The Alternating Knot 10₇₃

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Acknowledgement

KnotPlot

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

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

DT (Dowker-Thistlethwaite) Code: 4 8 10 14 2 18 16 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 - 27 + 20t - 7t² + t³

Conway Polynomial: 1 + z² - z⁴ + z⁶

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

Determinant and Signature: {83, -2}

Jones Polynomial: - q^-8 + 3q^-7 - 6q^-6 + 10q^-5 - 13q^-4 + 14q^-3 - 13q^-2 + 11q^-1 - 7 + 4q - q²

Other knots (up to mirrors) with the same Jones Polynomial: {10₈₃, ...}

A2 (sl(3)) Invariant: - q^-26 - q^-24 + 2q^-22 + 3q^-16 - 3q^-14 - q^-10 - q^-8 + 3q^-6 - 2q^-4 + 4q^-2 - q² + 2q⁴ - q⁶

HOMFLY-PT Polynomial: - z² - z⁴ + 3a² + 5a²z² + 3a²z⁴ + a²z⁶ - 4a⁴ - 6a⁴z² - 3a⁴z⁴ + 3a⁶ + 3a⁶z² - a⁸

Kauffman Polynomial: - a^-1z³ + a^-1z⁵ + 3z² - 7z⁴ + 4z⁶ - az + 4az³ - 10az⁵ + 6az⁷ - 3a² + 12a²z² - 16a²z⁴ + 2a²z⁶ + 4a²z⁸ - 3a³z + 16a³z³ - 26a³z⁵ + 12a³z⁷ + a³z⁹ - 4a⁴ + 17a⁴z² - 17a⁴z⁴ - 2a⁴z⁶ + 7a⁴z⁸ - 3a⁵z + 14a⁵z³ - 21a⁵z⁵ + 10a⁵z⁷ + a⁵z⁹ - 3a⁶ + 12a⁶z² - 14a⁶z⁴ + 3a⁶z⁶ + 3a⁶z⁸ + a⁷z³ - 5a⁷z⁵ + 4a⁷z⁷ - a⁸ + 4a⁸z² - 6a⁸z⁴ + 3a⁸z⁶ + a⁹z - 2a⁹z³ + a⁹z⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {1, -2}

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=-2 is the signature of 10₇₃. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3

j = 5 1

j = 3 3

j = 1 4 1

j = -1 7 3

j = -3 7 5

j = -5 7 6

j = -7 6 7

j = -9 4 7

j = -11 2 6

j = -13 1 4

j = -15 2

j = -17 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-23 - 3q^-22 + q^-21 + 9q^-20 - 17q^-19 + 2q^-18 + 36q^-17 - 50q^-16 - 6q^-15 + 90q^-14 - 90q^-13 - 32q^-12 + 151q^-11 - 111q^-10 - 66q^-9 + 182q^-8 - 100q^-7 - 86q^-6 + 166q^-5 - 64q^-4 - 83q^-3 + 113q^-2 - 24q^-1 - 57 + 52q - q² - 24q³ + 13q⁴ + 2q⁵ - 4q⁶ + q⁷

3 - q^-45 + 3q^-44 - q^-43 - 4q^-42 - 2q^-41 + 14q^-40 + q^-39 - 28q^-38 - 5q^-37 + 56q^-36 + 15q^-35 - 100q^-34 - 40q^-33 + 164q^-32 + 87q^-31 - 242q^-30 - 167q^-29 + 324q^-28 + 288q^-27 - 405q^-26 - 431q^-25 + 454q^-24 + 600q^-23 - 476q^-22 - 767q^-21 + 464q^-20 + 911q^-19 - 418q^-18 - 1019q^-17 + 344q^-16 + 1086q^-15 - 257q^-14 - 1098q^-13 + 152q^-12 + 1068q^-11 - 52q^-10 - 981q^-9 - 57q^-8 + 869q^-7 + 136q^-6 - 711q^-5 - 206q^-4 + 558q^-3 + 223q^-2 - 384q^-1 - 229 + 252q + 188q² - 136q³ - 144q⁴ + 66q⁵ + 90q⁶ - 20q⁷ - 54q⁸ + 7q⁹ + 23q¹⁰ - 8q¹² - 2q¹³ + 4q¹⁴ - q¹⁵

4 q^-74 - 3q^-73 + q^-72 + 4q^-71 - 3q^-70 + 5q^-69 - 17q^-68 + 7q^-67 + 24q^-66 - 16q^-65 + 10q^-64 - 69q^-63 + 28q^-62 + 108q^-61 - 28q^-60 - 9q^-59 - 249q^-58 + 50q^-57 + 360q^-56 + 68q^-55 - 48q^-54 - 751q^-53 - 85q^-52 + 859q^-51 + 543q^-50 + 89q^-49 - 1737q^-48 - 748q^-47 + 1401q^-46 + 1636q^-45 + 860q^-44 - 2996q^-43 - 2230q^-42 + 1445q^-41 + 3129q^-40 + 2562q^-39 - 3896q^-38 - 4246q^-37 + 599q^-36 + 4359q^-35 + 4846q^-34 - 3954q^-33 - 6039q^-32 - 896q^-31 + 4805q^-30 + 6905q^-29 - 3227q^-28 - 6996q^-27 - 2462q^-26 + 4423q^-25 + 8149q^-24 - 2056q^-23 - 6972q^-22 - 3688q^-21 + 3403q^-20 + 8387q^-19 - 684q^-18 - 6034q^-17 - 4414q^-16 + 1912q^-15 + 7619q^-14 + 673q^-13 - 4334q^-12 - 4465q^-11 + 247q^-10 + 5921q^-9 + 1625q^-8 - 2259q^-7 - 3700q^-6 - 1036q^-5 + 3695q^-4 + 1779q^-3 - 503q^-2 - 2333q^-1 - 1451 + 1680q + 1218q² + 376q³ - 1009q⁴ - 1085q⁵ + 480q⁶ + 503q⁷ + 440q⁸ - 242q⁹ - 508q¹⁰ + 60q¹¹ + 98q¹² + 212q¹³ - 8q¹⁴ - 152q¹⁵ - 5q¹⁷ + 55q¹⁸ + 11q¹⁹ - 29q²⁰ + q²¹ - 5q²² + 8q²³ + 2q²⁴ - 4q²⁵ + q²⁶

5 - q^-110 + 3q^-109 - q^-108 - 4q^-107 + 3q^-106 - 2q^-104 + 9q^-103 - 3q^-102 - 19q^-101 + 9q^-100 + 16q^-99 + 4q^-98 + 10q^-97 - 30q^-96 - 64q^-95 + 9q^-94 + 98q^-93 + 100q^-92 + 13q^-91 - 170q^-90 - 279q^-89 - 68q^-88 + 343q^-87 + 559q^-86 + 240q^-85 - 531q^-84 - 1086q^-83 - 662q^-82 + 717q^-81 + 1909q^-80 + 1510q^-79 - 735q^-78 - 3059q^-77 - 2979q^-76 + 299q^-75 + 4429q^-74 + 5298q^-73 + 923q^-72 - 5779q^-71 - 8502q^-70 - 3302q^-69 + 6661q^-68 + 12442q^-67 + 7145q^-66 - 6549q^-65 - 16808q^-64 - 12471q^-63 + 5075q^-62 + 20888q^-61 + 19035q^-60 - 1794q^-59 - 24226q^-58 - 26358q^-57 - 3051q^-56 + 26204q^-55 + 33617q^-54 + 9323q^-53 - 26595q^-52 - 40284q^-51 - 16288q^-50 + 25424q^-49 + 45717q^-48 + 23333q^-47 - 22892q^-46 - 49654q^-45 - 29929q^-44 + 19450q^-43 + 52032q^-42 + 35634q^-41 - 15480q^-40 - 52905q^-39 - 40256q^-38 + 11196q^-37 + 52489q^-36 + 43817q^-35 - 6851q^-34 - 50921q^-33 - 46231q^-32 + 2362q^-31 + 48204q^-30 + 47752q^-29 + 2144q^-28 - 44442q^-27 - 48110q^-26 - 6773q^-25 + 39498q^-24 + 47465q^-23 + 11246q^-22 - 33551q^-21 - 45385q^-20 - 15444q^-19 + 26651q^-18 + 42086q^-17 + 18749q^-16 - 19299q^-15 - 37171q^-14 - 21046q^-13 + 11920q^-12 + 31323q^-11 + 21650q^-10 - 5299q^-9 - 24457q^-8 - 20792q^-7 - 194q^-6 + 17773q^-5 + 18322q^-4 + 3868q^-3 - 11356q^-2 - 14951q^-1 - 5903 + 6278q + 11080q² + 6246q³ - 2442q⁴ - 7475q⁵ - 5555q⁶ + 221q⁷ + 4402q⁸ + 4257q⁹ + 925q¹⁰ - 2289q¹¹ - 2891q¹² - 1114q¹³ + 912q¹⁴ + 1696q¹⁵ + 1010q¹⁶ - 259q¹⁷ - 914q¹⁸ - 651q¹⁹ - 15q²⁰ + 391q²¹ + 390q²² + 97q²³ - 173q²⁴ - 196q²⁵ - 59q²⁶ + 58q²⁷ + 70q²⁸ + 46q²⁹ - 9q³⁰ - 46q³¹ - 12q³² + 11q³³ + 5q³⁴ + 4q³⁵ + 5q³⁶ - 8q³⁷ - 2q³⁸ + 4q³⁹ - q⁴⁰

6 q^-153 - 3q^-152 + q^-151 + 4q^-150 - 3q^-149 - 3q^-147 + 10q^-146 - 13q^-145 - 2q^-144 + 26q^-143 - 19q^-142 - 10q^-141 - 14q^-140 + 48q^-139 - 20q^-138 - 6q^-137 + 87q^-136 - 77q^-135 - 89q^-134 - 85q^-133 + 162q^-132 + 43q^-131 + 88q^-130 + 291q^-129 - 249q^-128 - 453q^-127 - 504q^-126 + 299q^-125 + 367q^-124 + 725q^-123 + 1179q^-122 - 432q^-121 - 1624q^-120 - 2299q^-119 - 360q^-118 + 889q^-117 + 3006q^-116 + 4639q^-115 + 824q^-114 - 3764q^-113 - 7712q^-112 - 4963q^-111 - 718q^-110 + 7515q^-109 + 14550q^-108 + 8773q^-107 - 3439q^-106 - 18099q^-105 - 19668q^-104 - 12557q^-103 + 9375q^-102 + 33170q^-101 + 32755q^-100 + 10952q^-99 - 26772q^-98 - 47814q^-97 - 47644q^-96 - 7043q^-95 + 51529q^-94 + 77222q^-93 + 56081q^-92 - 13719q^-91 - 77876q^-90 - 110955q^-89 - 61599q^-88 + 45298q^-87 + 127673q^-86 + 136431q^-85 + 43312q^-84 - 81848q^-83 - 184327q^-82 - 157047q^-81 - 8930q^-80 + 152717q^-79 + 229746q^-78 + 144548q^-77 - 36508q^-76 - 233351q^-75 - 266947q^-74 - 108265q^-73 + 129465q^-72 + 299383q^-71 + 260334q^-70 + 52358q^-69 - 235511q^-68 - 353478q^-67 - 221298q^-66 + 64742q^-65 + 323626q^-64 + 353656q^-63 + 153731q^-62 - 197255q^-61 - 396214q^-60 - 313815q^-59 - 13867q^-58 + 308004q^-57 + 406706q^-56 + 238295q^-55 - 141303q^-54 - 399730q^-53 - 371723q^-52 - 83847q^-51 + 270326q^-50 + 423848q^-49 + 296595q^-48 - 83357q^-47 - 377429q^-46 - 399768q^-45 - 140794q^-44 + 220484q^-43 + 414974q^-42 + 334111q^-41 - 23710q^-40 - 334412q^-39 - 404833q^-38 - 190614q^-37 + 155815q^-36 + 381039q^-35 + 354768q^-34 + 42953q^-33 - 265787q^-32 - 383771q^-31 - 233442q^-30 + 72448q^-29 + 314825q^-28 + 350715q^-27 + 111873q^-26 - 169328q^-25 - 326396q^-24 - 255369q^-23 - 19462q^-22 + 215010q^-21 + 307836q^-20 + 162216q^-19 - 60200q^-18 - 231371q^-17 - 237228q^-16 - 92576q^-15 + 100467q^-14 + 224103q^-13 + 169919q^-12 + 29122q^-11 - 120110q^-10 - 175737q^-9 - 119636q^-8 + 6659q^-7 + 122734q^-6 + 131292q^-5 + 70285q^-4 - 29381q^-3 - 95029q^-2 - 98262q^-1 - 39205 + 40497q + 71447q² + 63341q³ + 16101q⁴ - 30334q⁵ - 54756q⁶ - 40193q⁷ - 1149q⁸ + 23437q⁹ + 34991q¹⁰ + 21987q¹¹ + 803q¹² - 19489q¹³ - 22247q¹⁴ - 9581q¹⁵ + 1241q¹⁶ + 11873q¹⁷ + 12056q¹⁸ + 6610q¹⁹ - 3334q²⁰ - 7543q²¹ - 5348q²² - 2962q²³ + 1917q²⁴ + 3768q²⁵ + 3740q²⁶ + 365q²⁷ - 1478q²⁸ - 1455q²⁹ - 1615q³⁰ - 195q³¹ + 621q³² + 1215q³³ + 324q³⁴ - 135q³⁵ - 157q³⁶ - 448q³⁷ - 178q³⁸ + 12q³⁹ + 286q⁴⁰ + 61q⁴¹ - 5q⁴² + 22q⁴³ - 80q⁴⁴ - 40q⁴⁵ - 21q⁴⁶ + 60q⁴⁷ + 3q⁴⁸ - 10q⁴⁹ + 13q⁵⁰ - 10q⁵¹ - 4q⁵² - 5q⁵³ + 8q⁵⁴ + 2q⁵⁵ - 4q⁵⁶ + q⁵⁷

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, 73]]

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, 14, 17, 13], X[14, 7, 15, 8], X[6, 15, 7, 16], X[20, 17, 1, 18], > X[18, 11, 19, 12], X[12, 19, 13, 20]]

In[3]:=
GaussCode[Knot[10, 73]]

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

In[4]:=
DTCode[Knot[10, 73]]

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

In[5]:=
br = BR[Knot[10, 73]]

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

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

Out[6]=
{5, 12}

In[7]:=
BraidIndex[Knot[10, 73]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[10, 73]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 73]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 73]][t]

Out[10]=
-3 7 20 2 3 -27 + t - -- + -- + 20 t - 7 t + t 2 t t

In[11]:=
Conway[Knot[10, 73]][z]

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

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

Out[12]=
{Knot[10, 73]}

In[13]:=
{KnotDet[Knot[10, 73]], KnotSignature[Knot[10, 73]]}

Out[13]=
{83, -2}

In[14]:=
Jones[Knot[10, 73]][q]

Out[14]=
-8 3 6 10 13 14 13 11 2 -7 - q + -- - -- + -- - -- + -- - -- + -- + 4 q - q 7 6 5 4 3 2 q q q q q q q

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

Out[15]=
{Knot[10, 73], Knot[10, 83]}

In[16]:=
A2Invariant[Knot[10, 73]][q]

Out[16]=
-26 -24 2 3 3 -10 -8 3 2 4 2 4 6 -q - q + --- + --- - --- - q - q + -- - -- + -- - q + 2 q - q 22 16 14 6 4 2 q q q q q q

In[17]:=
HOMFLYPT[Knot[10, 73]][a, z]

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

In[18]:=
Kauffman[Knot[10, 73]][a, z]

Out[18]=
2 4 6 8 3 5 9 2 2 2 -3 a - 4 a - 3 a - a - a z - 3 a z - 3 a z + a z + 3 z + 12 a z + 3 4 2 6 2 8 2 z 3 3 3 5 3 7 3 > 17 a z + 12 a z + 4 a z - -- + 4 a z + 16 a z + 14 a z + a z - a 5 9 3 4 2 4 4 4 6 4 8 4 z 5 > 2 a z - 7 z - 16 a z - 17 a z - 14 a z - 6 a z + -- - 10 a z - a 3 5 5 5 7 5 9 5 6 2 6 4 6 > 26 a z - 21 a z - 5 a z + a z + 4 z + 2 a z - 2 a z + 6 6 8 6 7 3 7 5 7 7 7 2 8 > 3 a z + 3 a z + 6 a z + 12 a z + 10 a z + 4 a z + 4 a z + 4 8 6 8 3 9 5 9 > 7 a z + 3 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 73]], Vassiliev[3][Knot[10, 73]]}

Out[19]=
{1, -2}

In[20]:=
Kh[Knot[10, 73]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 73], 2][q]

Out[21]=
-23 3 -21 9 17 2 36 50 6 90 90 32 -57 + q - --- + q + --- - --- + --- + --- - --- - --- + --- - --- - --- + 22 20 19 18 17 16 15 14 13 12 q q q q q q q q q q 151 111 66 182 100 86 166 64 83 113 24 2 > --- - --- - -- + --- - --- - -- + --- - -- - -- + --- - -- + 52 q - q - 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 3 4 5 6 7 > 24 q + 13 q + 2 q - 4 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₇₃

10₇₂
10₇₄

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-23 - 3q^-22 + q^-21 + 9q^-20 - 17q^-19 + 2q^-18 + 36q^-17 - 50q^-16 - 6q^-15 + 90q^-14 - 90q^-13 - 32q^-12 + 151q^-11 - 111q^-10 - 66q^-9 + 182q^-8 - 100q^-7 - 86q^-6 + 166q^-5 - 64q^-4 - 83q^-3 + 113q^-2 - 24q^-1 - 57 + 52q - q² - 24q³ + 13q⁴ + 2q⁵ - 4q⁶ + q⁷
3	- q^-45 + 3q^-44 - q^-43 - 4q^-42 - 2q^-41 + 14q^-40 + q^-39 - 28q^-38 - 5q^-37 + 56q^-36 + 15q^-35 - 100q^-34 - 40q^-33 + 164q^-32 + 87q^-31 - 242q^-30 - 167q^-29 + 324q^-28 + 288q^-27 - 405q^-26 - 431q^-25 + 454q^-24 + 600q^-23 - 476q^-22 - 767q^-21 + 464q^-20 + 911q^-19 - 418q^-18 - 1019q^-17 + 344q^-16 + 1086q^-15 - 257q^-14 - 1098q^-13 + 152q^-12 + 1068q^-11 - 52q^-10 - 981q^-9 - 57q^-8 + 869q^-7 + 136q^-6 - 711q^-5 - 206q^-4 + 558q^-3 + 223q^-2 - 384q^-1 - 229 + 252q + 188q² - 136q³ - 144q⁴ + 66q⁵ + 90q⁶ - 20q⁷ - 54q⁸ + 7q⁹ + 23q¹⁰ - 8q¹² - 2q¹³ + 4q¹⁴ - q¹⁵
4	q^-74 - 3q^-73 + q^-72 + 4q^-71 - 3q^-70 + 5q^-69 - 17q^-68 + 7q^-67 + 24q^-66 - 16q^-65 + 10q^-64 - 69q^-63 + 28q^-62 + 108q^-61 - 28q^-60 - 9q^-59 - 249q^-58 + 50q^-57 + 360q^-56 + 68q^-55 - 48q^-54 - 751q^-53 - 85q^-52 + 859q^-51 + 543q^-50 + 89q^-49 - 1737q^-48 - 748q^-47 + 1401q^-46 + 1636q^-45 + 860q^-44 - 2996q^-43 - 2230q^-42 + 1445q^-41 + 3129q^-40 + 2562q^-39 - 3896q^-38 - 4246q^-37 + 599q^-36 + 4359q^-35 + 4846q^-34 - 3954q^-33 - 6039q^-32 - 896q^-31 + 4805q^-30 + 6905q^-29 - 3227q^-28 - 6996q^-27 - 2462q^-26 + 4423q^-25 + 8149q^-24 - 2056q^-23 - 6972q^-22 - 3688q^-21 + 3403q^-20 + 8387q^-19 - 684q^-18 - 6034q^-17 - 4414q^-16 + 1912q^-15 + 7619q^-14 + 673q^-13 - 4334q^-12 - 4465q^-11 + 247q^-10 + 5921q^-9 + 1625q^-8 - 2259q^-7 - 3700q^-6 - 1036q^-5 + 3695q^-4 + 1779q^-3 - 503q^-2 - 2333q^-1 - 1451 + 1680q + 1218q² + 376q³ - 1009q⁴ - 1085q⁵ + 480q⁶ + 503q⁷ + 440q⁸ - 242q⁹ - 508q¹⁰ + 60q¹¹ + 98q¹² + 212q¹³ - 8q¹⁴ - 152q¹⁵ - 5q¹⁷ + 55q¹⁸ + 11q¹⁹ - 29q²⁰ + q²¹ - 5q²² + 8q²³ + 2q²⁴ - 4q²⁵ + q²⁶
5	- q^-110 + 3q^-109 - q^-108 - 4q^-107 + 3q^-106 - 2q^-104 + 9q^-103 - 3q^-102 - 19q^-101 + 9q^-100 + 16q^-99 + 4q^-98 + 10q^-97 - 30q^-96 - 64q^-95 + 9q^-94 + 98q^-93 + 100q^-92 + 13q^-91 - 170q^-90 - 279q^-89 - 68q^-88 + 343q^-87 + 559q^-86 + 240q^-85 - 531q^-84 - 1086q^-83 - 662q^-82 + 717q^-81 + 1909q^-80 + 1510q^-79 - 735q^-78 - 3059q^-77 - 2979q^-76 + 299q^-75 + 4429q^-74 + 5298q^-73 + 923q^-72 - 5779q^-71 - 8502q^-70 - 3302q^-69 + 6661q^-68 + 12442q^-67 + 7145q^-66 - 6549q^-65 - 16808q^-64 - 12471q^-63 + 5075q^-62 + 20888q^-61 + 19035q^-60 - 1794q^-59 - 24226q^-58 - 26358q^-57 - 3051q^-56 + 26204q^-55 + 33617q^-54 + 9323q^-53 - 26595q^-52 - 40284q^-51 - 16288q^-50 + 25424q^-49 + 45717q^-48 + 23333q^-47 - 22892q^-46 - 49654q^-45 - 29929q^-44 + 19450q^-43 + 52032q^-42 + 35634q^-41 - 15480q^-40 - 52905q^-39 - 40256q^-38 + 11196q^-37 + 52489q^-36 + 43817q^-35 - 6851q^-34 - 50921q^-33 - 46231q^-32 + 2362q^-31 + 48204q^-30 + 47752q^-29 + 2144q^-28 - 44442q^-27 - 48110q^-26 - 6773q^-25 + 39498q^-24 + 47465q^-23 + 11246q^-22 - 33551q^-21 - 45385q^-20 - 15444q^-19 + 26651q^-18 + 42086q^-17 + 18749q^-16 - 19299q^-15 - 37171q^-14 - 21046q^-13 + 11920q^-12 + 31323q^-11 + 21650q^-10 - 5299q^-9 - 24457q^-8 - 20792q^-7 - 194q^-6 + 17773q^-5 + 18322q^-4 + 3868q^-3 - 11356q^-2 - 14951q^-1 - 5903 + 6278q + 11080q² + 6246q³ - 2442q⁴ - 7475q⁵ - 5555q⁶ + 221q⁷ + 4402q⁸ + 4257q⁹ + 925q¹⁰ - 2289q¹¹ - 2891q¹² - 1114q¹³ + 912q¹⁴ + 1696q¹⁵ + 1010q¹⁶ - 259q¹⁷ - 914q¹⁸ - 651q¹⁹ - 15q²⁰ + 391q²¹ + 390q²² + 97q²³ - 173q²⁴ - 196q²⁵ - 59q²⁶ + 58q²⁷ + 70q²⁸ + 46q²⁹ - 9q³⁰ - 46q³¹ - 12q³² + 11q³³ + 5q³⁴ + 4q³⁵ + 5q³⁶ - 8q³⁷ - 2q³⁸ + 4q³⁹ - q⁴⁰
6	q^-153 - 3q^-152 + q^-151 + 4q^-150 - 3q^-149 - 3q^-147 + 10q^-146 - 13q^-145 - 2q^-144 + 26q^-143 - 19q^-142 - 10q^-141 - 14q^-140 + 48q^-139 - 20q^-138 - 6q^-137 + 87q^-136 - 77q^-135 - 89q^-134 - 85q^-133 + 162q^-132 + 43q^-131 + 88q^-130 + 291q^-129 - 249q^-128 - 453q^-127 - 504q^-126 + 299q^-125 + 367q^-124 + 725q^-123 + 1179q^-122 - 432q^-121 - 1624q^-120 - 2299q^-119 - 360q^-118 + 889q^-117 + 3006q^-116 + 4639q^-115 + 824q^-114 - 3764q^-113 - 7712q^-112 - 4963q^-111 - 718q^-110 + 7515q^-109 + 14550q^-108 + 8773q^-107 - 3439q^-106 - 18099q^-105 - 19668q^-104 - 12557q^-103 + 9375q^-102 + 33170q^-101 + 32755q^-100 + 10952q^-99 - 26772q^-98 - 47814q^-97 - 47644q^-96 - 7043q^-95 + 51529q^-94 + 77222q^-93 + 56081q^-92 - 13719q^-91 - 77876q^-90 - 110955q^-89 - 61599q^-88 + 45298q^-87 + 127673q^-86 + 136431q^-85 + 43312q^-84 - 81848q^-83 - 184327q^-82 - 157047q^-81 - 8930q^-80 + 152717q^-79 + 229746q^-78 + 144548q^-77 - 36508q^-76 - 233351q^-75 - 266947q^-74 - 108265q^-73 + 129465q^-72 + 299383q^-71 + 260334q^-70 + 52358q^-69 - 235511q^-68 - 353478q^-67 - 221298q^-66 + 64742q^-65 + 323626q^-64 + 353656q^-63 + 153731q^-62 - 197255q^-61 - 396214q^-60 - 313815q^-59 - 13867q^-58 + 308004q^-57 + 406706q^-56 + 238295q^-55 - 141303q^-54 - 399730q^-53 - 371723q^-52 - 83847q^-51 + 270326q^-50 + 423848q^-49 + 296595q^-48 - 83357q^-47 - 377429q^-46 - 399768q^-45 - 140794q^-44 + 220484q^-43 + 414974q^-42 + 334111q^-41 - 23710q^-40 - 334412q^-39 - 404833q^-38 - 190614q^-37 + 155815q^-36 + 381039q^-35 + 354768q^-34 + 42953q^-33 - 265787q^-32 - 383771q^-31 - 233442q^-30 + 72448q^-29 + 314825q^-28 + 350715q^-27 + 111873q^-26 - 169328q^-25 - 326396q^-24 - 255369q^-23 - 19462q^-22 + 215010q^-21 + 307836q^-20 + 162216q^-19 - 60200q^-18 - 231371q^-17 - 237228q^-16 - 92576q^-15 + 100467q^-14 + 224103q^-13 + 169919q^-12 + 29122q^-11 - 120110q^-10 - 175737q^-9 - 119636q^-8 + 6659q^-7 + 122734q^-6 + 131292q^-5 + 70285q^-4 - 29381q^-3 - 95029q^-2 - 98262q^-1 - 39205 + 40497q + 71447q² + 63341q³ + 16101q⁴ - 30334q⁵ - 54756q⁶ - 40193q⁷ - 1149q⁸ + 23437q⁹ + 34991q¹⁰ + 21987q¹¹ + 803q¹² - 19489q¹³ - 22247q¹⁴ - 9581q¹⁵ + 1241q¹⁶ + 11873q¹⁷ + 12056q¹⁸ + 6610q¹⁹ - 3334q²⁰ - 7543q²¹ - 5348q²² - 2962q²³ + 1917q²⁴ + 3768q²⁵ + 3740q²⁶ + 365q²⁷ - 1478q²⁸ - 1455q²⁹ - 1615q³⁰ - 195q³¹ + 621q³² + 1215q³³ + 324q³⁴ - 135q³⁵ - 157q³⁶ - 448q³⁷ - 178q³⁸ + 12q³⁹ + 286q⁴⁰ + 61q⁴¹ - 5q⁴² + 22q⁴³ - 80q⁴⁴ - 40q⁴⁵ - 21q⁴⁶ + 60q⁴⁷ + 3q⁴⁸ - 10q⁴⁹ + 13q⁵⁰ - 10q⁵¹ - 4q⁵² - 5q⁵³ + 8q⁵⁴ + 2q⁵⁵ - 4q⁵⁶ + q⁵⁷

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 73]]
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, 14, 17, 13], X[14, 7, 15, 8], X[6, 15, 7, 16], X[20, 17, 1, 18], > X[18, 11, 19, 12], X[12, 19, 13, 20]]
In[3]:=	GaussCode[Knot[10, 73]]
Out[3]=	GaussCode[1, -4, 3, -1, 2, -7, 6, -3, 4, -2, 9, -10, 5, -6, 7, -5, 8, -9, 10, > -8]
In[4]:=	DTCode[Knot[10, 73]]
Out[4]=	DTCode[4, 8, 10, 14, 2, 18, 16, 6, 20, 12]
In[5]:=	br = BR[Knot[10, 73]]
Out[5]=	BR[5, {-1, -1, -2, 1, -2, -1, 3, -2, 3, -4, 3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 73]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 73]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 73]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 73]][t]
Out[10]=	-3 7 20 2 3 -27 + t - -- + -- + 20 t - 7 t + t 2 t t
In[11]:=	Conway[Knot[10, 73]][z]
Out[11]=	2 4 6 1 + z - z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 73]}
In[13]:=	{KnotDet[Knot[10, 73]], KnotSignature[Knot[10, 73]]}
Out[13]=	{83, -2}
In[14]:=	Jones[Knot[10, 73]][q]
Out[14]=	-8 3 6 10 13 14 13 11 2 -7 - q + -- - -- + -- - -- + -- - -- + -- + 4 q - q 7 6 5 4 3 2 q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 73], Knot[10, 83]}
In[16]:=	A2Invariant[Knot[10, 73]][q]
Out[16]=	-26 -24 2 3 3 -10 -8 3 2 4 2 4 6 -q - q + --- + --- - --- - q - q + -- - -- + -- - q + 2 q - q 22 16 14 6 4 2 q q q q q q
In[17]:=	HOMFLYPT[Knot[10, 73]][a, z]
Out[17]=	2 4 6 8 2 2 2 4 2 6 2 4 2 4 3 a - 4 a + 3 a - a - z + 5 a z - 6 a z + 3 a z - z + 3 a z - 4 4 2 6 > 3 a z + a z
In[18]:=	Kauffman[Knot[10, 73]][a, z]
Out[18]=	2 4 6 8 3 5 9 2 2 2 -3 a - 4 a - 3 a - a - a z - 3 a z - 3 a z + a z + 3 z + 12 a z + 3 4 2 6 2 8 2 z 3 3 3 5 3 7 3 > 17 a z + 12 a z + 4 a z - -- + 4 a z + 16 a z + 14 a z + a z - a 5 9 3 4 2 4 4 4 6 4 8 4 z 5 > 2 a z - 7 z - 16 a z - 17 a z - 14 a z - 6 a z + -- - 10 a z - a 3 5 5 5 7 5 9 5 6 2 6 4 6 > 26 a z - 21 a z - 5 a z + a z + 4 z + 2 a z - 2 a z + 6 6 8 6 7 3 7 5 7 7 7 2 8 > 3 a z + 3 a z + 6 a z + 12 a z + 10 a z + 4 a z + 4 a z + 4 8 6 8 3 9 5 9 > 7 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 73]], Vassiliev[3][Knot[10, 73]]}
Out[19]=	{1, -2}
In[20]:=	Kh[Knot[10, 73]][q, t]
Out[20]=	5 7 1 2 1 4 2 6 4 7 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 6 7 7 6 7 3 t 2 3 2 5 3 > ----- + ----- + ----- + ---- + ---- + --- + 4 q t + q t + 3 q t + q t 7 3 7 2 5 2 5 3 q q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 73], 2][q]
Out[21]=	-23 3 -21 9 17 2 36 50 6 90 90 32 -57 + q - --- + q + --- - --- + --- + --- - --- - --- + --- - --- - --- + 22 20 19 18 17 16 15 14 13 12 q q q q q q q q q q 151 111 66 182 100 86 166 64 83 113 24 2 > --- - --- - -- + --- - --- - -- + --- - -- - -- + --- - -- + 52 q - q - 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 3 4 5 6 7 > 24 q + 13 q + 2 q - 4 q + q

The Alternating Knot 1073

The Alternating Knot 10₇₃