The Non Alternating Knot 10₁₄₆

Visit 10₁₄₆'s page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10₁₄₆'s page at Knotilus!

Acknowledgement

KnotPlot

PD Presentation: X₄₂₅₁ X_5,18,6,19 X₈₃₉₄ X_2,9,3,10 X_11,17,12,16 X_7,12,8,13 X_15,6,16,7 X_17,11,18,10 X_13,1,14,20 X_19,15,20,14

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

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

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 2 3 / NotAvailable 1

Alexander Polynomial: 2t^-2 - 8t^-1 + 13 - 8t + 2t²

Conway Polynomial: 1 + 2z⁴

Other knots with the same Alexander/Conway Polynomial: {K11n18, K11n62, ...}

Determinant and Signature: {33, 0}

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

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

A2 (sl(3)) Invariant: - q^-16 + q^-14 + q^-12 - q^-10 + q^-8 - q^-6 + q^-2 + 2q² - q⁴ + q⁶ + q⁸ - q¹⁰

HOMFLY-PT Polynomial: - a^-2z² + 1 + z² + z⁴ + a²z² + a²z⁴ - a⁴z²

Kauffman Polynomial: - a^-3z + a^-3z³ - 3a^-2z² + 3a^-2z⁴ - 3a^-1z + 6a^-1z³ - 2a^-1z⁵ + a^-1z⁷ + 1 - 3z² + 5z⁴ - 2z⁶ + z⁸ - 3az + 12az³ - 11az⁵ + 4az⁷ + 3a²z² - 6a²z⁴ + a²z⁶ + a²z⁸ - a³z + 5a³z³ - 8a³z⁵ + 3a³z⁷ + 3a⁴z² - 8a⁴z⁴ + 3a⁴z⁶ - 2a⁵z³ + a⁵z⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {0, 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 10₁₄₆. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

j = 7 1

j = 5 2

j = 3 2 1

j = 1 4 2

j = -1 3 3

j = -3 2 3

j = -5 2 3

j = -7 1 2

j = -9 2

j = -11 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-15 - 3q^-14 + 9q^-12 - 9q^-11 - 7q^-10 + 20q^-9 - 9q^-8 - 19q^-7 + 28q^-6 - 2q^-5 - 29q^-4 + 30q^-3 + 4q^-2 - 33q^-1 + 24 + 8q - 25q² + 12q³ + 8q⁴ - 11q⁵ + 3q⁶ + 3q⁷ - 2q⁸

3 - q^-30 + 3q^-29 - 5q^-27 - 4q^-26 + 9q^-25 + 13q^-24 - 13q^-23 - 22q^-22 + 8q^-21 + 35q^-20 + 2q^-19 - 45q^-18 - 20q^-17 + 50q^-16 + 40q^-15 - 44q^-14 - 64q^-13 + 36q^-12 + 83q^-11 - 22q^-10 - 99q^-9 + 7q^-8 + 110q^-7 + 7q^-6 - 117q^-5 - 19q^-4 + 118q^-3 + 33q^-2 - 116q^-1 - 39 + 99q + 53q² - 86q³ - 52q⁴ + 59q⁵ + 53q⁶ - 38q⁷ - 42q⁸ + 15q⁹ + 33q¹⁰ - 5q¹¹ - 17q¹² - 3q¹³ + 8q¹⁴ + 4q¹⁵ - 3q¹⁶ - q¹⁷ - q¹⁸ + q¹⁹

4 q^-50 - 3q^-49 + 5q^-47 + 4q^-45 - 16q^-44 - 6q^-43 + 14q^-42 + 8q^-41 + 31q^-40 - 36q^-39 - 36q^-38 + q^-37 + 8q^-36 + 95q^-35 - 15q^-34 - 55q^-33 - 58q^-32 - 63q^-31 + 149q^-30 + 73q^-29 + 12q^-28 - 101q^-27 - 217q^-26 + 96q^-25 + 149q^-24 + 183q^-23 - 41q^-22 - 371q^-21 - 64q^-20 + 135q^-19 + 367q^-18 + 109q^-17 - 444q^-16 - 242q^-15 + 47q^-14 + 489q^-13 + 267q^-12 - 448q^-11 - 372q^-10 - 49q^-9 + 544q^-8 + 381q^-7 - 415q^-6 - 450q^-5 - 126q^-4 + 543q^-3 + 450q^-2 - 335q^-1 - 472 - 204q + 459q² + 475q³ - 187q⁴ - 413q⁵ - 271q⁶ + 286q⁷ + 415q⁸ - 20q⁹ - 257q¹⁰ - 265q¹¹ + 85q¹² + 262q¹³ + 76q¹⁴ - 81q¹⁵ - 169q¹⁶ - 30q¹⁷ + 99q¹⁸ + 64q¹⁹ + 11q²⁰ - 57q²¹ - 36q²² + 13q²³ + 18q²⁴ + 16q²⁵ - 5q²⁶ - 10q²⁷ - 2q²⁸ + 3q³⁰ + q³¹ - q³²

5 - q^-75 + 3q^-74 - 5q^-72 + 3q^-69 + 9q^-68 + 6q^-67 - 14q^-66 - 18q^-65 - 8q^-64 + 6q^-63 + 29q^-62 + 36q^-61 + 9q^-60 - 47q^-59 - 60q^-58 - 35q^-57 + 18q^-56 + 89q^-55 + 101q^-54 + 25q^-53 - 88q^-52 - 150q^-51 - 125q^-50 + 11q^-49 + 190q^-48 + 255q^-47 + 125q^-46 - 142q^-45 - 360q^-44 - 349q^-43 - 20q^-42 + 418q^-41 + 586q^-40 + 298q^-39 - 339q^-38 - 816q^-37 - 658q^-36 + 127q^-35 + 946q^-34 + 1063q^-33 + 215q^-32 - 974q^-31 - 1428q^-30 - 633q^-29 + 859q^-28 + 1737q^-27 + 1081q^-26 - 663q^-25 - 1946q^-24 - 1501q^-23 + 410q^-22 + 2067q^-21 + 1864q^-20 - 146q^-19 - 2122q^-18 - 2159q^-17 - 92q^-16 + 2138q^-15 + 2370q^-14 + 303q^-13 - 2118q^-12 - 2542q^-11 - 474q^-10 + 2093q^-9 + 2647q^-8 + 627q^-7 - 2016q^-6 - 2730q^-5 - 798q^-4 + 1932q^-3 + 2762q^-2 + 942q^-1 - 1730 - 2744q - 1150q² + 1503q³ + 2649q⁴ + 1296q⁵ - 1150q⁶ - 2435q⁷ - 1458q⁸ + 774q⁹ + 2121q¹⁰ + 1497q¹¹ - 348q¹² - 1706q¹³ - 1457q¹⁴ - 9q¹⁵ + 1238q¹⁶ + 1271q¹⁷ + 299q¹⁸ - 774q¹⁹ - 1034q²⁰ - 417q²¹ + 383q²² + 713q²³ + 444q²⁴ - 104q²⁵ - 443q²⁶ - 358q²⁷ - 37q²⁸ + 211q²⁹ + 239q³⁰ + 90q³¹ - 71q³² - 134q³³ - 76q³⁴ + 12q³⁵ + 49q³⁶ + 45q³⁷ + 15q³⁸ - 17q³⁹ - 22q⁴⁰ - 5q⁴¹ + 5q⁴³ + 5q⁴⁴ - 2q⁴⁶

6 q^-105 - 3q^-104 + 5q^-102 - 7q^-99 + 4q^-98 - 9q^-97 - 6q^-96 + 23q^-95 + 9q^-94 + 9q^-93 - 20q^-92 + q^-91 - 40q^-90 - 36q^-89 + 38q^-88 + 39q^-87 + 63q^-86 - 2q^-85 + 45q^-84 - 96q^-83 - 141q^-82 - 38q^-81 - q^-80 + 123q^-79 + 88q^-78 + 262q^-77 + 16q^-76 - 181q^-75 - 232q^-74 - 285q^-73 - 124q^-72 - 28q^-71 + 567q^-70 + 497q^-69 + 313q^-68 - 48q^-67 - 547q^-66 - 843q^-65 - 955q^-64 + 165q^-63 + 814q^-62 + 1377q^-61 + 1223q^-60 + 302q^-59 - 1152q^-58 - 2487q^-57 - 1682q^-56 - 378q^-55 + 1745q^-54 + 3087q^-53 + 2891q^-52 + 447q^-51 - 3012q^-50 - 4123q^-49 - 3501q^-48 - 115q^-47 + 3720q^-46 + 6014q^-45 + 4071q^-44 - 1130q^-43 - 5221q^-42 - 7074q^-41 - 3984q^-40 + 1982q^-39 + 7763q^-38 + 8045q^-37 + 2602q^-36 - 4130q^-35 - 9319q^-34 - 8082q^-33 - 1310q^-32 + 7563q^-31 + 10771q^-30 + 6409q^-29 - 1796q^-28 - 9870q^-27 - 11000q^-26 - 4511q^-25 + 6335q^-24 + 12003q^-23 + 9107q^-22 + 396q^-21 - 9519q^-20 - 12568q^-19 - 6725q^-18 + 5123q^-17 + 12370q^-16 + 10654q^-15 + 1882q^-14 - 8999q^-13 - 13313q^-12 - 8078q^-11 + 4174q^-10 + 12374q^-9 + 11590q^-8 + 2987q^-7 - 8352q^-6 - 13633q^-5 - 9143q^-4 + 3013q^-3 + 11916q^-2 + 12253q^-1 + 4337 - 7014q - 13327q² - 10185q³ + 1019q⁴ + 10342q⁵ + 12288q⁶ + 6077q⁷ - 4382q⁸ - 11619q⁹ - 10661q¹⁰ - 1757q¹¹ + 7097q¹² + 10770q¹³ + 7388q¹⁴ - 769q¹⁵ - 8029q¹⁶ - 9482q¹⁷ - 4116q¹⁸ + 2819q¹⁹ + 7300q²⁰ + 6962q²¹ + 2249q²² - 3522q²³ - 6351q²⁴ - 4539q²⁵ - 593q²⁶ + 3112q²⁷ + 4578q²⁸ + 3089q²⁹ - 157q³⁰ - 2693q³¹ - 2968q³² - 1692q³³ + 248q³⁴ + 1786q³⁵ + 1991q³⁶ + 915q³⁷ - 388q³⁸ - 1041q³⁹ - 1049q⁴⁰ - 547q⁴¹ + 206q⁴² + 645q⁴³ + 546q⁴⁴ + 205q⁴⁵ - 84q⁴⁶ - 262q⁴⁷ - 288q⁴⁸ - 114q⁴⁹ + 60q⁵⁰ + 114q⁵¹ + 90q⁵² + 52q⁵³ + 2q⁵⁴ - 51q⁵⁵ - 36q⁵⁶ - 11q⁵⁷ + 2q⁵⁸ + 5q⁵⁹ + 8q⁶⁰ + 10q⁶¹ - 3q⁶² - 2q⁶³ - q⁶⁶ - q⁶⁷ + q⁶⁸

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, 146]]

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

In[3]:=
GaussCode[Knot[10, 146]]

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

In[4]:=
DTCode[Knot[10, 146]]

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

In[5]:=
br = BR[Knot[10, 146]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 146]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 146]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 146]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 146]][t]

Out[10]=
2 8 2 13 + -- - - - 8 t + 2 t 2 t t

In[11]:=
Conway[Knot[10, 146]][z]

Out[11]=
4 1 + 2 z

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

Out[12]=
{Knot[10, 146], Knot[11, NonAlternating, 18], Knot[11, NonAlternating, 62]}

In[13]:=
{KnotDet[Knot[10, 146]], KnotSignature[Knot[10, 146]]}

Out[13]=
{33, 0}

In[14]:=
Jones[Knot[10, 146]][q]

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

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

Out[15]=
{Knot[10, 146]}

In[16]:=
A2Invariant[Knot[10, 146]][q]

Out[16]=
-16 -14 -12 -10 -8 -6 -2 2 4 6 8 10 -q + q + q - q + q - q + q + 2 q - q + q + q - q

In[17]:=
HOMFLYPT[Knot[10, 146]][a, z]

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

In[18]:=
Kauffman[Knot[10, 146]][a, z]

Out[18]=
2 3 3 z 3 z 3 2 3 z 2 2 4 2 z 6 z 1 - -- - --- - 3 a z - a z - 3 z - ---- + 3 a z + 3 a z + -- + ---- + 3 a 2 3 a a a a 4 5 3 3 3 5 3 4 3 z 2 4 4 4 2 z > 12 a z + 5 a z - 2 a z + 5 z + ---- - 6 a z - 8 a z - ---- - 2 a a 7 5 3 5 5 5 6 2 6 4 6 z 7 > 11 a z - 8 a z + a z - 2 z + a z + 3 a z + -- + 4 a z + a 3 7 8 2 8 > 3 a z + z + a z

In[19]:=
{Vassiliev[2][Knot[10, 146]], Vassiliev[3][Knot[10, 146]]}

Out[19]=
{0, 0}

In[20]:=
Kh[Knot[10, 146]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 146], 2][q]

Out[21]=
-15 3 9 9 7 20 9 19 28 2 29 30 4 24 + q - --- + --- - --- - --- + -- - -- - -- + -- - -- - -- + -- + -- - 14 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 33 2 3 4 5 6 7 8 > -- + 8 q - 25 q + 12 q + 8 q - 11 q + 3 q + 3 q - 2 q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₄₆

10₁₄₅
10₁₄₇

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-15 - 3q^-14 + 9q^-12 - 9q^-11 - 7q^-10 + 20q^-9 - 9q^-8 - 19q^-7 + 28q^-6 - 2q^-5 - 29q^-4 + 30q^-3 + 4q^-2 - 33q^-1 + 24 + 8q - 25q² + 12q³ + 8q⁴ - 11q⁵ + 3q⁶ + 3q⁷ - 2q⁸
3	- q^-30 + 3q^-29 - 5q^-27 - 4q^-26 + 9q^-25 + 13q^-24 - 13q^-23 - 22q^-22 + 8q^-21 + 35q^-20 + 2q^-19 - 45q^-18 - 20q^-17 + 50q^-16 + 40q^-15 - 44q^-14 - 64q^-13 + 36q^-12 + 83q^-11 - 22q^-10 - 99q^-9 + 7q^-8 + 110q^-7 + 7q^-6 - 117q^-5 - 19q^-4 + 118q^-3 + 33q^-2 - 116q^-1 - 39 + 99q + 53q² - 86q³ - 52q⁴ + 59q⁵ + 53q⁶ - 38q⁷ - 42q⁸ + 15q⁹ + 33q¹⁰ - 5q¹¹ - 17q¹² - 3q¹³ + 8q¹⁴ + 4q¹⁵ - 3q¹⁶ - q¹⁷ - q¹⁸ + q¹⁹
4	q^-50 - 3q^-49 + 5q^-47 + 4q^-45 - 16q^-44 - 6q^-43 + 14q^-42 + 8q^-41 + 31q^-40 - 36q^-39 - 36q^-38 + q^-37 + 8q^-36 + 95q^-35 - 15q^-34 - 55q^-33 - 58q^-32 - 63q^-31 + 149q^-30 + 73q^-29 + 12q^-28 - 101q^-27 - 217q^-26 + 96q^-25 + 149q^-24 + 183q^-23 - 41q^-22 - 371q^-21 - 64q^-20 + 135q^-19 + 367q^-18 + 109q^-17 - 444q^-16 - 242q^-15 + 47q^-14 + 489q^-13 + 267q^-12 - 448q^-11 - 372q^-10 - 49q^-9 + 544q^-8 + 381q^-7 - 415q^-6 - 450q^-5 - 126q^-4 + 543q^-3 + 450q^-2 - 335q^-1 - 472 - 204q + 459q² + 475q³ - 187q⁴ - 413q⁵ - 271q⁶ + 286q⁷ + 415q⁸ - 20q⁹ - 257q¹⁰ - 265q¹¹ + 85q¹² + 262q¹³ + 76q¹⁴ - 81q¹⁵ - 169q¹⁶ - 30q¹⁷ + 99q¹⁸ + 64q¹⁹ + 11q²⁰ - 57q²¹ - 36q²² + 13q²³ + 18q²⁴ + 16q²⁵ - 5q²⁶ - 10q²⁷ - 2q²⁸ + 3q³⁰ + q³¹ - q³²
5	- q^-75 + 3q^-74 - 5q^-72 + 3q^-69 + 9q^-68 + 6q^-67 - 14q^-66 - 18q^-65 - 8q^-64 + 6q^-63 + 29q^-62 + 36q^-61 + 9q^-60 - 47q^-59 - 60q^-58 - 35q^-57 + 18q^-56 + 89q^-55 + 101q^-54 + 25q^-53 - 88q^-52 - 150q^-51 - 125q^-50 + 11q^-49 + 190q^-48 + 255q^-47 + 125q^-46 - 142q^-45 - 360q^-44 - 349q^-43 - 20q^-42 + 418q^-41 + 586q^-40 + 298q^-39 - 339q^-38 - 816q^-37 - 658q^-36 + 127q^-35 + 946q^-34 + 1063q^-33 + 215q^-32 - 974q^-31 - 1428q^-30 - 633q^-29 + 859q^-28 + 1737q^-27 + 1081q^-26 - 663q^-25 - 1946q^-24 - 1501q^-23 + 410q^-22 + 2067q^-21 + 1864q^-20 - 146q^-19 - 2122q^-18 - 2159q^-17 - 92q^-16 + 2138q^-15 + 2370q^-14 + 303q^-13 - 2118q^-12 - 2542q^-11 - 474q^-10 + 2093q^-9 + 2647q^-8 + 627q^-7 - 2016q^-6 - 2730q^-5 - 798q^-4 + 1932q^-3 + 2762q^-2 + 942q^-1 - 1730 - 2744q - 1150q² + 1503q³ + 2649q⁴ + 1296q⁵ - 1150q⁶ - 2435q⁷ - 1458q⁸ + 774q⁹ + 2121q¹⁰ + 1497q¹¹ - 348q¹² - 1706q¹³ - 1457q¹⁴ - 9q¹⁵ + 1238q¹⁶ + 1271q¹⁷ + 299q¹⁸ - 774q¹⁹ - 1034q²⁰ - 417q²¹ + 383q²² + 713q²³ + 444q²⁴ - 104q²⁵ - 443q²⁶ - 358q²⁷ - 37q²⁸ + 211q²⁹ + 239q³⁰ + 90q³¹ - 71q³² - 134q³³ - 76q³⁴ + 12q³⁵ + 49q³⁶ + 45q³⁷ + 15q³⁸ - 17q³⁹ - 22q⁴⁰ - 5q⁴¹ + 5q⁴³ + 5q⁴⁴ - 2q⁴⁶
6	q^-105 - 3q^-104 + 5q^-102 - 7q^-99 + 4q^-98 - 9q^-97 - 6q^-96 + 23q^-95 + 9q^-94 + 9q^-93 - 20q^-92 + q^-91 - 40q^-90 - 36q^-89 + 38q^-88 + 39q^-87 + 63q^-86 - 2q^-85 + 45q^-84 - 96q^-83 - 141q^-82 - 38q^-81 - q^-80 + 123q^-79 + 88q^-78 + 262q^-77 + 16q^-76 - 181q^-75 - 232q^-74 - 285q^-73 - 124q^-72 - 28q^-71 + 567q^-70 + 497q^-69 + 313q^-68 - 48q^-67 - 547q^-66 - 843q^-65 - 955q^-64 + 165q^-63 + 814q^-62 + 1377q^-61 + 1223q^-60 + 302q^-59 - 1152q^-58 - 2487q^-57 - 1682q^-56 - 378q^-55 + 1745q^-54 + 3087q^-53 + 2891q^-52 + 447q^-51 - 3012q^-50 - 4123q^-49 - 3501q^-48 - 115q^-47 + 3720q^-46 + 6014q^-45 + 4071q^-44 - 1130q^-43 - 5221q^-42 - 7074q^-41 - 3984q^-40 + 1982q^-39 + 7763q^-38 + 8045q^-37 + 2602q^-36 - 4130q^-35 - 9319q^-34 - 8082q^-33 - 1310q^-32 + 7563q^-31 + 10771q^-30 + 6409q^-29 - 1796q^-28 - 9870q^-27 - 11000q^-26 - 4511q^-25 + 6335q^-24 + 12003q^-23 + 9107q^-22 + 396q^-21 - 9519q^-20 - 12568q^-19 - 6725q^-18 + 5123q^-17 + 12370q^-16 + 10654q^-15 + 1882q^-14 - 8999q^-13 - 13313q^-12 - 8078q^-11 + 4174q^-10 + 12374q^-9 + 11590q^-8 + 2987q^-7 - 8352q^-6 - 13633q^-5 - 9143q^-4 + 3013q^-3 + 11916q^-2 + 12253q^-1 + 4337 - 7014q - 13327q² - 10185q³ + 1019q⁴ + 10342q⁵ + 12288q⁶ + 6077q⁷ - 4382q⁸ - 11619q⁹ - 10661q¹⁰ - 1757q¹¹ + 7097q¹² + 10770q¹³ + 7388q¹⁴ - 769q¹⁵ - 8029q¹⁶ - 9482q¹⁷ - 4116q¹⁸ + 2819q¹⁹ + 7300q²⁰ + 6962q²¹ + 2249q²² - 3522q²³ - 6351q²⁴ - 4539q²⁵ - 593q²⁶ + 3112q²⁷ + 4578q²⁸ + 3089q²⁹ - 157q³⁰ - 2693q³¹ - 2968q³² - 1692q³³ + 248q³⁴ + 1786q³⁵ + 1991q³⁶ + 915q³⁷ - 388q³⁸ - 1041q³⁹ - 1049q⁴⁰ - 547q⁴¹ + 206q⁴² + 645q⁴³ + 546q⁴⁴ + 205q⁴⁵ - 84q⁴⁶ - 262q⁴⁷ - 288q⁴⁸ - 114q⁴⁹ + 60q⁵⁰ + 114q⁵¹ + 90q⁵² + 52q⁵³ + 2q⁵⁴ - 51q⁵⁵ - 36q⁵⁶ - 11q⁵⁷ + 2q⁵⁸ + 5q⁵⁹ + 8q⁶⁰ + 10q⁶¹ - 3q⁶² - 2q⁶³ - q⁶⁶ - q⁶⁷ + q⁶⁸

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 146]]
Out[2]=	PD[X[4, 2, 5, 1], X[5, 18, 6, 19], X[8, 3, 9, 4], X[2, 9, 3, 10], > X[11, 17, 12, 16], X[7, 12, 8, 13], X[15, 6, 16, 7], X[17, 11, 18, 10], > X[13, 1, 14, 20], X[19, 15, 20, 14]]
In[3]:=	GaussCode[Knot[10, 146]]
Out[3]=	GaussCode[1, -4, 3, -1, -2, 7, -6, -3, 4, 8, -5, 6, -9, 10, -7, 5, -8, 2, -10, > 9]
In[4]:=	DTCode[Knot[10, 146]]
Out[4]=	DTCode[4, 8, -18, -12, 2, -16, -20, -6, -10, -14]
In[5]:=	br = BR[Knot[10, 146]]
Out[5]=	BR[4, {-1, -1, 2, -1, 2, 1, -3, 2, -1, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 146]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 146]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 146]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 146]][t]
Out[10]=	2 8 2 13 + -- - - - 8 t + 2 t 2 t t
In[11]:=	Conway[Knot[10, 146]][z]
Out[11]=	4 1 + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 146], Knot[11, NonAlternating, 18], Knot[11, NonAlternating, 62]}
In[13]:=	{KnotDet[Knot[10, 146]], KnotSignature[Knot[10, 146]]}
Out[13]=	{33, 0}
In[14]:=	Jones[Knot[10, 146]][q]
Out[14]=	-5 3 4 5 6 2 3 6 - q + -- - -- + -- - - - 4 q + 3 q - q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 146]}
In[16]:=	A2Invariant[Knot[10, 146]][q]
Out[16]=	-16 -14 -12 -10 -8 -6 -2 2 4 6 8 10 -q + q + q - q + q - q + q + 2 q - q + q + q - q
In[17]:=	HOMFLYPT[Knot[10, 146]][a, z]
Out[17]=	2 2 z 2 2 4 2 4 2 4 1 + z - -- + a z - a z + z + a z 2 a
In[18]:=	Kauffman[Knot[10, 146]][a, z]
Out[18]=	2 3 3 z 3 z 3 2 3 z 2 2 4 2 z 6 z 1 - -- - --- - 3 a z - a z - 3 z - ---- + 3 a z + 3 a z + -- + ---- + 3 a 2 3 a a a a 4 5 3 3 3 5 3 4 3 z 2 4 4 4 2 z > 12 a z + 5 a z - 2 a z + 5 z + ---- - 6 a z - 8 a z - ---- - 2 a a 7 5 3 5 5 5 6 2 6 4 6 z 7 > 11 a z - 8 a z + a z - 2 z + a z + 3 a z + -- + 4 a z + a 3 7 8 2 8 > 3 a z + z + a z
In[19]:=	{Vassiliev[2][Knot[10, 146]], Vassiliev[3][Knot[10, 146]]}
Out[19]=	{0, 0}
In[20]:=	Kh[Knot[10, 146]][q, t]
Out[20]=	3 1 2 1 2 2 3 2 3 3 - + 4 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 7 3 > 2 q t + 2 q t + q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 146], 2][q]
Out[21]=	-15 3 9 9 7 20 9 19 28 2 29 30 4 24 + q - --- + --- - --- - --- + -- - -- - -- + -- - -- - -- + -- + -- - 14 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 33 2 3 4 5 6 7 8 > -- + 8 q - 25 q + 12 q + 8 q - 11 q + 3 q + 3 q - 2 q q

The Non Alternating Knot 10146

The Non Alternating Knot 10₁₄₆