The Alternating Knot 10₆₁

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Acknowledgement

KnotPlot

PD Presentation: X₈₂₉₁ X_10,4,11,3 X_2,10,3,9 X_18,12,19,11 X_14,7,15,8 X_16,5,17,6 X_6,15,7,16 X_4,17,5,18 X_20,14,1,13 X_12,20,13,19

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

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

Reversible 2--3 3 3 / NotAvailable 2

Alexander Polynomial: - 2t^-3 + 5t^-2 - 6t^-1 + 7 - 6t + 5t² - 2t³

Conway Polynomial: 1 - 4z² - 7z⁴ - 2z⁶

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

Determinant and Signature: {33, 4}

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

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

A2 (sl(3)) Invariant: q^-6 + q^-4 + 2q^-2 + 2 - q⁴ - 3q⁶ - 2q⁸ + 2q¹⁴ + q²⁴

HOMFLY-PT Polynomial: a^-6 + 3a^-6z² + a^-6z⁴ + a^-4 - 3a^-4z² - 4a^-4z⁴ - a^-4z⁶ - 5a^-2 - 8a^-2z² - 5a^-2z⁴ - a^-2z⁶ + 4 + 4z² + z⁴

Kauffman Polynomial: a^-10z² + 2a^-9z³ - 2a^-8z² + 3a^-8z⁴ - 6a^-7z³ + 4a^-7z⁵ - a^-6 + 6a^-6z² - 13a^-6z⁴ + 5a^-6z⁶ - 6a^-5z + 17a^-5z³ - 18a^-5z⁵ + 5a^-5z⁷ + a^-4 + a^-4z² + 5a^-4z⁴ - 10a^-4z⁶ + 3a^-4z⁸ - 8a^-3z + 26a^-3z³ - 16a^-3z⁵ + a^-3z⁹ + 5a^-2 - 24a^-2z² + 38a^-2z⁴ - 22a^-2z⁶ + 4a^-2z⁸ - 2a^-1z + a^-1z³ + 6a^-1z⁵ - 5a^-1z⁷ + a^-1z⁹ + 4 - 16z² + 17z⁴ - 7z⁶ + z⁸

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-4, -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=4 is the signature of 10₆₁. 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 = 17 1

j = 15 1

j = 13 2 1

j = 11 2 1

j = 9 3 2

j = 7 2 2

j = 5 2 3

j = 3 3 3

j = 1 1

j = -1 1 3

j = -3

j = -5 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-8 - q^-7 - q^-6 + 4q^-5 - q^-4 - 6q^-3 + 7q^-2 + q^-1 - 11 + 8q + 5q² - 13q³ + 7q⁴ + 9q⁵ - 13q⁶ + 3q⁷ + 9q⁸ - 10q⁹ + 7q¹¹ - 5q¹² - q¹³ + 5q¹⁴ - 4q¹⁵ + 3q¹⁷ - 4q¹⁸ + 2q¹⁹ + q²⁰ - 2q²¹ + q²²

3 q^-18 - q^-17 - q^-16 + 4q^-14 - 4q^-12 - 5q^-11 + 7q^-10 + 6q^-9 - 3q^-8 - 13q^-7 + 3q^-6 + 10q^-5 + 5q^-4 - 13q^-3 - 4q^-2 + 7q^-1 + 9 - 5q - 5q² + 5q⁴ + q⁵ - q⁶ - 3q⁷ - 3q⁸ + 4q⁹ + 5q¹⁰ - 3q¹¹ - 10q¹² + 4q¹³ + 14q¹⁴ - 2q¹⁵ - 21q¹⁶ + q¹⁷ + 25q¹⁸ + 3q¹⁹ - 29q²⁰ - 7q²¹ + 29q²² + 10q²³ - 25q²⁴ - 11q²⁵ + 20q²⁶ + 10q²⁷ - 13q²⁸ - 8q²⁹ + 8q³⁰ + 5q³¹ - 5q³² - 3q³³ + 3q³⁴ + 2q³⁵ - q³⁶ - 3q³⁷ + 2q³⁸ + q⁴⁰ - 2q⁴¹ + q⁴²

4 q^-32 - q^-31 - q^-30 + 5q^-27 - q^-26 - 3q^-25 - 3q^-24 - 6q^-23 + 11q^-22 + 5q^-21 + 2q^-20 - 5q^-19 - 20q^-18 + 7q^-17 + 5q^-16 + 15q^-15 + 9q^-14 - 25q^-13 - 2q^-12 - 13q^-11 + 13q^-10 + 26q^-9 - 11q^-8 + 10q^-7 - 29q^-6 - 10q^-5 + 19q^-4 - 4q^-3 + 39q^-2 - 16q^-1 - 26 - 4q - 24q² + 53q³ + 14q⁴ - 13q⁵ - 13q⁶ - 56q⁷ + 39q⁸ + 31q⁹ + 13q¹⁰ + 2q¹¹ - 73q¹² + 10q¹³ + 26q¹⁴ + 30q¹⁵ + 28q¹⁶ - 77q¹⁷ - 14q¹⁸ + 16q¹⁹ + 38q²⁰ + 50q²¹ - 77q²² - 36q²³ + 9q²⁴ + 52q²⁵ + 74q²⁶ - 79q²⁷ - 70q²⁸ - 5q²⁹ + 68q³⁰ + 107q³¹ - 66q³² - 98q³³ - 36q³⁴ + 61q³⁵ + 129q³⁶ - 31q³⁷ - 90q³⁸ - 58q³⁹ + 29q⁴⁰ + 110q⁴¹ - 3q⁴² - 52q⁴³ - 47q⁴⁴ + 3q⁴⁵ + 66q⁴⁶ - 2q⁴⁷ - 18q⁴⁸ - 20q⁴⁹ - 4q⁵⁰ + 30q⁵¹ - 9q⁵² - q⁵³ - 4q⁵⁴ - 2q⁵⁵ + 12q⁵⁶ - 12q⁵⁷ + 3q⁵⁸ + q⁵⁹ + 4q⁶¹ - 7q⁶² + 3q⁶³ + q⁶⁶ - 2q⁶⁷ + q⁶⁸

5 q^-50 - q^-49 - q^-48 + q^-45 + 4q^-44 - 4q^-42 - 3q^-41 - 3q^-40 - 2q^-39 + 9q^-38 + 9q^-37 + 2q^-36 - 4q^-35 - 11q^-34 - 16q^-33 + 12q^-31 + 17q^-30 + 15q^-29 - 22q^-27 - 21q^-26 - 14q^-25 + 3q^-24 + 27q^-23 + 30q^-22 + 9q^-21 - 6q^-20 - 28q^-19 - 39q^-18 - 12q^-17 + 19q^-16 + 33q^-15 + 41q^-14 + 19q^-13 - 30q^-12 - 52q^-11 - 42q^-10 - 10q^-9 + 40q^-8 + 71q^-7 + 42q^-6 - 13q^-5 - 63q^-4 - 74q^-3 - 33q^-2 + 42q^-1 + 83 + 69q + 5q² - 69q³ - 95q⁴ - 48q⁵ + 32q⁶ + 94q⁷ + 91q⁸ + 17q⁹ - 72q¹⁰ - 113q¹¹ - 68q¹² + 29q¹³ + 116q¹⁴ + 113q¹⁵ + 22q¹⁶ - 100q¹⁷ - 142q¹⁸ - 75q¹⁹ + 66q²⁰ + 156q²¹ + 125q²² - 26q²³ - 158q²⁴ - 158q²⁵ - 17q²⁶ + 144q²⁷ + 191q²⁸ + 49q²⁹ - 138q³⁰ - 201q³¹ - 78q³² + 129q³³ + 223q³⁴ + 94q³⁵ - 136q³⁶ - 244q³⁷ - 111q³⁸ + 148q³⁹ + 278q⁴⁰ + 138q⁴¹ - 160q⁴² - 326q⁴³ - 177q⁴⁴ + 166q⁴⁵ + 368q⁴⁶ + 228q⁴⁷ - 142q⁴⁸ - 400q⁴⁹ - 286q⁵⁰ + 105q⁵¹ + 401q⁵² + 324q⁵³ - 48q⁵⁴ - 369q⁵⁵ - 339q⁵⁶ - 6q⁵⁷ + 309q⁵⁸ + 325q⁵⁹ + 45q⁶⁰ - 241q⁶¹ - 277q⁶² - 67q⁶³ + 173q⁶⁴ + 222q⁶⁵ + 66q⁶⁶ - 119q⁶⁷ - 165q⁶⁸ - 52q⁶⁹ + 80q⁷⁰ + 112q⁷¹ + 36q⁷² - 52q⁷³ - 73q⁷⁴ - 21q⁷⁵ + 36q⁷⁶ + 42q⁷⁷ + 7q⁷⁸ - 19q⁷⁹ - 21q⁸⁰ - 6q⁸¹ + 13q⁸² + 11q⁸³ - 3q⁸⁴ - 3q⁸⁵ - 3q⁸⁷ + q⁸⁸ + 2q⁸⁹ - 2q⁹⁰ + q⁹¹ + 2q⁹² - 2q⁹³ - q⁹⁴ + q⁹⁵ + q⁹⁸ - 2q⁹⁹ + q¹⁰⁰

6 q^-72 - q^-71 - q^-70 + q^-67 + 5q^-65 - q^-64 - 4q^-63 - 3q^-62 - 3q^-61 - 3q^-59 + 14q^-58 + 7q^-57 + 2q^-56 - 3q^-55 - 9q^-54 - 11q^-53 - 23q^-52 + 11q^-51 + 11q^-50 + 20q^-49 + 18q^-48 + 12q^-47 - 4q^-46 - 45q^-45 - 17q^-44 - 26q^-43 + 2q^-42 + 19q^-41 + 49q^-40 + 48q^-39 - 5q^-38 + 4q^-37 - 53q^-36 - 52q^-35 - 56q^-34 + 4q^-33 + 47q^-32 + 44q^-31 + 95q^-30 + 29q^-29 - 8q^-28 - 89q^-27 - 84q^-26 - 68q^-25 - 48q^-24 + 83q^-23 + 102q^-22 + 129q^-21 + 43q^-20 - 16q^-19 - 102q^-18 - 184q^-17 - 87q^-16 - 25q^-15 + 115q^-14 + 150q^-13 + 169q^-12 + 87q^-11 - 104q^-10 - 153q^-9 - 216q^-8 - 104q^-7 + 3q^-6 + 165q^-5 + 265q^-4 + 150q^-3 + 57q^-2 - 167q^-1 - 231 - 260q - 111q² + 151q³ + 247q⁴ + 326q⁵ + 126q⁶ - 45q⁷ - 311q⁸ - 389q⁹ - 180q¹⁰ + 33q¹¹ + 355q¹² + 373q¹³ + 318q¹⁴ - 59q¹⁵ - 406q¹⁶ - 438q¹⁷ - 328q¹⁸ + 100q¹⁹ + 363q²⁰ + 594q²¹ + 315q²² - 162q²³ - 455q²⁴ - 601q²⁵ - 268q²⁶ + 119q²⁷ + 653q²⁸ + 612q²⁹ + 184q³⁰ - 258q³¹ - 681q³² - 578q³³ - 220q³⁴ + 520q³⁵ + 740q³⁶ + 483q³⁷ + 32q³⁸ - 589q³⁹ - 740q⁴⁰ - 516q⁴¹ + 307q⁴² + 716q⁴³ + 649q⁴⁴ + 266q⁴⁵ - 444q⁴⁶ - 773q⁴⁷ - 677q⁴⁸ + 170q⁴⁹ + 681q⁵⁰ + 718q⁵¹ + 355q⁵² - 417q⁵³ - 841q⁵⁴ - 769q⁵⁵ + 171q⁵⁶ + 801q⁵⁷ + 881q⁵⁸ + 420q⁵⁹ - 516q⁶⁰ - 1084q⁶¹ - 996q⁶² + 126q⁶³ + 990q⁶⁴ + 1211q⁶⁵ + 672q⁶⁶ - 475q⁶⁷ - 1309q⁶⁸ - 1348q⁶⁹ - 169q⁷⁰ + 922q⁷¹ + 1417q⁷² + 1024q⁷³ - 139q⁷⁴ - 1176q⁷⁵ - 1478q⁷⁶ - 537q⁷⁷ + 526q⁷⁸ + 1205q⁷⁹ + 1114q⁸⁰ + 232q⁸¹ - 730q⁸² - 1192q⁸³ - 629q⁸⁴ + 135q⁸⁵ + 739q⁸⁶ + 840q⁸⁷ + 330q⁸⁸ - 331q⁸⁹ - 728q⁹⁰ - 428q⁹¹ - 9q⁹² + 355q⁹³ + 463q⁹⁴ + 203q⁹⁵ - 143q⁹⁶ - 368q⁹⁷ - 178q⁹⁸ + 10q⁹⁹ + 151q¹⁰⁰ + 197q¹⁰¹ + 60q¹⁰² - 76q¹⁰³ - 165q¹⁰⁴ - 31q¹⁰⁵ + 35q¹⁰⁶ + 55q¹⁰⁷ + 64q¹⁰⁸ - 5q¹⁰⁹ - 35q¹¹⁰ - 64q¹¹¹ + 17q¹¹² + 25q¹¹³ + 11q¹¹⁴ + 15q¹¹⁵ - 14q¹¹⁶ - 6q¹¹⁷ - 22q¹¹⁸ + 16q¹¹⁹ + 7q¹²⁰ - 2q¹²¹ + 5q¹²² - 8q¹²³ + 5q¹²⁴ - 8q¹²⁵ + 6q¹²⁶ - 2q¹²⁸ + 3q¹²⁹ - 4q¹³⁰ + 4q¹³¹ - 3q¹³² + q¹³³ + q¹³⁶ - 2q¹³⁷ + q¹³⁸

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, 61]]

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

In[3]:=
GaussCode[Knot[10, 61]]

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

In[4]:=
DTCode[Knot[10, 61]]

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

In[5]:=
br = BR[Knot[10, 61]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 61]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 61]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 61]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 61]][t]

Out[10]=
2 5 6 2 3 7 - -- + -- - - - 6 t + 5 t - 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 61]][z]

Out[11]=
2 4 6 1 - 4 z - 7 z - 2 z

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

Out[12]=
{Knot[10, 61]}

In[13]:=
{KnotDet[Knot[10, 61]], KnotSignature[Knot[10, 61]]}

Out[13]=
{33, 4}

In[14]:=
Jones[Knot[10, 61]][q]

Out[14]=
-2 1 2 3 4 5 6 7 8 3 + q - - - 4 q + 4 q - 5 q + 5 q - 4 q + 3 q - 2 q + q q

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

Out[15]=
{Knot[10, 61]}

In[16]:=
A2Invariant[Knot[10, 61]][q]

Out[16]=
-6 -4 2 4 6 8 14 24 2 + q + q + -- - q - 3 q - 2 q + 2 q + q 2 q

In[17]:=
HOMFLYPT[Knot[10, 61]][a, z]

Out[17]=
2 2 2 4 4 4 6 6 -6 -4 5 2 3 z 3 z 8 z 4 z 4 z 5 z z z 4 + a + a - -- + 4 z + ---- - ---- - ---- + z + -- - ---- - ---- - -- - -- 2 6 4 2 6 4 2 4 2 a a a a a a a a a

In[18]:=
Kauffman[Knot[10, 61]][a, z]

Out[18]=
2 2 2 2 2 -6 -4 5 6 z 8 z 2 z 2 z 2 z 6 z z 24 z 4 - a + a + -- - --- - --- - --- - 16 z + --- - ---- + ---- + -- - ----- + 2 5 3 a 10 8 6 4 2 a a a a a a a a 3 3 3 3 3 4 4 4 4 2 z 6 z 17 z 26 z z 4 3 z 13 z 5 z 38 z > ---- - ---- + ----- + ----- + -- + 17 z + ---- - ----- + ---- + ----- + 9 7 5 3 a 8 6 4 2 a a a a a a a a 5 5 5 5 6 6 6 7 7 4 z 18 z 16 z 6 z 6 5 z 10 z 22 z 5 z 5 z > ---- - ----- - ----- + ---- - 7 z + ---- - ----- - ----- + ---- - ---- + 7 5 3 a 6 4 2 5 a a a a a a a a 8 8 9 9 8 3 z 4 z z z > z + ---- + ---- + -- + -- 4 2 3 a a a a

In[19]:=
{Vassiliev[2][Knot[10, 61]], Vassiliev[3][Knot[10, 61]]}

Out[19]=
{-4, -5}

In[20]:=
Kh[Knot[10, 61]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 61], 2][q]

Out[21]=
-8 -7 -6 4 -4 6 7 1 2 3 4 -11 + q - q - q + -- - q - -- + -- + - + 8 q + 5 q - 13 q + 7 q + 5 3 2 q q q q 5 6 7 8 9 11 12 13 14 15 > 9 q - 13 q + 3 q + 9 q - 10 q + 7 q - 5 q - q + 5 q - 4 q + 17 18 19 20 21 22 > 3 q - 4 q + 2 q + q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₆₁

10₆₀
10₆₂

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-8 - q^-7 - q^-6 + 4q^-5 - q^-4 - 6q^-3 + 7q^-2 + q^-1 - 11 + 8q + 5q² - 13q³ + 7q⁴ + 9q⁵ - 13q⁶ + 3q⁷ + 9q⁸ - 10q⁹ + 7q¹¹ - 5q¹² - q¹³ + 5q¹⁴ - 4q¹⁵ + 3q¹⁷ - 4q¹⁸ + 2q¹⁹ + q²⁰ - 2q²¹ + q²²
3	q^-18 - q^-17 - q^-16 + 4q^-14 - 4q^-12 - 5q^-11 + 7q^-10 + 6q^-9 - 3q^-8 - 13q^-7 + 3q^-6 + 10q^-5 + 5q^-4 - 13q^-3 - 4q^-2 + 7q^-1 + 9 - 5q - 5q² + 5q⁴ + q⁵ - q⁶ - 3q⁷ - 3q⁸ + 4q⁹ + 5q¹⁰ - 3q¹¹ - 10q¹² + 4q¹³ + 14q¹⁴ - 2q¹⁵ - 21q¹⁶ + q¹⁷ + 25q¹⁸ + 3q¹⁹ - 29q²⁰ - 7q²¹ + 29q²² + 10q²³ - 25q²⁴ - 11q²⁵ + 20q²⁶ + 10q²⁷ - 13q²⁸ - 8q²⁹ + 8q³⁰ + 5q³¹ - 5q³² - 3q³³ + 3q³⁴ + 2q³⁵ - q³⁶ - 3q³⁷ + 2q³⁸ + q⁴⁰ - 2q⁴¹ + q⁴²
4	q^-32 - q^-31 - q^-30 + 5q^-27 - q^-26 - 3q^-25 - 3q^-24 - 6q^-23 + 11q^-22 + 5q^-21 + 2q^-20 - 5q^-19 - 20q^-18 + 7q^-17 + 5q^-16 + 15q^-15 + 9q^-14 - 25q^-13 - 2q^-12 - 13q^-11 + 13q^-10 + 26q^-9 - 11q^-8 + 10q^-7 - 29q^-6 - 10q^-5 + 19q^-4 - 4q^-3 + 39q^-2 - 16q^-1 - 26 - 4q - 24q² + 53q³ + 14q⁴ - 13q⁵ - 13q⁶ - 56q⁷ + 39q⁸ + 31q⁹ + 13q¹⁰ + 2q¹¹ - 73q¹² + 10q¹³ + 26q¹⁴ + 30q¹⁵ + 28q¹⁶ - 77q¹⁷ - 14q¹⁸ + 16q¹⁹ + 38q²⁰ + 50q²¹ - 77q²² - 36q²³ + 9q²⁴ + 52q²⁵ + 74q²⁶ - 79q²⁷ - 70q²⁸ - 5q²⁹ + 68q³⁰ + 107q³¹ - 66q³² - 98q³³ - 36q³⁴ + 61q³⁵ + 129q³⁶ - 31q³⁷ - 90q³⁸ - 58q³⁹ + 29q⁴⁰ + 110q⁴¹ - 3q⁴² - 52q⁴³ - 47q⁴⁴ + 3q⁴⁵ + 66q⁴⁶ - 2q⁴⁷ - 18q⁴⁸ - 20q⁴⁹ - 4q⁵⁰ + 30q⁵¹ - 9q⁵² - q⁵³ - 4q⁵⁴ - 2q⁵⁵ + 12q⁵⁶ - 12q⁵⁷ + 3q⁵⁸ + q⁵⁹ + 4q⁶¹ - 7q⁶² + 3q⁶³ + q⁶⁶ - 2q⁶⁷ + q⁶⁸
5	q^-50 - q^-49 - q^-48 + q^-45 + 4q^-44 - 4q^-42 - 3q^-41 - 3q^-40 - 2q^-39 + 9q^-38 + 9q^-37 + 2q^-36 - 4q^-35 - 11q^-34 - 16q^-33 + 12q^-31 + 17q^-30 + 15q^-29 - 22q^-27 - 21q^-26 - 14q^-25 + 3q^-24 + 27q^-23 + 30q^-22 + 9q^-21 - 6q^-20 - 28q^-19 - 39q^-18 - 12q^-17 + 19q^-16 + 33q^-15 + 41q^-14 + 19q^-13 - 30q^-12 - 52q^-11 - 42q^-10 - 10q^-9 + 40q^-8 + 71q^-7 + 42q^-6 - 13q^-5 - 63q^-4 - 74q^-3 - 33q^-2 + 42q^-1 + 83 + 69q + 5q² - 69q³ - 95q⁴ - 48q⁵ + 32q⁶ + 94q⁷ + 91q⁸ + 17q⁹ - 72q¹⁰ - 113q¹¹ - 68q¹² + 29q¹³ + 116q¹⁴ + 113q¹⁵ + 22q¹⁶ - 100q¹⁷ - 142q¹⁸ - 75q¹⁹ + 66q²⁰ + 156q²¹ + 125q²² - 26q²³ - 158q²⁴ - 158q²⁵ - 17q²⁶ + 144q²⁷ + 191q²⁸ + 49q²⁹ - 138q³⁰ - 201q³¹ - 78q³² + 129q³³ + 223q³⁴ + 94q³⁵ - 136q³⁶ - 244q³⁷ - 111q³⁸ + 148q³⁹ + 278q⁴⁰ + 138q⁴¹ - 160q⁴² - 326q⁴³ - 177q⁴⁴ + 166q⁴⁵ + 368q⁴⁶ + 228q⁴⁷ - 142q⁴⁸ - 400q⁴⁹ - 286q⁵⁰ + 105q⁵¹ + 401q⁵² + 324q⁵³ - 48q⁵⁴ - 369q⁵⁵ - 339q⁵⁶ - 6q⁵⁷ + 309q⁵⁸ + 325q⁵⁹ + 45q⁶⁰ - 241q⁶¹ - 277q⁶² - 67q⁶³ + 173q⁶⁴ + 222q⁶⁵ + 66q⁶⁶ - 119q⁶⁷ - 165q⁶⁸ - 52q⁶⁹ + 80q⁷⁰ + 112q⁷¹ + 36q⁷² - 52q⁷³ - 73q⁷⁴ - 21q⁷⁵ + 36q⁷⁶ + 42q⁷⁷ + 7q⁷⁸ - 19q⁷⁹ - 21q⁸⁰ - 6q⁸¹ + 13q⁸² + 11q⁸³ - 3q⁸⁴ - 3q⁸⁵ - 3q⁸⁷ + q⁸⁸ + 2q⁸⁹ - 2q⁹⁰ + q⁹¹ + 2q⁹² - 2q⁹³ - q⁹⁴ + q⁹⁵ + q⁹⁸ - 2q⁹⁹ + q¹⁰⁰
6	q^-72 - q^-71 - q^-70 + q^-67 + 5q^-65 - q^-64 - 4q^-63 - 3q^-62 - 3q^-61 - 3q^-59 + 14q^-58 + 7q^-57 + 2q^-56 - 3q^-55 - 9q^-54 - 11q^-53 - 23q^-52 + 11q^-51 + 11q^-50 + 20q^-49 + 18q^-48 + 12q^-47 - 4q^-46 - 45q^-45 - 17q^-44 - 26q^-43 + 2q^-42 + 19q^-41 + 49q^-40 + 48q^-39 - 5q^-38 + 4q^-37 - 53q^-36 - 52q^-35 - 56q^-34 + 4q^-33 + 47q^-32 + 44q^-31 + 95q^-30 + 29q^-29 - 8q^-28 - 89q^-27 - 84q^-26 - 68q^-25 - 48q^-24 + 83q^-23 + 102q^-22 + 129q^-21 + 43q^-20 - 16q^-19 - 102q^-18 - 184q^-17 - 87q^-16 - 25q^-15 + 115q^-14 + 150q^-13 + 169q^-12 + 87q^-11 - 104q^-10 - 153q^-9 - 216q^-8 - 104q^-7 + 3q^-6 + 165q^-5 + 265q^-4 + 150q^-3 + 57q^-2 - 167q^-1 - 231 - 260q - 111q² + 151q³ + 247q⁴ + 326q⁵ + 126q⁶ - 45q⁷ - 311q⁸ - 389q⁹ - 180q¹⁰ + 33q¹¹ + 355q¹² + 373q¹³ + 318q¹⁴ - 59q¹⁵ - 406q¹⁶ - 438q¹⁷ - 328q¹⁸ + 100q¹⁹ + 363q²⁰ + 594q²¹ + 315q²² - 162q²³ - 455q²⁴ - 601q²⁵ - 268q²⁶ + 119q²⁷ + 653q²⁸ + 612q²⁹ + 184q³⁰ - 258q³¹ - 681q³² - 578q³³ - 220q³⁴ + 520q³⁵ + 740q³⁶ + 483q³⁷ + 32q³⁸ - 589q³⁹ - 740q⁴⁰ - 516q⁴¹ + 307q⁴² + 716q⁴³ + 649q⁴⁴ + 266q⁴⁵ - 444q⁴⁶ - 773q⁴⁷ - 677q⁴⁸ + 170q⁴⁹ + 681q⁵⁰ + 718q⁵¹ + 355q⁵² - 417q⁵³ - 841q⁵⁴ - 769q⁵⁵ + 171q⁵⁶ + 801q⁵⁷ + 881q⁵⁸ + 420q⁵⁹ - 516q⁶⁰ - 1084q⁶¹ - 996q⁶² + 126q⁶³ + 990q⁶⁴ + 1211q⁶⁵ + 672q⁶⁶ - 475q⁶⁷ - 1309q⁶⁸ - 1348q⁶⁹ - 169q⁷⁰ + 922q⁷¹ + 1417q⁷² + 1024q⁷³ - 139q⁷⁴ - 1176q⁷⁵ - 1478q⁷⁶ - 537q⁷⁷ + 526q⁷⁸ + 1205q⁷⁹ + 1114q⁸⁰ + 232q⁸¹ - 730q⁸² - 1192q⁸³ - 629q⁸⁴ + 135q⁸⁵ + 739q⁸⁶ + 840q⁸⁷ + 330q⁸⁸ - 331q⁸⁹ - 728q⁹⁰ - 428q⁹¹ - 9q⁹² + 355q⁹³ + 463q⁹⁴ + 203q⁹⁵ - 143q⁹⁶ - 368q⁹⁷ - 178q⁹⁸ + 10q⁹⁹ + 151q¹⁰⁰ + 197q¹⁰¹ + 60q¹⁰² - 76q¹⁰³ - 165q¹⁰⁴ - 31q¹⁰⁵ + 35q¹⁰⁶ + 55q¹⁰⁷ + 64q¹⁰⁸ - 5q¹⁰⁹ - 35q¹¹⁰ - 64q¹¹¹ + 17q¹¹² + 25q¹¹³ + 11q¹¹⁴ + 15q¹¹⁵ - 14q¹¹⁶ - 6q¹¹⁷ - 22q¹¹⁸ + 16q¹¹⁹ + 7q¹²⁰ - 2q¹²¹ + 5q¹²² - 8q¹²³ + 5q¹²⁴ - 8q¹²⁵ + 6q¹²⁶ - 2q¹²⁸ + 3q¹²⁹ - 4q¹³⁰ + 4q¹³¹ - 3q¹³² + q¹³³ + q¹³⁶ - 2q¹³⁷ + q¹³⁸

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 61]]
Out[2]=	PD[X[8, 2, 9, 1], X[10, 4, 11, 3], X[2, 10, 3, 9], X[18, 12, 19, 11], > X[14, 7, 15, 8], X[16, 5, 17, 6], X[6, 15, 7, 16], X[4, 17, 5, 18], > X[20, 14, 1, 13], X[12, 20, 13, 19]]
In[3]:=	GaussCode[Knot[10, 61]]
Out[3]=	GaussCode[1, -3, 2, -8, 6, -7, 5, -1, 3, -2, 4, -10, 9, -5, 7, -6, 8, -4, 10, > -9]
In[4]:=	DTCode[Knot[10, 61]]
Out[4]=	DTCode[8, 10, 16, 14, 2, 18, 20, 6, 4, 12]
In[5]:=	br = BR[Knot[10, 61]]
Out[5]=	BR[4, {1, 1, 1, -2, 1, 1, 1, -2, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 61]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 61]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 61]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, {2, 3}, 3, 3, NotAvailable, 2}
In[10]:=	alex = Alexander[Knot[10, 61]][t]
Out[10]=	2 5 6 2 3 7 - -- + -- - - - 6 t + 5 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 61]][z]
Out[11]=	2 4 6 1 - 4 z - 7 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 61]}
In[13]:=	{KnotDet[Knot[10, 61]], KnotSignature[Knot[10, 61]]}
Out[13]=	{33, 4}
In[14]:=	Jones[Knot[10, 61]][q]
Out[14]=	-2 1 2 3 4 5 6 7 8 3 + q - - - 4 q + 4 q - 5 q + 5 q - 4 q + 3 q - 2 q + q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 61]}
In[16]:=	A2Invariant[Knot[10, 61]][q]
Out[16]=	-6 -4 2 4 6 8 14 24 2 + q + q + -- - q - 3 q - 2 q + 2 q + q 2 q
In[17]:=	HOMFLYPT[Knot[10, 61]][a, z]
Out[17]=	2 2 2 4 4 4 6 6 -6 -4 5 2 3 z 3 z 8 z 4 z 4 z 5 z z z 4 + a + a - -- + 4 z + ---- - ---- - ---- + z + -- - ---- - ---- - -- - -- 2 6 4 2 6 4 2 4 2 a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 61]][a, z]
Out[18]=	2 2 2 2 2 -6 -4 5 6 z 8 z 2 z 2 z 2 z 6 z z 24 z 4 - a + a + -- - --- - --- - --- - 16 z + --- - ---- + ---- + -- - ----- + 2 5 3 a 10 8 6 4 2 a a a a a a a a 3 3 3 3 3 4 4 4 4 2 z 6 z 17 z 26 z z 4 3 z 13 z 5 z 38 z > ---- - ---- + ----- + ----- + -- + 17 z + ---- - ----- + ---- + ----- + 9 7 5 3 a 8 6 4 2 a a a a a a a a 5 5 5 5 6 6 6 7 7 4 z 18 z 16 z 6 z 6 5 z 10 z 22 z 5 z 5 z > ---- - ----- - ----- + ---- - 7 z + ---- - ----- - ----- + ---- - ---- + 7 5 3 a 6 4 2 5 a a a a a a a a 8 8 9 9 8 3 z 4 z z z > z + ---- + ---- + -- + -- 4 2 3 a a a a
In[19]:=	{Vassiliev[2][Knot[10, 61]], Vassiliev[3][Knot[10, 61]]}
Out[19]=	{-4, -5}
In[20]:=	Kh[Knot[10, 61]][q, t]
Out[20]=	3 3 5 1 1 3 q 3 q 5 7 7 2 3 q + 2 q + ----- + ---- + ---- + - + ---- + 3 q t + 2 q t + 2 q t + 5 4 3 2 t t q t q t q t 9 2 9 3 11 3 11 4 13 4 13 5 15 5 17 6 > 3 q t + 2 q t + 2 q t + q t + 2 q t + q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 61], 2][q]
Out[21]=	-8 -7 -6 4 -4 6 7 1 2 3 4 -11 + q - q - q + -- - q - -- + -- + - + 8 q + 5 q - 13 q + 7 q + 5 3 2 q q q q 5 6 7 8 9 11 12 13 14 15 > 9 q - 13 q + 3 q + 9 q - 10 q + 7 q - 5 q - q + 5 q - 4 q + 17 18 19 20 21 22 > 3 q - 4 q + 2 q + q - 2 q + q

The Alternating Knot 1061

The Alternating Knot 10₆₁