The Alternating Knot 10₆₀

Visit 10₆₀'s page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

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Acknowledgement

KnotPlot

PD Presentation: X₄₂₅₁ X_10,6,11,5 X₈₃₉₄ X_2,9,3,10 X_16,12,17,11 X_14,7,15,8 X_6,15,7,16 X_20,18,1,17 X_18,13,19,14 X_12,19,13,20

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

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

Minimum Braid Representative:

Length is 12, width is 5
Braid index is 5

A Morse Link Presentation:

3D Invariants:

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

Reversible 1 3 3 / NotAvailable 1

Alexander Polynomial: - t^-3 + 7t^-2 - 20t^-1 + 29 - 20t + 7t² - t³

Conway Polynomial: 1 - z² + z⁴ - z⁶

Other knots with the same Alexander/Conway Polynomial: {K11n165, ...}

Determinant and Signature: {85, 0}

Jones Polynomial: q^-6 - 3q^-5 + 6q^-4 - 10q^-3 + 13q^-2 - 14q^-1 + 14 - 11q + 8q² - 4q³ + q⁴

Other knots (up to mirrors) with the same Jones Polynomial: {10₈₆, ...}

A2 (sl(3)) Invariant: q^-20 + q^-18 - 2q^-16 - 3q^-10 + 3q^-8 + q^-4 + 2q^-2 - 2 + 3q² - 3q⁴ + q⁶ + 2q⁸ - 2q¹⁰ + q¹²

HOMFLY-PT Polynomial: a^-2 + a^-2z² + a^-2z⁴ - 2 - 5z² - 3z⁴ - z⁶ + 4a² + 6a²z² + 3a²z⁴ - 3a⁴ - 3a⁴z² + a⁶

Kauffman Polynomial: a^-4z⁴ - 2a^-3z³ + 4a^-3z⁵ - a^-2 + 4a^-2z² - 9a^-2z⁴ + 8a^-2z⁶ - 2a^-1z + 5a^-1z³ - 11a^-1z⁵ + 9a^-1z⁷ - 2 + 14z² - 22z⁴ + 5z⁶ + 5z⁸ - 6az + 25az³ - 38az⁵ + 16az⁷ + az⁹ - 4a² + 18a²z² - 17a²z⁴ - 7a²z⁶ + 8a²z⁸ - 7a³z + 27a³z³ - 32a³z⁵ + 10a³z⁷ + a³z⁹ - 3a⁴ + 11a⁴z² - 8a⁴z⁴ - 3a⁴z⁶ + 3a⁴z⁸ - 3a⁵z + 9a⁵z³ - 9a⁵z⁵ + 3a⁵z⁷ - a⁶ + 3a⁶z² - 3a⁶z⁴ + a⁶z⁶

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-1, 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 10₆₀. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

j = 9 1

j = 7 3

j = 5 5 1

j = 3 6 3

j = 1 8 5

j = -1 7 7

j = -3 6 7

j = -5 4 7

j = -7 2 6

j = -9 1 4

j = -11 2

j = -13 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-18 - 3q^-17 + 11q^-15 - 15q^-14 - 8q^-13 + 43q^-12 - 34q^-11 - 39q^-10 + 97q^-9 - 42q^-8 - 93q^-7 + 147q^-6 - 28q^-5 - 144q^-4 + 165q^-3 - q^-2 - 162q^-1 + 143 + 22q - 136q² + 91q³ + 28q⁴ - 80q⁵ + 39q⁶ + 17q⁷ - 28q⁸ + 9q⁹ + 4q¹⁰ - 4q¹¹ + q¹²

3 q^-36 - 3q^-35 + 5q^-33 + 6q^-32 - 15q^-31 - 15q^-30 + 26q^-29 + 39q^-28 - 40q^-27 - 81q^-26 + 46q^-25 + 149q^-24 - 37q^-23 - 237q^-22 - 5q^-21 + 340q^-20 + 89q^-19 - 446q^-18 - 207q^-17 + 525q^-16 + 367q^-15 - 580q^-14 - 540q^-13 + 594q^-12 + 712q^-11 - 566q^-10 - 872q^-9 + 510q^-8 + 996q^-7 - 422q^-6 - 1087q^-5 + 328q^-4 + 1115q^-3 - 205q^-2 - 1111q^-1 + 105 + 1028q + 10q² - 920q³ - 81q⁴ + 752q⁵ + 144q⁶ - 585q⁷ - 157q⁸ + 411q⁹ + 150q¹⁰ - 268q¹¹ - 116q¹² + 154q¹³ + 80q¹⁴ - 80q¹⁵ - 49q¹⁶ + 41q¹⁷ + 20q¹⁸ - 15q¹⁹ - 8q²⁰ + 5q²¹ + 4q²² - 4q²³ + q²⁴

4 q^-60 - 3q^-59 + 5q^-57 + 6q^-55 - 22q^-54 - 8q^-53 + 26q^-52 + 15q^-51 + 42q^-50 - 88q^-49 - 75q^-48 + 51q^-47 + 88q^-46 + 215q^-45 - 192q^-44 - 308q^-43 - 65q^-42 + 192q^-41 + 736q^-40 - 119q^-39 - 712q^-38 - 623q^-37 + 7q^-36 + 1659q^-35 + 550q^-34 - 881q^-33 - 1717q^-32 - 1016q^-31 + 2512q^-30 + 1950q^-29 - 158q^-28 - 2834q^-27 - 3029q^-26 + 2540q^-25 + 3546q^-24 + 1664q^-23 - 3221q^-22 - 5449q^-21 + 1473q^-20 + 4583q^-19 + 4048q^-18 - 2620q^-17 - 7461q^-16 - 281q^-15 + 4757q^-14 + 6248q^-13 - 1359q^-12 - 8618q^-11 - 2126q^-10 + 4205q^-9 + 7773q^-8 + 153q^-7 - 8782q^-6 - 3679q^-5 + 3084q^-4 + 8346q^-3 + 1655q^-2 - 7858q^-1 - 4619 + 1513q + 7705q² + 2829q³ - 5888q⁴ - 4591q⁵ - 126q⁶ + 5875q⁷ + 3210q⁸ - 3439q⁹ - 3513q¹⁰ - 1172q¹¹ + 3491q¹² + 2619q¹³ - 1425q¹⁴ - 1950q¹⁵ - 1280q¹⁶ + 1528q¹⁷ + 1520q¹⁸ - 375q¹⁹ - 714q²⁰ - 800q²¹ + 473q²² + 615q²³ - 63q²⁴ - 143q²⁵ - 320q²⁶ + 110q²⁷ + 169q²⁸ - 22q²⁹ - q³⁰ - 82q³¹ + 22q³² + 32q³³ - 12q³⁴ + 5q³⁵ - 12q³⁶ + 5q³⁷ + 4q³⁸ - 4q³⁹ + q⁴⁰

5 q^-90 - 3q^-89 + 5q^-87 - q^-84 - 15q^-83 - 8q^-82 + 26q^-81 + 24q^-80 + 9q^-79 - 15q^-78 - 73q^-77 - 67q^-76 + 48q^-75 + 147q^-74 + 138q^-73 - q^-72 - 251q^-71 - 366q^-70 - 103q^-69 + 384q^-68 + 691q^-67 + 429q^-66 - 412q^-65 - 1206q^-64 - 1078q^-63 + 197q^-62 + 1772q^-61 + 2177q^-60 + 535q^-59 - 2191q^-58 - 3687q^-57 - 2081q^-56 + 2059q^-55 + 5464q^-54 + 4579q^-53 - 935q^-52 - 7003q^-51 - 7988q^-50 - 1657q^-49 + 7727q^-48 + 11967q^-47 + 5868q^-46 - 6997q^-45 - 15802q^-44 - 11549q^-43 + 4212q^-42 + 18798q^-41 + 18265q^-40 + 607q^-39 - 20175q^-38 - 25111q^-37 - 7415q^-36 + 19480q^-35 + 31482q^-34 + 15479q^-33 - 16696q^-32 - 36536q^-31 - 24126q^-30 + 11998q^-29 + 40014q^-28 + 32659q^-27 - 6032q^-26 - 41809q^-25 - 40432q^-24 - 674q^-23 + 42059q^-22 + 47206q^-21 + 7582q^-20 - 41185q^-19 - 52724q^-18 - 14307q^-17 + 39304q^-16 + 57144q^-15 + 20653q^-14 - 36755q^-13 - 60287q^-12 - 26575q^-11 + 33340q^-10 + 62395q^-9 + 31978q^-8 - 29276q^-7 - 62884q^-6 - 36876q^-5 + 24100q^-4 + 62017q^-3 + 40926q^-2 - 18273q^-1 - 58974 - 43822q + 11476q² + 54206q³ + 45119q⁴ - 4751q⁵ - 47270q⁶ - 44414q⁷ - 1816q⁸ + 39116q⁹ + 41591q¹⁰ + 7035q¹¹ - 30057q¹² - 36811q¹³ - 10740q¹⁴ + 21362q¹⁵ + 30612q¹⁶ + 12335q¹⁷ - 13540q¹⁸ - 23812q¹⁹ - 12237q²⁰ + 7522q²¹ + 17198q²² + 10649q²³ - 3261q²⁴ - 11461q²⁵ - 8430q²⁶ + 788q²⁷ + 7057q²⁸ + 5989q²⁹ + 379q³⁰ - 3950q³¹ - 3870q³² - 760q³³ + 2043q³⁴ + 2314q³⁵ + 629q³⁶ - 955q³⁷ - 1214q³⁸ - 448q³⁹ + 390q⁴⁰ + 628q⁴¹ + 243q⁴² - 173q⁴³ - 268q⁴⁴ - 107q⁴⁵ + 58q⁴⁶ + 98q⁴⁷ + 63q⁴⁸ - 28q⁴⁹ - 53q⁵⁰ - q⁵¹ + 13q⁵² + 8q⁵⁴ + q⁵⁵ - 12q⁵⁶ + 5q⁵⁷ + 4q⁵⁸ - 4q⁵⁹ + q⁶⁰

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, 60]]

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

In[3]:=
GaussCode[Knot[10, 60]]

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

In[4]:=
DTCode[Knot[10, 60]]

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

In[5]:=
br = BR[Knot[10, 60]]

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

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

Out[6]=
{5, 12}

In[7]:=
BraidIndex[Knot[10, 60]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[10, 60]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 60]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 60]][t]

Out[10]=
-3 7 20 2 3 29 - t + -- - -- - 20 t + 7 t - t 2 t t

In[11]:=
Conway[Knot[10, 60]][z]

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

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

Out[12]=
{Knot[10, 60], Knot[11, NonAlternating, 165]}

In[13]:=
{KnotDet[Knot[10, 60]], KnotSignature[Knot[10, 60]]}

Out[13]=
{85, 0}

In[14]:=
Jones[Knot[10, 60]][q]

Out[14]=
-6 3 6 10 13 14 2 3 4 14 + q - -- + -- - -- + -- - -- - 11 q + 8 q - 4 q + q 5 4 3 2 q q q q q

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

Out[15]=
{Knot[10, 60], Knot[10, 86]}

In[16]:=
A2Invariant[Knot[10, 60]][q]

Out[16]=
-20 -18 2 3 3 -4 2 2 4 6 8 -2 + q + q - --- - --- + -- + q + -- + 3 q - 3 q + q + 2 q - 16 10 8 2 q q q q 10 12 > 2 q + q

In[17]:=
HOMFLYPT[Knot[10, 60]][a, z]

Out[17]=
2 4 -2 2 4 6 2 z 2 2 4 2 4 z -2 + a + 4 a - 3 a + a - 5 z + -- + 6 a z - 3 a z - 3 z + -- + 2 2 a a 2 4 6 > 3 a z - z

In[18]:=
Kauffman[Knot[10, 60]][a, z]

Out[18]=
2 -2 2 4 6 2 z 3 5 2 4 z -2 - a - 4 a - 3 a - a - --- - 6 a z - 7 a z - 3 a z + 14 z + ---- + a 2 a 3 3 2 2 4 2 6 2 2 z 5 z 3 3 3 > 18 a z + 11 a z + 3 a z - ---- + ---- + 25 a z + 27 a z + 3 a a 4 4 5 5 5 3 4 z 9 z 2 4 4 4 6 4 4 z 11 z > 9 a z - 22 z + -- - ---- - 17 a z - 8 a z - 3 a z + ---- - ----- - 4 2 3 a a a a 6 5 3 5 5 5 6 8 z 2 6 4 6 6 6 > 38 a z - 32 a z - 9 a z + 5 z + ---- - 7 a z - 3 a z + a z + 2 a 7 9 z 7 3 7 5 7 8 2 8 4 8 9 > ---- + 16 a z + 10 a z + 3 a z + 5 z + 8 a z + 3 a z + a z + a 3 9 > a z

In[19]:=
{Vassiliev[2][Knot[10, 60]], Vassiliev[3][Knot[10, 60]]}

Out[19]=
{-1, 1}

In[20]:=
Kh[Knot[10, 60]][q, t]

Out[20]=
7 1 2 1 4 2 6 4 7 - + 8 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 6 7 7 3 3 2 5 2 5 3 7 3 > ----- + ---- + --- + 5 q t + 6 q t + 3 q t + 5 q t + q t + 3 q t + 3 2 3 q t q t q t 9 4 > q t

In[21]:=
ColouredJones[Knot[10, 60], 2][q]

Out[21]=
-18 3 11 15 8 43 34 39 97 42 93 147 143 + q - --- + --- - --- - --- + --- - --- - --- + -- - -- - -- + --- - 17 15 14 13 12 11 10 9 8 7 6 q q q q q q q q q q q 28 144 165 -2 162 2 3 4 5 > -- - --- + --- - q - --- + 22 q - 136 q + 91 q + 28 q - 80 q + 5 4 3 q q q q 6 7 8 9 10 11 12 > 39 q + 17 q - 28 q + 9 q + 4 q - 4 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₆₀

10₅₉
10₆₁

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-18 - 3q^-17 + 11q^-15 - 15q^-14 - 8q^-13 + 43q^-12 - 34q^-11 - 39q^-10 + 97q^-9 - 42q^-8 - 93q^-7 + 147q^-6 - 28q^-5 - 144q^-4 + 165q^-3 - q^-2 - 162q^-1 + 143 + 22q - 136q² + 91q³ + 28q⁴ - 80q⁵ + 39q⁶ + 17q⁷ - 28q⁸ + 9q⁹ + 4q¹⁰ - 4q¹¹ + q¹²
3	q^-36 - 3q^-35 + 5q^-33 + 6q^-32 - 15q^-31 - 15q^-30 + 26q^-29 + 39q^-28 - 40q^-27 - 81q^-26 + 46q^-25 + 149q^-24 - 37q^-23 - 237q^-22 - 5q^-21 + 340q^-20 + 89q^-19 - 446q^-18 - 207q^-17 + 525q^-16 + 367q^-15 - 580q^-14 - 540q^-13 + 594q^-12 + 712q^-11 - 566q^-10 - 872q^-9 + 510q^-8 + 996q^-7 - 422q^-6 - 1087q^-5 + 328q^-4 + 1115q^-3 - 205q^-2 - 1111q^-1 + 105 + 1028q + 10q² - 920q³ - 81q⁴ + 752q⁵ + 144q⁶ - 585q⁷ - 157q⁸ + 411q⁹ + 150q¹⁰ - 268q¹¹ - 116q¹² + 154q¹³ + 80q¹⁴ - 80q¹⁵ - 49q¹⁶ + 41q¹⁷ + 20q¹⁸ - 15q¹⁹ - 8q²⁰ + 5q²¹ + 4q²² - 4q²³ + q²⁴
4	q^-60 - 3q^-59 + 5q^-57 + 6q^-55 - 22q^-54 - 8q^-53 + 26q^-52 + 15q^-51 + 42q^-50 - 88q^-49 - 75q^-48 + 51q^-47 + 88q^-46 + 215q^-45 - 192q^-44 - 308q^-43 - 65q^-42 + 192q^-41 + 736q^-40 - 119q^-39 - 712q^-38 - 623q^-37 + 7q^-36 + 1659q^-35 + 550q^-34 - 881q^-33 - 1717q^-32 - 1016q^-31 + 2512q^-30 + 1950q^-29 - 158q^-28 - 2834q^-27 - 3029q^-26 + 2540q^-25 + 3546q^-24 + 1664q^-23 - 3221q^-22 - 5449q^-21 + 1473q^-20 + 4583q^-19 + 4048q^-18 - 2620q^-17 - 7461q^-16 - 281q^-15 + 4757q^-14 + 6248q^-13 - 1359q^-12 - 8618q^-11 - 2126q^-10 + 4205q^-9 + 7773q^-8 + 153q^-7 - 8782q^-6 - 3679q^-5 + 3084q^-4 + 8346q^-3 + 1655q^-2 - 7858q^-1 - 4619 + 1513q + 7705q² + 2829q³ - 5888q⁴ - 4591q⁵ - 126q⁶ + 5875q⁷ + 3210q⁸ - 3439q⁹ - 3513q¹⁰ - 1172q¹¹ + 3491q¹² + 2619q¹³ - 1425q¹⁴ - 1950q¹⁵ - 1280q¹⁶ + 1528q¹⁷ + 1520q¹⁸ - 375q¹⁹ - 714q²⁰ - 800q²¹ + 473q²² + 615q²³ - 63q²⁴ - 143q²⁵ - 320q²⁶ + 110q²⁷ + 169q²⁸ - 22q²⁹ - q³⁰ - 82q³¹ + 22q³² + 32q³³ - 12q³⁴ + 5q³⁵ - 12q³⁶ + 5q³⁷ + 4q³⁸ - 4q³⁹ + q⁴⁰
5	q^-90 - 3q^-89 + 5q^-87 - q^-84 - 15q^-83 - 8q^-82 + 26q^-81 + 24q^-80 + 9q^-79 - 15q^-78 - 73q^-77 - 67q^-76 + 48q^-75 + 147q^-74 + 138q^-73 - q^-72 - 251q^-71 - 366q^-70 - 103q^-69 + 384q^-68 + 691q^-67 + 429q^-66 - 412q^-65 - 1206q^-64 - 1078q^-63 + 197q^-62 + 1772q^-61 + 2177q^-60 + 535q^-59 - 2191q^-58 - 3687q^-57 - 2081q^-56 + 2059q^-55 + 5464q^-54 + 4579q^-53 - 935q^-52 - 7003q^-51 - 7988q^-50 - 1657q^-49 + 7727q^-48 + 11967q^-47 + 5868q^-46 - 6997q^-45 - 15802q^-44 - 11549q^-43 + 4212q^-42 + 18798q^-41 + 18265q^-40 + 607q^-39 - 20175q^-38 - 25111q^-37 - 7415q^-36 + 19480q^-35 + 31482q^-34 + 15479q^-33 - 16696q^-32 - 36536q^-31 - 24126q^-30 + 11998q^-29 + 40014q^-28 + 32659q^-27 - 6032q^-26 - 41809q^-25 - 40432q^-24 - 674q^-23 + 42059q^-22 + 47206q^-21 + 7582q^-20 - 41185q^-19 - 52724q^-18 - 14307q^-17 + 39304q^-16 + 57144q^-15 + 20653q^-14 - 36755q^-13 - 60287q^-12 - 26575q^-11 + 33340q^-10 + 62395q^-9 + 31978q^-8 - 29276q^-7 - 62884q^-6 - 36876q^-5 + 24100q^-4 + 62017q^-3 + 40926q^-2 - 18273q^-1 - 58974 - 43822q + 11476q² + 54206q³ + 45119q⁴ - 4751q⁵ - 47270q⁶ - 44414q⁷ - 1816q⁸ + 39116q⁹ + 41591q¹⁰ + 7035q¹¹ - 30057q¹² - 36811q¹³ - 10740q¹⁴ + 21362q¹⁵ + 30612q¹⁶ + 12335q¹⁷ - 13540q¹⁸ - 23812q¹⁹ - 12237q²⁰ + 7522q²¹ + 17198q²² + 10649q²³ - 3261q²⁴ - 11461q²⁵ - 8430q²⁶ + 788q²⁷ + 7057q²⁸ + 5989q²⁹ + 379q³⁰ - 3950q³¹ - 3870q³² - 760q³³ + 2043q³⁴ + 2314q³⁵ + 629q³⁶ - 955q³⁷ - 1214q³⁸ - 448q³⁹ + 390q⁴⁰ + 628q⁴¹ + 243q⁴² - 173q⁴³ - 268q⁴⁴ - 107q⁴⁵ + 58q⁴⁶ + 98q⁴⁷ + 63q⁴⁸ - 28q⁴⁹ - 53q⁵⁰ - q⁵¹ + 13q⁵² + 8q⁵⁴ + q⁵⁵ - 12q⁵⁶ + 5q⁵⁷ + 4q⁵⁸ - 4q⁵⁹ + q⁶⁰

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 60]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], > X[16, 12, 17, 11], X[14, 7, 15, 8], X[6, 15, 7, 16], X[20, 18, 1, 17], > X[18, 13, 19, 14], X[12, 19, 13, 20]]
In[3]:=	GaussCode[Knot[10, 60]]
Out[3]=	GaussCode[1, -4, 3, -1, 2, -7, 6, -3, 4, -2, 5, -10, 9, -6, 7, -5, 8, -9, 10, > -8]
In[4]:=	DTCode[Knot[10, 60]]
Out[4]=	DTCode[4, 8, 10, 14, 2, 16, 18, 6, 20, 12]
In[5]:=	br = BR[Knot[10, 60]]
Out[5]=	BR[5, {-1, 2, -1, 2, 2, -3, 2, -3, -2, -4, 3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 60]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 60]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 60]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 60]][t]
Out[10]=	-3 7 20 2 3 29 - t + -- - -- - 20 t + 7 t - t 2 t t
In[11]:=	Conway[Knot[10, 60]][z]
Out[11]=	2 4 6 1 - z + z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 60], Knot[11, NonAlternating, 165]}
In[13]:=	{KnotDet[Knot[10, 60]], KnotSignature[Knot[10, 60]]}
Out[13]=	{85, 0}
In[14]:=	Jones[Knot[10, 60]][q]
Out[14]=	-6 3 6 10 13 14 2 3 4 14 + q - -- + -- - -- + -- - -- - 11 q + 8 q - 4 q + q 5 4 3 2 q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 60], Knot[10, 86]}
In[16]:=	A2Invariant[Knot[10, 60]][q]
Out[16]=	-20 -18 2 3 3 -4 2 2 4 6 8 -2 + q + q - --- - --- + -- + q + -- + 3 q - 3 q + q + 2 q - 16 10 8 2 q q q q 10 12 > 2 q + q
In[17]:=	HOMFLYPT[Knot[10, 60]][a, z]
Out[17]=	2 4 -2 2 4 6 2 z 2 2 4 2 4 z -2 + a + 4 a - 3 a + a - 5 z + -- + 6 a z - 3 a z - 3 z + -- + 2 2 a a 2 4 6 > 3 a z - z
In[18]:=	Kauffman[Knot[10, 60]][a, z]
Out[18]=	2 -2 2 4 6 2 z 3 5 2 4 z -2 - a - 4 a - 3 a - a - --- - 6 a z - 7 a z - 3 a z + 14 z + ---- + a 2 a 3 3 2 2 4 2 6 2 2 z 5 z 3 3 3 > 18 a z + 11 a z + 3 a z - ---- + ---- + 25 a z + 27 a z + 3 a a 4 4 5 5 5 3 4 z 9 z 2 4 4 4 6 4 4 z 11 z > 9 a z - 22 z + -- - ---- - 17 a z - 8 a z - 3 a z + ---- - ----- - 4 2 3 a a a a 6 5 3 5 5 5 6 8 z 2 6 4 6 6 6 > 38 a z - 32 a z - 9 a z + 5 z + ---- - 7 a z - 3 a z + a z + 2 a 7 9 z 7 3 7 5 7 8 2 8 4 8 9 > ---- + 16 a z + 10 a z + 3 a z + 5 z + 8 a z + 3 a z + a z + a 3 9 > a z
In[19]:=	{Vassiliev[2][Knot[10, 60]], Vassiliev[3][Knot[10, 60]]}
Out[19]=	{-1, 1}
In[20]:=	Kh[Knot[10, 60]][q, t]
Out[20]=	7 1 2 1 4 2 6 4 7 - + 8 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 6 7 7 3 3 2 5 2 5 3 7 3 > ----- + ---- + --- + 5 q t + 6 q t + 3 q t + 5 q t + q t + 3 q t + 3 2 3 q t q t q t 9 4 > q t
In[21]:=	ColouredJones[Knot[10, 60], 2][q]
Out[21]=	-18 3 11 15 8 43 34 39 97 42 93 147 143 + q - --- + --- - --- - --- + --- - --- - --- + -- - -- - -- + --- - 17 15 14 13 12 11 10 9 8 7 6 q q q q q q q q q q q 28 144 165 -2 162 2 3 4 5 > -- - --- + --- - q - --- + 22 q - 136 q + 91 q + 28 q - 80 q + 5 4 3 q q q q 6 7 8 9 10 11 12 > 39 q + 17 q - 28 q + 9 q + 4 q - 4 q + q

The Alternating Knot 1060

The Alternating Knot 10₆₀