The Alternating Knot 10₈₇

Visit 10₈₇'s page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

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Acknowledgement

KnotPlot

PD Presentation: X₄₂₅₁ X_10,4,11,3 X_14,6,15,5 X_20,16,1,15 X_16,7,17,8 X_6,19,7,20 X_8,12,9,11 X_18,13,19,14 X_12,17,13,18 X_2,10,3,9

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

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

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 2 3 3 / NotAvailable 1

Alexander Polynomial: - 2t^-3 + 9t^-2 - 18t^-1 + 23 - 18t + 9t² - 2t³

Conway Polynomial: 1 - 3z⁴ - 2z⁶

Other knots with the same Alexander/Conway Polynomial: {10₉₈, K11a58, K11a165, K11n72, ...}

Determinant and Signature: {81, 0}

Jones Polynomial: q^-4 - 3q^-3 + 6q^-2 - 10q^-1 + 13 - 13q + 13q² - 10q³ + 7q⁴ - 4q⁵ + q⁶

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

A2 (sl(3)) Invariant: q^-12 - q^-10 + q^-8 + q^-6 - 3q^-4 + 2q^-2 - 2 + q² + 2q⁴ + 4q⁸ - 2q¹⁰ - 2q¹⁶ + q¹⁸

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

Kauffman Polynomial: a^-6z² - 2a^-6z⁴ + a^-6z⁶ + 7a^-5z³ - 11a^-5z⁵ + 4a^-5z⁷ - a^-4 + a^-4z² + 5a^-4z⁴ - 12a^-4z⁶ + 5a^-4z⁸ - a^-3z + 15a^-3z³ - 23a^-3z⁵ + 5a^-3z⁷ + 2a^-3z⁹ - 3a^-2 + 3a^-2z² + 8a^-2z⁴ - 21a^-2z⁶ + 10a^-2z⁸ - a^-1z + 13a^-1z³ - 21a^-1z⁵ + 7a^-1z⁷ + 2a^-1z⁹ - 2 + 7z² - 5z⁴ - 3z⁶ + 5z⁸ + az + 2az³ - 6az⁵ + 6az⁷ - a² + 3a²z² - 5a²z⁴ + 5a²z⁶ + a³z - 3a³z³ + 3a³z⁵ - a⁴z² + a⁴z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {0, 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 10₈₇. 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 3

j = 9 4 1

j = 7 6 3

j = 5 7 4

j = 3 6 6

j = 1 7 7

j = -1 4 7

j = -3 2 6

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 - 16q^-8 + 12q^-7 + 18q^-6 - 51q^-5 + 33q^-4 + 47q^-3 - 105q^-2 + 46q^-1 + 90 - 143q + 35q² + 120q³ - 141q⁴ + 7q⁵ + 123q⁶ - 105q⁷ - 20q⁸ + 97q⁹ - 54q¹⁰ - 31q¹¹ + 53q¹² - 13q¹³ - 20q¹⁴ + 15q¹⁵ + q¹⁶ - 4q¹⁷ + q¹⁸

3 q^-24 - 3q^-23 + 2q^-22 + 2q^-21 - 8q^-19 + 6q^-18 + 10q^-17 - 16q^-16 - 13q^-15 + 39q^-14 + 20q^-13 - 83q^-12 - 39q^-11 + 154q^-10 + 80q^-9 - 245q^-8 - 154q^-7 + 336q^-6 + 272q^-5 - 419q^-4 - 410q^-3 + 464q^-2 + 549q^-1 - 453 - 685q + 419q² + 767q³ - 329q⁴ - 830q⁵ + 242q⁶ + 827q⁷ - 116q⁸ - 815q⁹ + 11q¹⁰ + 747q¹¹ + 113q¹² - 669q¹³ - 205q¹⁴ + 546q¹⁵ + 287q¹⁶ - 413q¹⁷ - 326q¹⁸ + 271q¹⁹ + 317q²⁰ - 135q²¹ - 275q²² + 36q²³ + 202q²⁴ + 26q²⁵ - 127q²⁶ - 44q²⁷ + 60q²⁸ + 43q²⁹ - 25q³⁰ - 23q³¹ + 5q³² + 9q³³ + q³⁴ - 4q³⁵ + q³⁶

4 q^-40 - 3q^-39 + 2q^-38 + 2q^-37 - 4q^-36 + 8q^-35 - 14q^-34 + 8q^-33 + 5q^-32 - 20q^-31 + 40q^-30 - 29q^-29 + 16q^-28 - 22q^-27 - 88q^-26 + 145q^-25 + 37q^-24 + 60q^-23 - 188q^-22 - 385q^-21 + 326q^-20 + 411q^-19 + 396q^-18 - 537q^-17 - 1302q^-16 + 251q^-15 + 1209q^-14 + 1544q^-13 - 655q^-12 - 2969q^-11 - 709q^-10 + 1882q^-9 + 3592q^-8 + 189q^-7 - 4632q^-6 - 2600q^-5 + 1575q^-4 + 5610q^-3 + 1987q^-2 - 5236q^-1 - 4448 + 234q + 6521q² + 3810q³ - 4613q⁴ - 5337q⁵ - 1368q⁶ + 6185q⁷ + 4901q⁸ - 3322q⁹ - 5219q¹⁰ - 2692q¹¹ + 5060q¹² + 5267q¹³ - 1756q¹⁴ - 4449q¹⁵ - 3695q¹⁶ + 3417q¹⁷ + 5059q¹⁸ - 28q¹⁹ - 3117q²⁰ - 4268q²¹ + 1389q²² + 4126q²³ + 1478q²⁴ - 1275q²⁵ - 3959q²⁶ - 500q²⁷ + 2401q²⁸ + 2056q²⁹ + 512q³⁰ - 2611q³¹ - 1411q³² + 544q³³ + 1450q³⁴ + 1348q³⁵ - 948q³⁶ - 1099q³⁷ - 470q³⁸ + 406q³⁹ + 1047q⁴⁰ + 20q⁴¹ - 343q⁴² - 466q⁴³ - 146q⁴⁴ + 391q⁴⁵ + 156q⁴⁶ + 41q⁴⁷ - 149q⁴⁸ - 142q⁴⁹ + 59q⁵⁰ + 38q⁵¹ + 50q⁵² - 10q⁵³ - 35q⁵⁴ + 2q⁵⁵ - q⁵⁶ + 9q⁵⁷ + q⁵⁸ - 4q⁵⁹ + q⁶⁰

5 q^-60 - 3q^-59 + 2q^-58 + 2q^-57 - 4q^-56 + 4q^-55 + 2q^-54 - 12q^-53 + 3q^-52 + 11q^-51 - 7q^-50 + 14q^-49 + 9q^-48 - 41q^-47 - 27q^-46 + 11q^-45 + 40q^-44 + 92q^-43 + 55q^-42 - 129q^-41 - 255q^-40 - 137q^-39 + 213q^-38 + 563q^-37 + 453q^-36 - 306q^-35 - 1170q^-34 - 1133q^-33 + 284q^-32 + 2115q^-31 + 2460q^-30 + 174q^-29 - 3377q^-28 - 4774q^-27 - 1508q^-26 + 4732q^-25 + 8230q^-24 + 4252q^-23 - 5576q^-22 - 12742q^-21 - 8907q^-20 + 5172q^-19 + 17807q^-18 + 15585q^-17 - 2828q^-16 - 22464q^-15 - 23822q^-14 - 1989q^-13 + 25672q^-12 + 32799q^-11 + 9036q^-10 - 26662q^-9 - 41084q^-8 - 17662q^-7 + 25016q^-6 + 47626q^-5 + 26730q^-4 - 21076q^-3 - 51744q^-2 - 34918q^-1 + 15615 + 53054q + 41535q² - 9434q³ - 52288q⁴ - 46021q⁵ + 3628q⁶ + 49590q⁷ + 48605q⁸ + 1776q⁹ - 46135q¹⁰ - 49591q¹¹ - 6259q¹² + 41817q¹³ + 49507q¹⁴ + 10504q¹⁵ - 37297q¹⁶ - 48648q¹⁷ - 14339q¹⁸ + 31969q¹⁹ + 47190q²⁰ + 18380q²¹ - 26171q²² - 44870q²³ - 22173q²⁴ + 19260q²⁵ + 41502q²⁶ + 25764q²⁷ - 11811q²⁸ - 36688q²⁹ - 28199q³⁰ + 3785q³¹ + 30312q³² + 29233q³³ + 3740q³⁴ - 22616q³⁵ - 28002q³⁶ - 10120q³⁷ + 14131q³⁸ + 24602q³⁹ + 14406q⁴⁰ - 5945q⁴¹ - 19243q⁴² - 16028q⁴³ - 987q⁴⁴ + 12790q⁴⁵ + 15075q⁴⁶ + 5733q⁴⁷ - 6459q⁴⁸ - 12002q⁴⁹ - 7936q⁵⁰ + 1232q⁵¹ + 7945q⁵² + 7801q⁵³ + 2142q⁵⁴ - 3944q⁵⁵ - 6123q⁵⁶ - 3570q⁵⁷ + 944q⁵⁸ + 3784q⁵⁹ + 3444q⁶⁰ + 845q⁶¹ - 1744q⁶² - 2524q⁶³ - 1364q⁶⁴ + 362q⁶⁵ + 1361q⁶⁶ + 1236q⁶⁷ + 300q⁶⁸ - 571q⁶⁹ - 767q⁷⁰ - 395q⁷¹ + 81q⁷² + 351q⁷³ + 320q⁷⁴ + 63q⁷⁵ - 129q⁷⁶ - 151q⁷⁷ - 65q⁷⁸ + 9q⁷⁹ + 57q⁸⁰ + 53q⁸¹ - 3q⁸² - 20q⁸³ - 10q⁸⁴ - 4q⁸⁵ - q⁸⁶ + 9q⁸⁷ + q⁸⁸ - 4q⁸⁹ + q⁹⁰

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, 87]]

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

In[3]:=
GaussCode[Knot[10, 87]]

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

In[4]:=
DTCode[Knot[10, 87]]

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

In[5]:=
br = BR[Knot[10, 87]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 87]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 87]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 87]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 87]][t]

Out[10]=
2 9 18 2 3 23 - -- + -- - -- - 18 t + 9 t - 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 87]][z]

Out[11]=
4 6 1 - 3 z - 2 z

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

Out[12]=
{Knot[10, 87], Knot[10, 98], Knot[11, Alternating, 58], > Knot[11, Alternating, 165], Knot[11, NonAlternating, 72]}

In[13]:=
{KnotDet[Knot[10, 87]], KnotSignature[Knot[10, 87]]}

Out[13]=
{81, 0}

In[14]:=
Jones[Knot[10, 87]][q]

Out[14]=
-4 3 6 10 2 3 4 5 6 13 + q - -- + -- - -- - 13 q + 13 q - 10 q + 7 q - 4 q + q 3 2 q q q

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

Out[15]=
{Knot[10, 87]}

In[16]:=
A2Invariant[Knot[10, 87]][q]

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

In[17]:=
HOMFLYPT[Knot[10, 87]][a, z]

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

In[18]:=
Kauffman[Knot[10, 87]][a, z]

Out[18]=
2 2 2 -4 3 2 z z 3 2 z z 3 z 2 2 -2 - a - -- - a - -- - - + a z + a z + 7 z + -- + -- + ---- + 3 a z - 2 3 a 6 4 2 a a a a a 3 3 3 4 4 4 2 7 z 15 z 13 z 3 3 3 4 2 z 5 z > a z + ---- + ----- + ----- + 2 a z - 3 a z - 5 z - ---- + ---- + 5 3 a 6 4 a a a a 4 5 5 5 8 z 2 4 4 4 11 z 23 z 21 z 5 3 5 6 > ---- - 5 a z + a z - ----- - ----- - ----- - 6 a z + 3 a z - 3 z + 2 5 3 a a a a 6 6 6 7 7 7 8 z 12 z 21 z 2 6 4 z 5 z 7 z 7 8 5 z > -- - ----- - ----- + 5 a z + ---- + ---- + ---- + 6 a z + 5 z + ---- + 6 4 2 5 3 a 4 a a a a a a 8 9 9 10 z 2 z 2 z > ----- + ---- + ---- 2 3 a a a

In[19]:=
{Vassiliev[2][Knot[10, 87]], Vassiliev[3][Knot[10, 87]]}

Out[19]=
{0, 1}

In[20]:=
Kh[Knot[10, 87]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 87], 2][q]

Out[21]=
-12 3 2 6 16 12 18 51 33 47 105 46 90 + q - --- + --- + -- - -- + -- + -- - -- + -- + -- - --- + -- - 143 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 > 35 q + 120 q - 141 q + 7 q + 123 q - 105 q - 20 q + 97 q - 54 q - 11 12 13 14 15 16 17 18 > 31 q + 53 q - 13 q - 20 q + 15 q + q - 4 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₈₇

10₈₆
10₈₈

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-12 - 3q^-11 + 2q^-10 + 6q^-9 - 16q^-8 + 12q^-7 + 18q^-6 - 51q^-5 + 33q^-4 + 47q^-3 - 105q^-2 + 46q^-1 + 90 - 143q + 35q² + 120q³ - 141q⁴ + 7q⁵ + 123q⁶ - 105q⁷ - 20q⁸ + 97q⁹ - 54q¹⁰ - 31q¹¹ + 53q¹² - 13q¹³ - 20q¹⁴ + 15q¹⁵ + q¹⁶ - 4q¹⁷ + q¹⁸
3	q^-24 - 3q^-23 + 2q^-22 + 2q^-21 - 8q^-19 + 6q^-18 + 10q^-17 - 16q^-16 - 13q^-15 + 39q^-14 + 20q^-13 - 83q^-12 - 39q^-11 + 154q^-10 + 80q^-9 - 245q^-8 - 154q^-7 + 336q^-6 + 272q^-5 - 419q^-4 - 410q^-3 + 464q^-2 + 549q^-1 - 453 - 685q + 419q² + 767q³ - 329q⁴ - 830q⁵ + 242q⁶ + 827q⁷ - 116q⁸ - 815q⁹ + 11q¹⁰ + 747q¹¹ + 113q¹² - 669q¹³ - 205q¹⁴ + 546q¹⁵ + 287q¹⁶ - 413q¹⁷ - 326q¹⁸ + 271q¹⁹ + 317q²⁰ - 135q²¹ - 275q²² + 36q²³ + 202q²⁴ + 26q²⁵ - 127q²⁶ - 44q²⁷ + 60q²⁸ + 43q²⁹ - 25q³⁰ - 23q³¹ + 5q³² + 9q³³ + q³⁴ - 4q³⁵ + q³⁶
4	q^-40 - 3q^-39 + 2q^-38 + 2q^-37 - 4q^-36 + 8q^-35 - 14q^-34 + 8q^-33 + 5q^-32 - 20q^-31 + 40q^-30 - 29q^-29 + 16q^-28 - 22q^-27 - 88q^-26 + 145q^-25 + 37q^-24 + 60q^-23 - 188q^-22 - 385q^-21 + 326q^-20 + 411q^-19 + 396q^-18 - 537q^-17 - 1302q^-16 + 251q^-15 + 1209q^-14 + 1544q^-13 - 655q^-12 - 2969q^-11 - 709q^-10 + 1882q^-9 + 3592q^-8 + 189q^-7 - 4632q^-6 - 2600q^-5 + 1575q^-4 + 5610q^-3 + 1987q^-2 - 5236q^-1 - 4448 + 234q + 6521q² + 3810q³ - 4613q⁴ - 5337q⁵ - 1368q⁶ + 6185q⁷ + 4901q⁸ - 3322q⁹ - 5219q¹⁰ - 2692q¹¹ + 5060q¹² + 5267q¹³ - 1756q¹⁴ - 4449q¹⁵ - 3695q¹⁶ + 3417q¹⁷ + 5059q¹⁸ - 28q¹⁹ - 3117q²⁰ - 4268q²¹ + 1389q²² + 4126q²³ + 1478q²⁴ - 1275q²⁵ - 3959q²⁶ - 500q²⁷ + 2401q²⁸ + 2056q²⁹ + 512q³⁰ - 2611q³¹ - 1411q³² + 544q³³ + 1450q³⁴ + 1348q³⁵ - 948q³⁶ - 1099q³⁷ - 470q³⁸ + 406q³⁹ + 1047q⁴⁰ + 20q⁴¹ - 343q⁴² - 466q⁴³ - 146q⁴⁴ + 391q⁴⁵ + 156q⁴⁶ + 41q⁴⁷ - 149q⁴⁸ - 142q⁴⁹ + 59q⁵⁰ + 38q⁵¹ + 50q⁵² - 10q⁵³ - 35q⁵⁴ + 2q⁵⁵ - q⁵⁶ + 9q⁵⁷ + q⁵⁸ - 4q⁵⁹ + q⁶⁰
5	q^-60 - 3q^-59 + 2q^-58 + 2q^-57 - 4q^-56 + 4q^-55 + 2q^-54 - 12q^-53 + 3q^-52 + 11q^-51 - 7q^-50 + 14q^-49 + 9q^-48 - 41q^-47 - 27q^-46 + 11q^-45 + 40q^-44 + 92q^-43 + 55q^-42 - 129q^-41 - 255q^-40 - 137q^-39 + 213q^-38 + 563q^-37 + 453q^-36 - 306q^-35 - 1170q^-34 - 1133q^-33 + 284q^-32 + 2115q^-31 + 2460q^-30 + 174q^-29 - 3377q^-28 - 4774q^-27 - 1508q^-26 + 4732q^-25 + 8230q^-24 + 4252q^-23 - 5576q^-22 - 12742q^-21 - 8907q^-20 + 5172q^-19 + 17807q^-18 + 15585q^-17 - 2828q^-16 - 22464q^-15 - 23822q^-14 - 1989q^-13 + 25672q^-12 + 32799q^-11 + 9036q^-10 - 26662q^-9 - 41084q^-8 - 17662q^-7 + 25016q^-6 + 47626q^-5 + 26730q^-4 - 21076q^-3 - 51744q^-2 - 34918q^-1 + 15615 + 53054q + 41535q² - 9434q³ - 52288q⁴ - 46021q⁵ + 3628q⁶ + 49590q⁷ + 48605q⁸ + 1776q⁹ - 46135q¹⁰ - 49591q¹¹ - 6259q¹² + 41817q¹³ + 49507q¹⁴ + 10504q¹⁵ - 37297q¹⁶ - 48648q¹⁷ - 14339q¹⁸ + 31969q¹⁹ + 47190q²⁰ + 18380q²¹ - 26171q²² - 44870q²³ - 22173q²⁴ + 19260q²⁵ + 41502q²⁶ + 25764q²⁷ - 11811q²⁸ - 36688q²⁹ - 28199q³⁰ + 3785q³¹ + 30312q³² + 29233q³³ + 3740q³⁴ - 22616q³⁵ - 28002q³⁶ - 10120q³⁷ + 14131q³⁸ + 24602q³⁹ + 14406q⁴⁰ - 5945q⁴¹ - 19243q⁴² - 16028q⁴³ - 987q⁴⁴ + 12790q⁴⁵ + 15075q⁴⁶ + 5733q⁴⁷ - 6459q⁴⁸ - 12002q⁴⁹ - 7936q⁵⁰ + 1232q⁵¹ + 7945q⁵² + 7801q⁵³ + 2142q⁵⁴ - 3944q⁵⁵ - 6123q⁵⁶ - 3570q⁵⁷ + 944q⁵⁸ + 3784q⁵⁹ + 3444q⁶⁰ + 845q⁶¹ - 1744q⁶² - 2524q⁶³ - 1364q⁶⁴ + 362q⁶⁵ + 1361q⁶⁶ + 1236q⁶⁷ + 300q⁶⁸ - 571q⁶⁹ - 767q⁷⁰ - 395q⁷¹ + 81q⁷² + 351q⁷³ + 320q⁷⁴ + 63q⁷⁵ - 129q⁷⁶ - 151q⁷⁷ - 65q⁷⁸ + 9q⁷⁹ + 57q⁸⁰ + 53q⁸¹ - 3q⁸² - 20q⁸³ - 10q⁸⁴ - 4q⁸⁵ - q⁸⁶ + 9q⁸⁷ + q⁸⁸ - 4q⁸⁹ + q⁹⁰

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 87]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[20, 16, 1, 15], > X[16, 7, 17, 8], X[6, 19, 7, 20], X[8, 12, 9, 11], X[18, 13, 19, 14], > X[12, 17, 13, 18], X[2, 10, 3, 9]]
In[3]:=	GaussCode[Knot[10, 87]]
Out[3]=	GaussCode[1, -10, 2, -1, 3, -6, 5, -7, 10, -2, 7, -9, 8, -3, 4, -5, 9, -8, 6, > -4]
In[4]:=	DTCode[Knot[10, 87]]
Out[4]=	DTCode[4, 10, 14, 16, 2, 8, 18, 20, 12, 6]
In[5]:=	br = BR[Knot[10, 87]]
Out[5]=	BR[4, {1, 1, 1, 2, -1, -3, 2, -3, 2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 87]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 87]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 87]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 87]][t]
Out[10]=	2 9 18 2 3 23 - -- + -- - -- - 18 t + 9 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 87]][z]
Out[11]=	4 6 1 - 3 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 87], Knot[10, 98], Knot[11, Alternating, 58], > Knot[11, Alternating, 165], Knot[11, NonAlternating, 72]}
In[13]:=	{KnotDet[Knot[10, 87]], KnotSignature[Knot[10, 87]]}
Out[13]=	{81, 0}
In[14]:=	Jones[Knot[10, 87]][q]
Out[14]=	-4 3 6 10 2 3 4 5 6 13 + q - -- + -- - -- - 13 q + 13 q - 10 q + 7 q - 4 q + q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 87]}
In[16]:=	A2Invariant[Knot[10, 87]][q]
Out[16]=	-12 -10 -8 -6 3 2 2 4 8 10 16 18 -2 + q - q + q + q - -- + -- + q + 2 q + 4 q - 2 q - 2 q + q 4 2 q q
In[17]:=	HOMFLYPT[Knot[10, 87]][a, z]
Out[17]=	2 2 4 4 -4 3 2 2 z z 2 2 4 z 2 z 2 4 6 -2 - a + -- + a - 4 z + -- + -- + 2 a z - 3 z + -- - ---- + a z - z - 2 4 2 4 2 a a a a a 6 z > -- 2 a
In[18]:=	Kauffman[Knot[10, 87]][a, z]
Out[18]=	2 2 2 -4 3 2 z z 3 2 z z 3 z 2 2 -2 - a - -- - a - -- - - + a z + a z + 7 z + -- + -- + ---- + 3 a z - 2 3 a 6 4 2 a a a a a 3 3 3 4 4 4 2 7 z 15 z 13 z 3 3 3 4 2 z 5 z > a z + ---- + ----- + ----- + 2 a z - 3 a z - 5 z - ---- + ---- + 5 3 a 6 4 a a a a 4 5 5 5 8 z 2 4 4 4 11 z 23 z 21 z 5 3 5 6 > ---- - 5 a z + a z - ----- - ----- - ----- - 6 a z + 3 a z - 3 z + 2 5 3 a a a a 6 6 6 7 7 7 8 z 12 z 21 z 2 6 4 z 5 z 7 z 7 8 5 z > -- - ----- - ----- + 5 a z + ---- + ---- + ---- + 6 a z + 5 z + ---- + 6 4 2 5 3 a 4 a a a a a a 8 9 9 10 z 2 z 2 z > ----- + ---- + ---- 2 3 a a a
In[19]:=	{Vassiliev[2][Knot[10, 87]], Vassiliev[3][Knot[10, 87]]}
Out[19]=	{0, 1}
In[20]:=	Kh[Knot[10, 87]][q, t]
Out[20]=	7 1 2 1 4 2 6 4 3 - + 7 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 7 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 > 6 q t + 7 q t + 4 q t + 6 q t + 3 q t + 4 q t + q t + 11 5 13 6 > 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 87], 2][q]
Out[21]=	-12 3 2 6 16 12 18 51 33 47 105 46 90 + q - --- + --- + -- - -- + -- + -- - -- + -- + -- - --- + -- - 143 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 > 35 q + 120 q - 141 q + 7 q + 123 q - 105 q - 20 q + 97 q - 54 q - 11 12 13 14 15 16 17 18 > 31 q + 53 q - 13 q - 20 q + 15 q + q - 4 q + q

The Alternating Knot 1087

The Alternating Knot 10₈₇