The Alternating Knot 10₁₁₄

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X₈₃₉₄ X_18,13,19,14 X_20,11,1,12 X_12,19,13,20 X_2,16,3,15 X_4,17,5,18 X_10,6,11,5 X_14,7,15,8 X_16,10,17,9

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

DT (Dowker-Thistlethwaite) Code: 6 8 10 14 16 20 18 2 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 1 3 3 / NotAvailable 1

Alexander Polynomial: - 2t^-3 + 10t^-2 - 21t^-1 + 27 - 21t + 10t² - 2t³

Conway Polynomial: 1 + z² - 2z⁴ - 2z⁶

Other knots with the same Alexander/Conway Polynomial: {K11a93, ...}

Determinant and Signature: {93, 0}

Jones Polynomial: q^-6 - 4q^-5 + 7q^-4 - 11q^-3 + 15q^-2 - 15q^-1 + 15 - 12q + 8q² - 4q³ + q⁴

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

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

HOMFLY-PT Polynomial: a^-2z² + a^-2z⁴ - z² - 2z⁴ - z⁶ + 2a² - 2a²z⁴ - a²z⁶ - a⁴ + a⁴z² + a⁴z⁴

Kauffman Polynomial: a^-4z⁴ - 2a^-3z³ + 4a^-3z⁵ + 2a^-2z² - 8a^-2z⁴ + 8a^-2z⁶ - a^-1z + 5a^-1z³ - 13a^-1z⁵ + 10a^-1z⁷ + z⁴ - 9z⁶ + 8z⁸ - 3az + 18az³ - 27az⁵ + 8az⁷ + 3az⁹ - 2a² - 5a²z² + 26a²z⁴ - 35a²z⁶ + 14a²z⁸ - 2a³z + 18a³z³ - 21a³z⁵ + 2a³z⁷ + 3a³z⁹ - a⁴ - 3a⁴z² + 14a⁴z⁴ - 17a⁴z⁶ + 6a⁴z⁸ + 7a⁵z³ - 11a⁵z⁵ + 4a⁵z⁷ - 2a⁶z⁴ + 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 10₁₁₄. 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 7 3

j = 1 8 5

j = -1 8 8

j = -3 7 7

j = -5 4 8

j = -7 3 7

j = -9 1 4

j = -11 3

j = -13 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-18 - 4q^-17 + 2q^-16 + 14q^-15 - 24q^-14 - 7q^-13 + 58q^-12 - 47q^-11 - 50q^-10 + 121q^-9 - 47q^-8 - 117q^-7 + 170q^-6 - 20q^-5 - 173q^-4 + 179q^-3 + 18q^-2 - 188q^-1 + 144 + 44q - 150q² + 82q³ + 42q⁴ - 81q⁵ + 31q⁶ + 20q⁷ - 26q⁸ + 8q⁹ + 4q¹⁰ - 4q¹¹ + q¹²

3 q^-36 - 4q^-35 + 2q^-34 + 9q^-33 - 27q^-31 - 12q^-30 + 62q^-29 + 43q^-28 - 95q^-27 - 122q^-26 + 120q^-25 + 237q^-24 - 93q^-23 - 388q^-22 + 9q^-21 + 524q^-20 + 158q^-19 - 637q^-18 - 363q^-17 + 679q^-16 + 600q^-15 - 659q^-14 - 837q^-13 + 594q^-12 + 1027q^-11 - 463q^-10 - 1208q^-9 + 337q^-8 + 1309q^-7 - 162q^-6 - 1388q^-5 + 15q^-4 + 1368q^-3 + 167q^-2 - 1309q^-1 - 296 + 1147q + 412q² - 946q³ - 453q⁴ + 701q⁵ + 437q⁶ - 469q⁷ - 365q⁸ + 281q⁹ + 262q¹⁰ - 151q¹¹ - 162q¹² + 76q¹³ + 86q¹⁴ - 37q¹⁵ - 44q¹⁶ + 25q¹⁷ + 15q¹⁸ - 11q¹⁹ - 6q²⁰ + 4q²¹ + 4q²² - 4q²³ + q²⁴

4 q^-60 - 4q^-59 + 2q^-58 + 9q^-57 - 5q^-56 - 3q^-55 - 33q^-54 + 12q^-53 + 74q^-52 + 16q^-51 - 12q^-50 - 209q^-49 - 84q^-48 + 248q^-47 + 275q^-46 + 230q^-45 - 586q^-44 - 686q^-43 + 83q^-42 + 757q^-41 + 1365q^-40 - 416q^-39 - 1696q^-38 - 1278q^-37 + 346q^-36 + 3137q^-35 + 1349q^-34 - 1602q^-33 - 3397q^-32 - 2150q^-31 + 3720q^-30 + 4074q^-29 + 864q^-28 - 4273q^-27 - 5907q^-26 + 1853q^-25 + 5699q^-24 + 4794q^-23 - 2794q^-22 - 8871q^-21 - 1562q^-20 + 5263q^-19 + 8295q^-18 + 133q^-17 - 10112q^-16 - 4917q^-15 + 3532q^-14 + 10543q^-13 + 3156q^-12 - 10036q^-11 - 7529q^-10 + 1385q^-9 + 11623q^-8 + 5828q^-7 - 8912q^-6 - 9327q^-5 - 1105q^-4 + 11304q^-3 + 8019q^-2 - 6410q^-1 - 9709 - 3804q + 8925q² + 8870q³ - 2743q⁴ - 7825q⁵ - 5521q⁶ + 4897q⁷ + 7366q⁸ + 395q⁹ - 4227q¹⁰ - 4984q¹¹ + 1296q¹² + 4180q¹³ + 1405q¹⁴ - 1111q¹⁵ - 2840q¹⁶ - 229q¹⁷ + 1465q¹⁸ + 803q¹⁹ + 139q²⁰ - 1004q²¹ - 242q²² + 303q²³ + 155q²⁴ + 184q²⁵ - 239q²⁶ - 30q²⁷ + 58q²⁸ - 25q²⁹ + 54q³⁰ - 52q³¹ + 11q³² + 19q³³ - 16q³⁴ + 9q³⁵ - 10q³⁶ + 4q³⁷ + 4q³⁸ - 4q³⁹ + q⁴⁰

5 q^-90 - 4q^-89 + 2q^-88 + 9q^-87 - 5q^-86 - 8q^-85 - 9q^-84 - 9q^-83 + 23q^-82 + 67q^-81 + 22q^-80 - 80q^-79 - 144q^-78 - 124q^-77 + 78q^-76 + 366q^-75 + 438q^-74 + 25q^-73 - 639q^-72 - 1010q^-71 - 600q^-70 + 674q^-69 + 1959q^-68 + 1889q^-67 - 76q^-66 - 2730q^-65 - 3928q^-64 - 1991q^-63 + 2581q^-62 + 6372q^-61 + 5600q^-60 - 384q^-59 - 7781q^-58 - 10497q^-57 - 4692q^-56 + 6743q^-55 + 15061q^-54 + 12367q^-53 - 1599q^-52 - 17244q^-51 - 21262q^-50 - 7856q^-49 + 14745q^-48 + 28908q^-47 + 20774q^-46 - 6610q^-45 - 32603q^-44 - 34561q^-43 - 7141q^-42 + 30397q^-41 + 46683q^-40 + 24422q^-39 - 21831q^-38 - 54326q^-37 - 42787q^-36 + 7696q^-35 + 56447q^-34 + 59629q^-33 + 9722q^-32 - 53030q^-31 - 72840q^-30 - 28292q^-29 + 45115q^-28 + 82137q^-27 + 45796q^-26 - 34862q^-25 - 87225q^-24 - 61069q^-23 + 23379q^-22 + 89682q^-21 + 73812q^-20 - 12620q^-19 - 89921q^-18 - 84237q^-17 + 2241q^-16 + 89436q^-15 + 93094q^-14 + 7185q^-13 - 87747q^-12 - 100836q^-11 - 17118q^-10 + 85150q^-9 + 107579q^-8 + 27453q^-7 - 79863q^-6 - 112765q^-5 - 39254q^-4 + 71610q^-3 + 115139q^-2 + 51245q^-1 - 58967 - 113105q - 62759q² + 42813q³ + 105530q⁴ + 71094q⁵ - 24148q⁶ - 91983q⁷ - 74673q⁸ + 5716q⁹ + 73678q¹⁰ + 72019q¹¹ + 9792q¹² - 52999q¹³ - 63424q¹⁴ - 20091q¹⁵ + 32984q¹⁶ + 50614q¹⁷ + 24395q¹⁸ - 16501q¹⁹ - 36246q²⁰ - 23319q²¹ + 5034q²² + 22958q²³ + 18846q²⁴ + 1323q²⁵ - 12644q²⁶ - 13168q²⁷ - 3588q²⁸ + 5819q²⁹ + 7938q³⁰ + 3533q³¹ - 2019q³² - 4230q³³ - 2480q³⁴ + 430q³⁵ + 1861q³⁶ + 1397q³⁷ + 142q³⁸ - 717q³⁹ - 681q⁴⁰ - 139q⁴¹ + 221q⁴² + 235q⁴³ + 88q⁴⁴ - 35q⁴⁵ - 76q⁴⁶ - 47q⁴⁷ + 25q⁴⁸ + 14q⁴⁹ - 12q⁵⁰ + 13q⁵¹ + 5q⁵² - 12q⁵³ + 4q⁵⁴ + 5q⁵⁵ - 10q⁵⁶ + 4q⁵⁷ + 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, 114]]

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

In[3]:=
GaussCode[Knot[10, 114]]

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

In[4]:=
DTCode[Knot[10, 114]]

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

In[5]:=
br = BR[Knot[10, 114]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 114]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 114]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 114]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 114]][t]

Out[10]=
2 10 21 2 3 27 - -- + -- - -- - 21 t + 10 t - 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 114]][z]

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

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

Out[12]=
{Knot[10, 114], Knot[11, Alternating, 93]}

In[13]:=
{KnotDet[Knot[10, 114]], KnotSignature[Knot[10, 114]]}

Out[13]=
{93, 0}

In[14]:=
Jones[Knot[10, 114]][q]

Out[14]=
-6 4 7 11 15 15 2 3 4 15 + q - -- + -- - -- + -- - -- - 12 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, 114]}

In[16]:=
A2Invariant[Knot[10, 114]][q]

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

In[17]:=
HOMFLYPT[Knot[10, 114]][a, z]

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

In[18]:=
Kauffman[Knot[10, 114]][a, z]

Out[18]=
2 3 3 2 4 z 3 2 z 2 2 4 2 2 z 5 z -2 a - a - - - 3 a z - 2 a z + ---- - 5 a z - 3 a z - ---- + ---- + a 2 3 a a a 4 4 3 3 3 5 3 4 z 8 z 2 4 4 4 > 18 a z + 18 a z + 7 a z + z + -- - ---- + 26 a z + 14 a z - 4 2 a a 5 5 6 6 4 4 z 13 z 5 3 5 5 5 6 8 z > 2 a z + ---- - ----- - 27 a z - 21 a z - 11 a z - 9 z + ---- - 3 a 2 a a 7 2 6 4 6 6 6 10 z 7 3 7 5 7 8 > 35 a z - 17 a z + a z + ----- + 8 a z + 2 a z + 4 a z + 8 z + a 2 8 4 8 9 3 9 > 14 a z + 6 a z + 3 a z + 3 a z

In[19]:=
{Vassiliev[2][Knot[10, 114]], Vassiliev[3][Knot[10, 114]]}

Out[19]=
{1, -1}

In[20]:=
Kh[Knot[10, 114]][q, t]

Out[20]=
8 1 3 1 4 3 7 4 8 - + 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 7 7 8 3 3 2 5 2 5 3 7 3 > ----- + ---- + --- + 5 q t + 7 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, 114], 2][q]

Out[21]=
-18 4 2 14 24 7 58 47 50 121 47 117 144 + q - --- + --- + --- - --- - --- + --- - --- - --- + --- - -- - --- + 17 16 15 14 13 12 11 10 9 8 7 q q q q q q q q q q q 170 20 173 179 18 188 2 3 4 5 > --- - -- - --- + --- + -- - --- + 44 q - 150 q + 82 q + 42 q - 81 q + 6 5 4 3 2 q q q q q q 6 7 8 9 10 11 12 > 31 q + 20 q - 26 q + 8 q + 4 q - 4 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₁₄

10₁₁₃
10₁₁₅

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-18 - 4q^-17 + 2q^-16 + 14q^-15 - 24q^-14 - 7q^-13 + 58q^-12 - 47q^-11 - 50q^-10 + 121q^-9 - 47q^-8 - 117q^-7 + 170q^-6 - 20q^-5 - 173q^-4 + 179q^-3 + 18q^-2 - 188q^-1 + 144 + 44q - 150q² + 82q³ + 42q⁴ - 81q⁵ + 31q⁶ + 20q⁷ - 26q⁸ + 8q⁹ + 4q¹⁰ - 4q¹¹ + q¹²
3	q^-36 - 4q^-35 + 2q^-34 + 9q^-33 - 27q^-31 - 12q^-30 + 62q^-29 + 43q^-28 - 95q^-27 - 122q^-26 + 120q^-25 + 237q^-24 - 93q^-23 - 388q^-22 + 9q^-21 + 524q^-20 + 158q^-19 - 637q^-18 - 363q^-17 + 679q^-16 + 600q^-15 - 659q^-14 - 837q^-13 + 594q^-12 + 1027q^-11 - 463q^-10 - 1208q^-9 + 337q^-8 + 1309q^-7 - 162q^-6 - 1388q^-5 + 15q^-4 + 1368q^-3 + 167q^-2 - 1309q^-1 - 296 + 1147q + 412q² - 946q³ - 453q⁴ + 701q⁵ + 437q⁶ - 469q⁷ - 365q⁸ + 281q⁹ + 262q¹⁰ - 151q¹¹ - 162q¹² + 76q¹³ + 86q¹⁴ - 37q¹⁵ - 44q¹⁶ + 25q¹⁷ + 15q¹⁸ - 11q¹⁹ - 6q²⁰ + 4q²¹ + 4q²² - 4q²³ + q²⁴
4	q^-60 - 4q^-59 + 2q^-58 + 9q^-57 - 5q^-56 - 3q^-55 - 33q^-54 + 12q^-53 + 74q^-52 + 16q^-51 - 12q^-50 - 209q^-49 - 84q^-48 + 248q^-47 + 275q^-46 + 230q^-45 - 586q^-44 - 686q^-43 + 83q^-42 + 757q^-41 + 1365q^-40 - 416q^-39 - 1696q^-38 - 1278q^-37 + 346q^-36 + 3137q^-35 + 1349q^-34 - 1602q^-33 - 3397q^-32 - 2150q^-31 + 3720q^-30 + 4074q^-29 + 864q^-28 - 4273q^-27 - 5907q^-26 + 1853q^-25 + 5699q^-24 + 4794q^-23 - 2794q^-22 - 8871q^-21 - 1562q^-20 + 5263q^-19 + 8295q^-18 + 133q^-17 - 10112q^-16 - 4917q^-15 + 3532q^-14 + 10543q^-13 + 3156q^-12 - 10036q^-11 - 7529q^-10 + 1385q^-9 + 11623q^-8 + 5828q^-7 - 8912q^-6 - 9327q^-5 - 1105q^-4 + 11304q^-3 + 8019q^-2 - 6410q^-1 - 9709 - 3804q + 8925q² + 8870q³ - 2743q⁴ - 7825q⁵ - 5521q⁶ + 4897q⁷ + 7366q⁸ + 395q⁹ - 4227q¹⁰ - 4984q¹¹ + 1296q¹² + 4180q¹³ + 1405q¹⁴ - 1111q¹⁵ - 2840q¹⁶ - 229q¹⁷ + 1465q¹⁸ + 803q¹⁹ + 139q²⁰ - 1004q²¹ - 242q²² + 303q²³ + 155q²⁴ + 184q²⁵ - 239q²⁶ - 30q²⁷ + 58q²⁸ - 25q²⁹ + 54q³⁰ - 52q³¹ + 11q³² + 19q³³ - 16q³⁴ + 9q³⁵ - 10q³⁶ + 4q³⁷ + 4q³⁸ - 4q³⁹ + q⁴⁰
5	q^-90 - 4q^-89 + 2q^-88 + 9q^-87 - 5q^-86 - 8q^-85 - 9q^-84 - 9q^-83 + 23q^-82 + 67q^-81 + 22q^-80 - 80q^-79 - 144q^-78 - 124q^-77 + 78q^-76 + 366q^-75 + 438q^-74 + 25q^-73 - 639q^-72 - 1010q^-71 - 600q^-70 + 674q^-69 + 1959q^-68 + 1889q^-67 - 76q^-66 - 2730q^-65 - 3928q^-64 - 1991q^-63 + 2581q^-62 + 6372q^-61 + 5600q^-60 - 384q^-59 - 7781q^-58 - 10497q^-57 - 4692q^-56 + 6743q^-55 + 15061q^-54 + 12367q^-53 - 1599q^-52 - 17244q^-51 - 21262q^-50 - 7856q^-49 + 14745q^-48 + 28908q^-47 + 20774q^-46 - 6610q^-45 - 32603q^-44 - 34561q^-43 - 7141q^-42 + 30397q^-41 + 46683q^-40 + 24422q^-39 - 21831q^-38 - 54326q^-37 - 42787q^-36 + 7696q^-35 + 56447q^-34 + 59629q^-33 + 9722q^-32 - 53030q^-31 - 72840q^-30 - 28292q^-29 + 45115q^-28 + 82137q^-27 + 45796q^-26 - 34862q^-25 - 87225q^-24 - 61069q^-23 + 23379q^-22 + 89682q^-21 + 73812q^-20 - 12620q^-19 - 89921q^-18 - 84237q^-17 + 2241q^-16 + 89436q^-15 + 93094q^-14 + 7185q^-13 - 87747q^-12 - 100836q^-11 - 17118q^-10 + 85150q^-9 + 107579q^-8 + 27453q^-7 - 79863q^-6 - 112765q^-5 - 39254q^-4 + 71610q^-3 + 115139q^-2 + 51245q^-1 - 58967 - 113105q - 62759q² + 42813q³ + 105530q⁴ + 71094q⁵ - 24148q⁶ - 91983q⁷ - 74673q⁸ + 5716q⁹ + 73678q¹⁰ + 72019q¹¹ + 9792q¹² - 52999q¹³ - 63424q¹⁴ - 20091q¹⁵ + 32984q¹⁶ + 50614q¹⁷ + 24395q¹⁸ - 16501q¹⁹ - 36246q²⁰ - 23319q²¹ + 5034q²² + 22958q²³ + 18846q²⁴ + 1323q²⁵ - 12644q²⁶ - 13168q²⁷ - 3588q²⁸ + 5819q²⁹ + 7938q³⁰ + 3533q³¹ - 2019q³² - 4230q³³ - 2480q³⁴ + 430q³⁵ + 1861q³⁶ + 1397q³⁷ + 142q³⁸ - 717q³⁹ - 681q⁴⁰ - 139q⁴¹ + 221q⁴² + 235q⁴³ + 88q⁴⁴ - 35q⁴⁵ - 76q⁴⁶ - 47q⁴⁷ + 25q⁴⁸ + 14q⁴⁹ - 12q⁵⁰ + 13q⁵¹ + 5q⁵² - 12q⁵³ + 4q⁵⁴ + 5q⁵⁵ - 10q⁵⁶ + 4q⁵⁷ + 4q⁵⁸ - 4q⁵⁹ + q⁶⁰

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 114]]
Out[2]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[18, 13, 19, 14], X[20, 11, 1, 12], > X[12, 19, 13, 20], X[2, 16, 3, 15], X[4, 17, 5, 18], X[10, 6, 11, 5], > X[14, 7, 15, 8], X[16, 10, 17, 9]]
In[3]:=	GaussCode[Knot[10, 114]]
Out[3]=	GaussCode[1, -6, 2, -7, 8, -1, 9, -2, 10, -8, 4, -5, 3, -9, 6, -10, 7, -3, 5, > -4]
In[4]:=	DTCode[Knot[10, 114]]
Out[4]=	DTCode[6, 8, 10, 14, 16, 20, 18, 2, 4, 12]
In[5]:=	br = BR[Knot[10, 114]]
Out[5]=	BR[4, {-1, -1, -2, 1, 3, -2, 3, -2, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 114]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 114]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 114]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 114]][t]
Out[10]=	2 10 21 2 3 27 - -- + -- - -- - 21 t + 10 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 114]][z]
Out[11]=	2 4 6 1 + z - 2 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 114], Knot[11, Alternating, 93]}
In[13]:=	{KnotDet[Knot[10, 114]], KnotSignature[Knot[10, 114]]}
Out[13]=	{93, 0}
In[14]:=	Jones[Knot[10, 114]][q]
Out[14]=	-6 4 7 11 15 15 2 3 4 15 + q - -- + -- - -- + -- - -- - 12 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, 114]}
In[16]:=	A2Invariant[Knot[10, 114]][q]
Out[16]=	-18 2 3 4 2 2 2 4 6 8 10 12 -2 + q - --- - --- + -- + -- + -- + 3 q - 3 q + q + q - 2 q + q 16 10 8 4 2 q q q q q
In[17]:=	HOMFLYPT[Knot[10, 114]][a, z]
Out[17]=	2 4 2 4 2 z 4 2 4 z 2 4 4 4 6 2 6 2 a - a - z + -- + a z - 2 z + -- - 2 a z + a z - z - a z 2 2 a a
In[18]:=	Kauffman[Knot[10, 114]][a, z]
Out[18]=	2 3 3 2 4 z 3 2 z 2 2 4 2 2 z 5 z -2 a - a - - - 3 a z - 2 a z + ---- - 5 a z - 3 a z - ---- + ---- + a 2 3 a a a 4 4 3 3 3 5 3 4 z 8 z 2 4 4 4 > 18 a z + 18 a z + 7 a z + z + -- - ---- + 26 a z + 14 a z - 4 2 a a 5 5 6 6 4 4 z 13 z 5 3 5 5 5 6 8 z > 2 a z + ---- - ----- - 27 a z - 21 a z - 11 a z - 9 z + ---- - 3 a 2 a a 7 2 6 4 6 6 6 10 z 7 3 7 5 7 8 > 35 a z - 17 a z + a z + ----- + 8 a z + 2 a z + 4 a z + 8 z + a 2 8 4 8 9 3 9 > 14 a z + 6 a z + 3 a z + 3 a z
In[19]:=	{Vassiliev[2][Knot[10, 114]], Vassiliev[3][Knot[10, 114]]}
Out[19]=	{1, -1}
In[20]:=	Kh[Knot[10, 114]][q, t]
Out[20]=	8 1 3 1 4 3 7 4 8 - + 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 7 7 8 3 3 2 5 2 5 3 7 3 > ----- + ---- + --- + 5 q t + 7 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, 114], 2][q]
Out[21]=	-18 4 2 14 24 7 58 47 50 121 47 117 144 + q - --- + --- + --- - --- - --- + --- - --- - --- + --- - -- - --- + 17 16 15 14 13 12 11 10 9 8 7 q q q q q q q q q q q 170 20 173 179 18 188 2 3 4 5 > --- - -- - --- + --- + -- - --- + 44 q - 150 q + 82 q + 42 q - 81 q + 6 5 4 3 2 q q q q q q 6 7 8 9 10 11 12 > 31 q + 20 q - 26 q + 8 q + 4 q - 4 q + q

The Alternating Knot 10114

The Alternating Knot 10₁₁₄