-59 + q^-58 + 5q^-57 - 3q^-56 - 3q^-55 - 2q^-54 - 5q^-53 + 8q^-52 + 26q^-51 + 4q^-50 - 30q^-49 - 43q^-48 - 34q^-47 + 35q^-46 + 112q^-45 + 107q^-44 - 31q^-43 - 197q^-42 - 237q^-41 - 60q^-40 + 270q^-39 + 462q^-38 + 264q^-37 - 285q^-36 - 728q^-35 - 603q^-34 + 141q^-33 + 976q^-32 + 1094q^-31 + 186q^-30 - 1134q^-29 - 1650q^-28 - 699q^-27 + 1099q^-26 + 2200q^-25 + 1387q^-24 - 888q^-23 - 2662q^-22 - 2125q^-21 + 494q^-20 + 2955q^-19 + 2877q^-18 + 9q^-17 - 3114q^-16 - 3506q^-15 - 568q^-14 + 3121q^-13 + 4033q^-12 + 1086q^-11 - 3040q^-10 - 4387q^-9 - 1560q^-8 + 2881q^-7 + 4660q^-6 + 1920q^-5 - 2707q^-4 - 4762q^-3 - 2247q^-2 + 2479q^-1 + 4841 + 2479q - 2247q² - 4762q³ - 2707q⁴ + 1920q⁵ + 4660q⁶ + 2881q⁷ - 1560q⁸ - 4387q⁹ - 3040q¹⁰ + 1086q¹¹ + 4033q¹² + 3121q¹³ - 568q¹⁴ - 3506q¹⁵ - 3114q¹⁶ + 9q¹⁷ + 2877q¹⁸ + 2955q¹⁹ + 494q²⁰ - 2125q²¹ - 2662q²² - 888q²³ + 1387q²⁴ + 2200q²⁵ + 1099q²⁶ - 699q²⁷ - 1650q²⁸ - 1134q²⁹ + 186q³⁰ + 1094q³¹ + 976q³² + 141q³³ - 603q³⁴ - 728q³⁵ - 285q³⁶ + 264q³⁷ + 462q³⁸ + 270q³⁹ - 60q⁴⁰ - 237q⁴¹ - 197q⁴² - 31q⁴³ + 107q⁴⁴ + 112q⁴⁵ + 35q⁴⁶ - 34q⁴⁷ - 43q⁴⁸ - 30q⁴⁹ + 4q⁵⁰ + 26q⁵¹ + 8q⁵² - 5q⁵³ - 2q⁵⁴ - 3q⁵⁵ - 3q⁵⁶ + 5q⁵⁷ + q⁵⁸ - 3q⁵⁹ + q⁶⁰

6 q^-84 - 3q^-83 + q^-82 + 5q^-81 - 3q^-80 - 3q^-79 - 6q^-78 + 10q^-77 - 3q^-76 + 3q^-75 + 29q^-74 - 15q^-73 - 31q^-72 - 49q^-71 + 14q^-70 + 16q^-69 + 61q^-68 + 153q^-67 + 14q^-66 - 117q^-65 - 273q^-64 - 149q^-63 - 92q^-62 + 203q^-61 + 641q^-60 + 463q^-59 + 57q^-58 - 691q^-57 - 870q^-56 - 1005q^-55 - 189q^-54 + 1343q^-53 + 1882q^-52 + 1543q^-51 - 247q^-50 - 1725q^-49 - 3355q^-48 - 2544q^-47 + 658q^-46 + 3470q^-45 + 4894q^-44 + 2812q^-43 - 590q^-42 - 5842q^-41 - 7188q^-40 - 3328q^-39 + 2689q^-38 + 8288q^-37 + 8456q^-36 + 4312q^-35 - 5674q^-34 - 11801q^-33 - 10070q^-32 - 1954q^-31 + 8878q^-30 + 14013q^-29 + 11845q^-28 - 1776q^-27 - 13643q^-26 - 16608q^-25 - 8872q^-24 + 6032q^-23 + 16953q^-22 + 18906q^-21 + 4000q^-20 - 12400q^-19 - 20628q^-18 - 15121q^-17 + 1645q^-16 + 17146q^-15 + 23466q^-14 + 9075q^-13 - 9763q^-12 - 22071q^-11 - 19122q^-10 - 2210q^-9 + 15962q^-8 + 25565q^-7 + 12389q^-6 - 7226q^-5 - 22014q^-4 - 21134q^-3 - 4957q^-2 + 14414q^-1 + 26111 + 14414q - 4957q² - 21134q³ - 22014q⁴ - 7226q⁵ + 12389q⁶ + 25565q⁷ + 15962q⁸ - 2210q⁹ - 19122q¹⁰ - 22071q¹¹ - 9763q¹² + 9075q¹³ + 23466q¹⁴ + 17146q¹⁵ + 1645q¹⁶ - 15121q¹⁷ - 20628q¹⁸ - 12400q¹⁹ + 4000q²⁰ + 18906q²¹ + 16953q²² + 6032q²³ - 8872q²⁴ - 16608q²⁵ - 13643q²⁶ - 1776q²⁷ + 11845q²⁸ + 14013q²⁹ + 8878q³⁰ - 1954q³¹ - 10070q³² - 11801q³³ - 5674q³⁴ + 4312q³⁵ + 8456q³⁶ + 8288q³⁷ + 2689q³⁸ - 3328q³⁹ - 7188q⁴⁰ - 5842q⁴¹ - 590q⁴² + 2812q⁴³ + 4894q⁴⁴ + 3470q⁴⁵ + 658q⁴⁶ - 2544q⁴⁷ - 3355q⁴⁸ - 1725q⁴⁹ - 247q⁵⁰ + 1543q⁵¹ + 1882q⁵² + 1343q⁵³ - 189q⁵⁴ - 1005q⁵⁵ - 870q⁵⁶ - 691q⁵⁷ + 57q⁵⁸ + 463q⁵⁹ + 641q⁶⁰ + 203q⁶¹ - 92q⁶² - 149q⁶³ - 273q⁶⁴ - 117q⁶⁵ + 14q⁶⁶ + 153q⁶⁷ + 61q⁶⁸ + 16q⁶⁹ + 14q⁷⁰ - 49q⁷¹ - 31q⁷² - 15q⁷³ + 29q⁷⁴ + 3q⁷⁵ - 3q⁷⁶ + 10q⁷⁷ - 6q⁷⁸ - 3q⁷⁹ - 3q⁸⁰ + 5q⁸¹ + q⁸² - 3q⁸³ + q⁸⁴

7 q^-112 - 3q^-111 + q^-110 + 5q^-109 - 3q^-108 - 3q^-107 - 6q^-106 + 6q^-105 + 12q^-104 - 8q^-103 + 6q^-102 + 10q^-101 - 16q^-100 - 26q^-99 - 41q^-98 + 7q^-97 + 74q^-96 + 44q^-95 + 71q^-94 + 43q^-93 - 78q^-92 - 159q^-91 - 283q^-90 - 154q^-89 + 143q^-88 + 317q^-87 + 550q^-86 + 516q^-85 + 93q^-84 - 417q^-83 - 1159q^-82 - 1332q^-81 - 683q^-80 + 256q^-79 + 1725q^-78 + 2573q^-77 + 2216q^-76 + 836q^-75 - 1934q^-74 - 4278q^-73 - 4774q^-72 - 3301q^-71 + 913q^-70 + 5542q^-69 + 8189q^-68 + 7815q^-67 + 2389q^-66 - 5364q^-65 - 11792q^-64 - 14143q^-63 - 8598q^-62 + 2327q^-61 + 13890q^-60 + 21402q^-59 + 18063q^-58 + 4775q^-57 - 12979q^-56 - 28137q^-55 - 29695q^-54 - 16138q^-53 + 7462q^-52 + 32099q^-51 + 41960q^-50 + 31319q^-49 + 3221q^-48 - 31814q^-47 - 52797q^-46 - 48414q^-45 - 18546q^-44 + 26269q^-43 + 60101q^-42 + 65456q^-41 + 37209q^-40 - 15734q^-39 - 62982q^-38 - 80416q^-37 - 56819q^-36 + 1416q^-35 + 61028q^-34 + 91744q^-33 + 75657q^-32 + 14955q^-31 - 55290q^-30 - 98994q^-29 - 91866q^-28 - 31357q^-27 + 46837q^-26 + 102372q^-25 + 104758q^-24 + 46339q^-23 - 37376q^-22 - 102713q^-21 - 113992q^-20 - 58991q^-19 + 28020q^-18 + 101063q^-17 + 120209q^-16 + 68876q^-15 - 19792q^-14 - 98300q^-13 - 123826q^-12 - 76313q^-11 + 12722q^-10 + 95254q^-9 + 125943q^-8 + 81699q^-7 - 7156q^-6 - 92151q^-5 - 126720q^-4 - 85773q^-3 + 2177q^-2 + 89089q^-1 + 127145 + 89089q + 2177q² - 85773q³ - 126720q⁴ - 92151q⁵ - 7156q⁶ + 81699q⁷ + 125943q⁸ + 95254q⁹ + 12722q¹⁰ - 76313q¹¹ - 123826q¹² - 98300q¹³ - 19792q¹⁴ + 68876q¹⁵ + 120209q¹⁶ + 101063q¹⁷ + 28020q¹⁸ - 58991q¹⁹ - 113992q²⁰ - 102713q²¹ - 37376q²² + 46339q²³ + 104758q²⁴ + 102372q²⁵ + 46837q²⁶ - 31357q²⁷ - 91866q²⁸ - 98994q²⁹ - 55290q³⁰ + 14955q³¹ + 75657q³² + 91744q³³ + 61028q³⁴ + 1416q³⁵ - 56819q³⁶ - 80416q³⁷ - 62982q³⁸ - 15734q³⁹ + 37209q⁴⁰ + 65456q⁴¹ + 60101q⁴² + 26269q⁴³ - 18546q⁴⁴ - 48414q⁴⁵ - 52797q⁴⁶ - 31814q⁴⁷ + 3221q⁴⁸ + 31319q⁴⁹ + 41960q⁵⁰ + 32099q⁵¹ + 7462q⁵² - 16138q⁵³ - 29695q⁵⁴ - 28137q⁵⁵ - 12979q⁵⁶ + 4775q⁵⁷ + 18063q⁵⁸ + 21402q⁵⁹ + 13890q⁶⁰ + 2327q⁶¹ - 8598q⁶² - 14143q⁶³ - 11792q⁶⁴ - 5364q⁶⁵ + 2389q⁶⁶ + 7815q⁶⁷ + 8189q⁶⁸ + 5542q⁶⁹ + 913q⁷⁰ - 3301q⁷¹ - 4774q⁷² - 4278q⁷³ - 1934q⁷⁴ + 836q⁷⁵ + 2216q⁷⁶ + 2573q⁷⁷ + 1725q⁷⁸ + 256q⁷⁹ - 683q⁸⁰ - 1332q⁸¹ - 1159q⁸² - 417q⁸³ + 93q⁸⁴ + 516q⁸⁵ + 550q⁸⁶ + 317q⁸⁷ + 143q⁸⁸ - 154q⁸⁹ - 283q⁹⁰ - 159q⁹¹ - 78q⁹² + 43q⁹³ + 71q⁹⁴ + 44q⁹⁵ + 74q⁹⁶ + 7q⁹⁷ - 41q⁹⁸ - 26q⁹⁹ - 16q¹⁰⁰ + 10q¹⁰¹ + 6q¹⁰² - 8q¹⁰³ + 12q¹⁰⁴ + 6q¹⁰⁵ - 6q¹⁰⁶ - 3q¹⁰⁷ - 3q¹⁰⁸ + 5q¹⁰⁹ + q¹¹⁰ - 3q¹¹¹ + q¹¹²

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[8, 17]]

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

In[3]:=
GaussCode[Knot[8, 17]]

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

In[4]:=
DTCode[Knot[8, 17]]

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

In[5]:=
br = BR[Knot[8, 17]]

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

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

Out[6]=
{3, 8}

In[7]:=
BraidIndex[Knot[8, 17]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[8, 17]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[8, 17]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{NegativeAmphicheiral, 1, 3, 3, 4, 1}

In[10]:=
alex = Alexander[Knot[8, 17]][t]

Out[10]=
-3 4 8 2 3 11 - t + -- - - - 8 t + 4 t - t 2 t t

In[11]:=
Conway[Knot[8, 17]][z]

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

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

Out[12]=
{Knot[8, 17], Knot[11, NonAlternating, 53]}

In[13]:=
{KnotDet[Knot[8, 17]], KnotSignature[Knot[8, 17]]}

Out[13]=
{37, 0}

In[14]:=
Jones[Knot[8, 17]][q]

Out[14]=
-4 3 5 6 2 3 4 7 + q - -- + -- - - - 6 q + 5 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[8, 17]}

In[16]:=
A2Invariant[Knot[8, 17]][q]

Out[16]=
-12 -10 -8 -4 2 2 4 8 10 12 -1 + q - q + q - q + -- + 2 q - q + q - q + q 2 q

In[17]:=
HOMFLYPT[Knot[8, 17]][a, z]

Out[17]=
2 4 -2 2 2 2 z 2 2 4 z 2 4 6 -1 + a + a - 5 z + ---- + 2 a z - 4 z + -- + a z - z 2 2 a a

In[18]:=
Kauffman[Knot[8, 17]][a, z]

Out[18]=
2 2 -2 2 z 2 z 3 2 z 3 z 2 2 4 2 -1 - a - a + -- + --- + 2 a z + a z + 8 z - -- + ---- + 3 a z - a z - 3 a 4 2 a a a 3 3 4 4 4 z 6 z 3 3 3 4 z 6 z 2 4 4 4 > ---- - ---- - 6 a z - 4 a z - 14 z + -- - ---- - 6 a z + a z + 3 a 4 2 a a a 5 5 6 7 3 z 2 z 5 3 5 6 4 z 2 6 2 z 7 > ---- + ---- + 2 a z + 3 a z + 8 z + ---- + 4 a z + ---- + 2 a z 3 a 2 a a a

In[19]:=
{Vassiliev[2][Knot[8, 17]], Vassiliev[3][Knot[8, 17]]}

Out[19]=
{-1, 0}

In[20]:=
Kh[Knot[8, 17]][q, t]

Out[20]=
4 1 2 1 3 2 3 3 3 - + 4 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 3 q t + 3 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 9 4 > 2 q t + 3 q t + q t + 2 q t + q t

In[21]:=
ColouredJones[Knot[8, 17], 2][q]

Out[21]=
-12 3 -10 9 14 3 28 25 14 47 29 25 55 + q - --- + q + -- - -- - -- + -- - -- - -- + -- - -- - -- - 25 q - 11 9 8 7 6 5 4 3 2 q q q q q q q q q q 2 3 4 5 6 7 8 9 10 11 > 29 q + 47 q - 14 q - 25 q + 28 q - 3 q - 14 q + 9 q + q - 3 q + 12 > q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 8₁₇

8₁₆
8₁₈

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-12 - 3q^-11 + q^-10 + 9q^-9 - 14q^-8 - 3q^-7 + 28q^-6 - 25q^-5 - 14q^-4 + 47q^-3 - 29q^-2 - 25q^-1 + 55 - 25q - 29q² + 47q³ - 14q⁴ - 25q⁵ + 28q⁶ - 3q⁷ - 14q⁸ + 9q⁹ + q¹⁰ - 3q¹¹ + q¹²
3	q^-24 - 3q^-23 + q^-22 + 5q^-21 + q^-20 - 14q^-19 - 6q^-18 + 29q^-17 + 17q^-16 - 43q^-15 - 40q^-14 + 55q^-13 + 73q^-12 - 64q^-11 - 108q^-10 + 61q^-9 + 146q^-8 - 53q^-7 - 177q^-6 + 38q^-5 + 205q^-4 - 26q^-3 - 216q^-2 + 6q^-1 + 225 + 6q - 216q² - 26q³ + 205q⁴ + 38q⁵ - 177q⁶ - 53q⁷ + 146q⁸ + 61q⁹ - 108q¹⁰ - 64q¹¹ + 73q¹² + 55q¹³ - 40q¹⁴ - 43q¹⁵ + 17q¹⁶ + 29q¹⁷ - 6q¹⁸ - 14q¹⁹ + q²⁰ + 5q²¹ + q²² - 3q²³ + q²⁴
4	q^-40 - 3q^-39 + q^-38 + 5q^-37 - 3q^-36 + q^-35 - 17q^-34 + 6q^-33 + 31q^-32 + q^-31 - 82q^-29 - 16q^-28 + 96q^-27 + 69q^-26 + 52q^-25 - 216q^-24 - 146q^-23 + 120q^-22 + 216q^-21 + 260q^-20 - 323q^-19 - 393q^-18 - 7q^-17 + 340q^-16 + 605q^-15 - 292q^-14 - 631q^-13 - 265q^-12 + 347q^-11 + 945q^-10 - 149q^-9 - 759q^-8 - 522q^-7 + 261q^-6 + 1161q^-5 + 11q^-4 - 771q^-3 - 694q^-2 + 144q^-1 + 1233 + 144q - 694q² - 771q³ + 11q⁴ + 1161q⁵ + 261q⁶ - 522q⁷ - 759q⁸ - 149q⁹ + 945q¹⁰ + 347q¹¹ - 265q¹² - 631q¹³ - 292q¹⁴ + 605q¹⁵ + 340q¹⁶ - 7q¹⁷ - 393q¹⁸ - 323q¹⁹ + 260q²⁰ + 216q²¹ + 120q²² - 146q²³ - 216q²⁴ + 52q²⁵ + 69q²⁶ + 96q²⁷ - 16q²⁸ - 82q²⁹ + q³¹ + 31q³² + 6q³³ - 17q³⁴ + q³⁵ - 3q³⁶ + 5q³⁷ + q³⁸ - 3q³⁹ + q⁴⁰
5	q^-60 - 3q^-59 + q^-58 + 5q^-57 - 3q^-56 - 3q^-55 - 2q^-54 - 5q^-53 + 8q^-52 + 26q^-51 + 4q^-50 - 30q^-49 - 43q^-48 - 34q^-47 + 35q^-46 + 112q^-45 + 107q^-44 - 31q^-43 - 197q^-42 - 237q^-41 - 60q^-40 + 270q^-39 + 462q^-38 + 264q^-37 - 285q^-36 - 728q^-35 - 603q^-34 + 141q^-33 + 976q^-32 + 1094q^-31 + 186q^-30 - 1134q^-29 - 1650q^-28 - 699q^-27 + 1099q^-26 + 2200q^-25 + 1387q^-24 - 888q^-23 - 2662q^-22 - 2125q^-21 + 494q^-20 + 2955q^-19 + 2877q^-18 + 9q^-17 - 3114q^-16 - 3506q^-15 - 568q^-14 + 3121q^-13 + 4033q^-12 + 1086q^-11 - 3040q^-10 - 4387q^-9 - 1560q^-8 + 2881q^-7 + 4660q^-6 + 1920q^-5 - 2707q^-4 - 4762q^-3 - 2247q^-2 + 2479q^-1 + 4841 + 2479q - 2247q² - 4762q³ - 2707q⁴ + 1920q⁵ + 4660q⁶ + 2881q⁷ - 1560q⁸ - 4387q⁹ - 3040q¹⁰ + 1086q¹¹ + 4033q¹² + 3121q¹³ - 568q¹⁴ - 3506q¹⁵ - 3114q¹⁶ + 9q¹⁷ + 2877q¹⁸ + 2955q¹⁹ + 494q²⁰ - 2125q²¹ - 2662q²² - 888q²³ + 1387q²⁴ + 2200q²⁵ + 1099q²⁶ - 699q²⁷ - 1650q²⁸ - 1134q²⁹ + 186q³⁰ + 1094q³¹ + 976q³² + 141q³³ - 603q³⁴ - 728q³⁵ - 285q³⁶ + 264q³⁷ + 462q³⁸ + 270q³⁹ - 60q⁴⁰ - 237q⁴¹ - 197q⁴² - 31q⁴³ + 107q⁴⁴ + 112q⁴⁵ + 35q⁴⁶ - 34q⁴⁷ - 43q⁴⁸ - 30q⁴⁹ + 4q⁵⁰ + 26q⁵¹ + 8q⁵² - 5q⁵³ - 2q⁵⁴ - 3q⁵⁵ - 3q⁵⁶ + 5q⁵⁷ + q⁵⁸ - 3q⁵⁹ + q⁶⁰
6	q^-84 - 3q^-83 + q^-82 + 5q^-81 - 3q^-80 - 3q^-79 - 6q^-78 + 10q^-77 - 3q^-76 + 3q^-75 + 29q^-74 - 15q^-73 - 31q^-72 - 49q^-71 + 14q^-70 + 16q^-69 + 61q^-68 + 153q^-67 + 14q^-66 - 117q^-65 - 273q^-64 - 149q^-63 - 92q^-62 + 203q^-61 + 641q^-60 + 463q^-59 + 57q^-58 - 691q^-57 - 870q^-56 - 1005q^-55 - 189q^-54 + 1343q^-53 + 1882q^-52 + 1543q^-51 - 247q^-50 - 1725q^-49 - 3355q^-48 - 2544q^-47 + 658q^-46 + 3470q^-45 + 4894q^-44 + 2812q^-43 - 590q^-42 - 5842q^-41 - 7188q^-40 - 3328q^-39 + 2689q^-38 + 8288q^-37 + 8456q^-36 + 4312q^-35 - 5674q^-34 - 11801q^-33 - 10070q^-32 - 1954q^-31 + 8878q^-30 + 14013q^-29 + 11845q^-28 - 1776q^-27 - 13643q^-26 - 16608q^-25 - 8872q^-24 + 6032q^-23 + 16953q^-22 + 18906q^-21 + 4000q^-20 - 12400q^-19 - 20628q^-18 - 15121q^-17 + 1645q^-16 + 17146q^-15 + 23466q^-14 + 9075q^-13 - 9763q^-12 - 22071q^-11 - 19122q^-10 - 2210q^-9 + 15962q^-8 + 25565q^-7 + 12389q^-6 - 7226q^-5 - 22014q^-4 - 21134q^-3 - 4957q^-2 + 14414q^-1 + 26111 + 14414q - 4957q² - 21134q³ - 22014q⁴ - 7226q⁵ + 12389q⁶ + 25565q⁷ + 15962q⁸ - 2210q⁹ - 19122q¹⁰ - 22071q¹¹ - 9763q¹² + 9075q¹³ + 23466q¹⁴ + 17146q¹⁵ + 1645q¹⁶ - 15121q¹⁷ - 20628q¹⁸ - 12400q¹⁹ + 4000q²⁰ + 18906q²¹ + 16953q²² + 6032q²³ - 8872q²⁴ - 16608q²⁵ - 13643q²⁶ - 1776q²⁷ + 11845q²⁸ + 14013q²⁹ + 8878q³⁰ - 1954q³¹ - 10070q³² - 11801q³³ - 5674q³⁴ + 4312q³⁵ + 8456q³⁶ + 8288q³⁷ + 2689q³⁸ - 3328q³⁹ - 7188q⁴⁰ - 5842q⁴¹ - 590q⁴² + 2812q⁴³ + 4894q⁴⁴ + 3470q⁴⁵ + 658q⁴⁶ - 2544q⁴⁷ - 3355q⁴⁸ - 1725q⁴⁹ - 247q⁵⁰ + 1543q⁵¹ + 1882q⁵² + 1343q⁵³ - 189q⁵⁴ - 1005q⁵⁵ - 870q⁵⁶ - 691q⁵⁷ + 57q⁵⁸ + 463q⁵⁹ + 641q⁶⁰ + 203q⁶¹ - 92q⁶² - 149q⁶³ - 273q⁶⁴ - 117q⁶⁵ + 14q⁶⁶ + 153q⁶⁷ + 61q⁶⁸ + 16q⁶⁹ + 14q⁷⁰ - 49q⁷¹ - 31q⁷² - 15q⁷³ + 29q⁷⁴ + 3q⁷⁵ - 3q⁷⁶ + 10q⁷⁷ - 6q⁷⁸ - 3q⁷⁹ - 3q⁸⁰ + 5q⁸¹ + q⁸² - 3q⁸³ + q⁸⁴
7	q^-112 - 3q^-111 + q^-110 + 5q^-109 - 3q^-108 - 3q^-107 - 6q^-106 + 6q^-105 + 12q^-104 - 8q^-103 + 6q^-102 + 10q^-101 - 16q^-100 - 26q^-99 - 41q^-98 + 7q^-97 + 74q^-96 + 44q^-95 + 71q^-94 + 43q^-93 - 78q^-92 - 159q^-91 - 283q^-90 - 154q^-89 + 143q^-88 + 317q^-87 + 550q^-86 + 516q^-85 + 93q^-84 - 417q^-83 - 1159q^-82 - 1332q^-81 - 683q^-80 + 256q^-79 + 1725q^-78 + 2573q^-77 + 2216q^-76 + 836q^-75 - 1934q^-74 - 4278q^-73 - 4774q^-72 - 3301q^-71 + 913q^-70 + 5542q^-69 + 8189q^-68 + 7815q^-67 + 2389q^-66 - 5364q^-65 - 11792q^-64 - 14143q^-63 - 8598q^-62 + 2327q^-61 + 13890q^-60 + 21402q^-59 + 18063q^-58 + 4775q^-57 - 12979q^-56 - 28137q^-55 - 29695q^-54 - 16138q^-53 + 7462q^-52 + 32099q^-51 + 41960q^-50 + 31319q^-49 + 3221q^-48 - 31814q^-47 - 52797q^-46 - 48414q^-45 - 18546q^-44 + 26269q^-43 + 60101q^-42 + 65456q^-41 + 37209q^-40 - 15734q^-39 - 62982q^-38 - 80416q^-37 - 56819q^-36 + 1416q^-35 + 61028q^-34 + 91744q^-33 + 75657q^-32 + 14955q^-31 - 55290q^-30 - 98994q^-29 - 91866q^-28 - 31357q^-27 + 46837q^-26 + 102372q^-25 + 104758q^-24 + 46339q^-23 - 37376q^-22 - 102713q^-21 - 113992q^-20 - 58991q^-19 + 28020q^-18 + 101063q^-17 + 120209q^-16 + 68876q^-15 - 19792q^-14 - 98300q^-13 - 123826q^-12 - 76313q^-11 + 12722q^-10 + 95254q^-9 + 125943q^-8 + 81699q^-7 - 7156q^-6 - 92151q^-5 - 126720q^-4 - 85773q^-3 + 2177q^-2 + 89089q^-1 + 127145 + 89089q + 2177q² - 85773q³ - 126720q⁴ - 92151q⁵ - 7156q⁶ + 81699q⁷ + 125943q⁸ + 95254q⁹ + 12722q¹⁰ - 76313q¹¹ - 123826q¹² - 98300q¹³ - 19792q¹⁴ + 68876q¹⁵ + 120209q¹⁶ + 101063q¹⁷ + 28020q¹⁸ - 58991q¹⁹ - 113992q²⁰ - 102713q²¹ - 37376q²² + 46339q²³ + 104758q²⁴ + 102372q²⁵ + 46837q²⁶ - 31357q²⁷ - 91866q²⁸ - 98994q²⁹ - 55290q³⁰ + 14955q³¹ + 75657q³² + 91744q³³ + 61028q³⁴ + 1416q³⁵ - 56819q³⁶ - 80416q³⁷ - 62982q³⁸ - 15734q³⁹ + 37209q⁴⁰ + 65456q⁴¹ + 60101q⁴² + 26269q⁴³ - 18546q⁴⁴ - 48414q⁴⁵ - 52797q⁴⁶ - 31814q⁴⁷ + 3221q⁴⁸ + 31319q⁴⁹ + 41960q⁵⁰ + 32099q⁵¹ + 7462q⁵² - 16138q⁵³ - 29695q⁵⁴ - 28137q⁵⁵ - 12979q⁵⁶ + 4775q⁵⁷ + 18063q⁵⁸ + 21402q⁵⁹ + 13890q⁶⁰ + 2327q⁶¹ - 8598q⁶² - 14143q⁶³ - 11792q⁶⁴ - 5364q⁶⁵ + 2389q⁶⁶ + 7815q⁶⁷ + 8189q⁶⁸ + 5542q⁶⁹ + 913q⁷⁰ - 3301q⁷¹ - 4774q⁷² - 4278q⁷³ - 1934q⁷⁴ + 836q⁷⁵ + 2216q⁷⁶ + 2573q⁷⁷ + 1725q⁷⁸ + 256q⁷⁹ - 683q⁸⁰ - 1332q⁸¹ - 1159q⁸² - 417q⁸³ + 93q⁸⁴ + 516q⁸⁵ + 550q⁸⁶ + 317q⁸⁷ + 143q⁸⁸ - 154q⁸⁹ - 283q⁹⁰ - 159q⁹¹ - 78q⁹² + 43q⁹³ + 71q⁹⁴ + 44q⁹⁵ + 74q⁹⁶ + 7q⁹⁷ - 41q⁹⁸ - 26q⁹⁹ - 16q¹⁰⁰ + 10q¹⁰¹ + 6q¹⁰² - 8q¹⁰³ + 12q¹⁰⁴ + 6q¹⁰⁵ - 6q¹⁰⁶ - 3q¹⁰⁷ - 3q¹⁰⁸ + 5q¹⁰⁹ + q¹¹⁰ - 3q¹¹¹ + q¹¹²

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

Out[8]=	-Graphics-
In[9]:=	#[Knot[8, 17]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{NegativeAmphicheiral, 1, 3, 3, 4, 1}
In[10]:=	alex = Alexander[Knot[8, 17]][t]
Out[10]=	-3 4 8 2 3 11 - t + -- - - - 8 t + 4 t - t 2 t t
In[11]:=	Conway[Knot[8, 17]][z]
Out[11]=	2 4 6 1 - z - 2 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 17], Knot[11, NonAlternating, 53]}
In[13]:=	{KnotDet[Knot[8, 17]], KnotSignature[Knot[8, 17]]}
Out[13]=	{37, 0}
In[14]:=	Jones[Knot[8, 17]][q]
Out[14]=	-4 3 5 6 2 3 4 7 + q - -- + -- - - - 6 q + 5 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[8, 17]}
In[16]:=	A2Invariant[Knot[8, 17]][q]
Out[16]=	-12 -10 -8 -4 2 2 4 8 10 12 -1 + q - q + q - q + -- + 2 q - q + q - q + q 2 q
In[17]:=	HOMFLYPT[Knot[8, 17]][a, z]
Out[17]=	2 4 -2 2 2 2 z 2 2 4 z 2 4 6 -1 + a + a - 5 z + ---- + 2 a z - 4 z + -- + a z - z 2 2 a a
In[18]:=	Kauffman[Knot[8, 17]][a, z]
Out[18]=	2 2 -2 2 z 2 z 3 2 z 3 z 2 2 4 2 -1 - a - a + -- + --- + 2 a z + a z + 8 z - -- + ---- + 3 a z - a z - 3 a 4 2 a a a 3 3 4 4 4 z 6 z 3 3 3 4 z 6 z 2 4 4 4 > ---- - ---- - 6 a z - 4 a z - 14 z + -- - ---- - 6 a z + a z + 3 a 4 2 a a a 5 5 6 7 3 z 2 z 5 3 5 6 4 z 2 6 2 z 7 > ---- + ---- + 2 a z + 3 a z + 8 z + ---- + 4 a z + ---- + 2 a z 3 a 2 a a a
In[19]:=	{Vassiliev[2][Knot[8, 17]], Vassiliev[3][Knot[8, 17]]}
Out[19]=	{-1, 0}
In[20]:=	Kh[Knot[8, 17]][q, t]
Out[20]=	4 1 2 1 3 2 3 3 3 - + 4 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 3 q t + 3 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 9 4 > 2 q t + 3 q t + q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[8, 17], 2][q]
Out[21]=	-12 3 -10 9 14 3 28 25 14 47 29 25 55 + q - --- + q + -- - -- - -- + -- - -- - -- + -- - -- - -- - 25 q - 11 9 8 7 6 5 4 3 2 q q q q q q q q q q 2 3 4 5 6 7 8 9 10 11 > 29 q + 47 q - 14 q - 25 q + 28 q - 3 q - 14 q + 9 q + q - 3 q + 12 > q

The Alternating Knot 817

The Alternating Knot 8₁₇