The Alternating Knot 10₅₈

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_7,10,8,11 X₃₉₄₈ X_9,3,10,2 X_5,14,6,15 X_11,19,12,18 X_15,20,16,1 X_19,16,20,17 X_17,13,18,12 X_13,6,14,7

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

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

Minimum Braid Representative:

Length is 11, width is 6
Braid index is 6

A Morse Link Presentation:

3D Invariants:

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

Reversible 2 2 3 / NotAvailable 1

Alexander Polynomial: 3t^-2 - 16t^-1 + 27 - 16t + 3t²

Conway Polynomial: 1 - 4z² + 3z⁴

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

Determinant and Signature: {65, 0}

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

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

A2 (sl(3)) Invariant: q^-20 + q^-18 - 2q^-16 + q^-14 - 2q^-10 + 3q^-8 + q^-4 - 2 + q² - 3q⁴ + q⁶ + 2q⁸ - q¹⁰ + q¹² + q¹⁴

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

Kauffman Polynomial: a^-4 - 2a^-4z² + a^-4z⁴ - 2a^-3z³ + 2a^-3z⁵ - 2a^-2z⁴ + 3a^-2z⁶ - 4a^-1z + 8a^-1z³ - 6a^-1z⁵ + 4a^-1z⁷ - 2 + 8z² - 5z⁴ - z⁶ + 3z⁸ - 6az + 21az³ - 22az⁵ + 7az⁷ + az⁹ - 3a² + 10a²z² - 4a²z⁴ - 10a²z⁶ + 6a²z⁸ - 4a³z + 18a³z³ - 23a³z⁵ + 6a³z⁷ + a³z⁹ - 2a⁴ + 7a⁴z² - 5a⁴z⁴ - 5a⁴z⁶ + 3a⁴z⁸ - 2a⁵z + 7a⁵z³ - 9a⁵z⁵ + 3a⁵z⁷ - a⁶ + 3a⁶z² - 3a⁶z⁴ + a⁶z⁶

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-4, 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 10₅₈. 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 1

j = 5 4 1

j = 3 4 1

j = 1 6 4

j = -1 6 5

j = -3 4 5

j = -5 4 6

j = -7 2 4

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 - 13q^-14 - 10q^-13 + 36q^-12 - 20q^-11 - 36q^-10 + 63q^-9 - 12q^-8 - 68q^-7 + 78q^-6 + 6q^-5 - 91q^-4 + 76q^-3 + 23q^-2 - 91q^-1 + 57 + 29q - 66q² + 30q³ + 20q⁴ - 32q⁵ + 12q⁶ + 7q⁷ - 10q⁸ + 4q⁹ + q¹⁰ - 2q¹¹ + q¹²

3 q^-36 - 3q^-35 + 5q^-33 + 6q^-32 - 13q^-31 - 17q^-30 + 20q^-29 + 38q^-28 - 20q^-27 - 69q^-26 + 5q^-25 + 109q^-24 + 22q^-23 - 136q^-22 - 77q^-21 + 157q^-20 + 138q^-19 - 151q^-18 - 210q^-17 + 130q^-16 + 269q^-15 - 81q^-14 - 328q^-13 + 31q^-12 + 366q^-11 + 31q^-10 - 392q^-9 - 92q^-8 + 403q^-7 + 146q^-6 - 393q^-5 - 198q^-4 + 370q^-3 + 229q^-2 - 320q^-1 - 247 + 259q + 242q² - 190q³ - 215q⁴ + 125q⁵ + 171q⁶ - 72q⁷ - 120q⁸ + 33q⁹ + 80q¹⁰ - 21q¹¹ - 37q¹² + 5q¹³ + 22q¹⁴ - 8q¹⁵ - 5q¹⁶ + 2q¹⁷ + 4q¹⁸ - 5q¹⁹ + 2q²⁰ + q²² - 2q²³ + q²⁴

4 q^-60 - 3q^-59 + 5q^-57 + 6q^-55 - 20q^-54 - 10q^-53 + 20q^-52 + 15q^-51 + 47q^-50 - 62q^-49 - 71q^-48 + 5q^-47 + 41q^-46 + 198q^-45 - 53q^-44 - 175q^-43 - 143q^-42 - 56q^-41 + 446q^-40 + 148q^-39 - 135q^-38 - 383q^-37 - 456q^-36 + 529q^-35 + 482q^-34 + 277q^-33 - 412q^-32 - 1071q^-31 + 188q^-30 + 608q^-29 + 963q^-28 + 26q^-27 - 1520q^-26 - 490q^-25 + 276q^-24 + 1571q^-23 + 812q^-22 - 1561q^-21 - 1172q^-20 - 392q^-19 + 1873q^-18 + 1629q^-17 - 1274q^-16 - 1659q^-15 - 1119q^-14 + 1905q^-13 + 2279q^-12 - 840q^-11 - 1928q^-10 - 1742q^-9 + 1722q^-8 + 2691q^-7 - 311q^-6 - 1939q^-5 - 2192q^-4 + 1280q^-3 + 2743q^-2 + 273q^-1 - 1568 - 2318q + 600q² + 2296q³ + 712q⁴ - 867q⁵ - 1959q⁶ - 29q⁷ + 1456q⁸ + 755q⁹ - 182q¹⁰ - 1234q¹¹ - 300q¹² + 639q¹³ + 464q¹⁴ + 148q¹⁵ - 550q¹⁶ - 229q¹⁷ + 179q¹⁸ + 152q¹⁹ + 155q²⁰ - 173q²¹ - 85q²² + 36q²³ + 10q²⁴ + 73q²⁵ - 45q²⁶ - 15q²⁷ + 11q²⁸ - 15q²⁹ + 25q³⁰ - 11q³¹ - q³² + 4q³³ - 8q³⁴ + 7q³⁵ - 2q³⁶ + q³⁸ - 2q³⁹ + q⁴⁰

5 q^-90 - 3q^-89 + 5q^-87 - q^-84 - 13q^-83 - 10q^-82 + 20q^-81 + 24q^-80 + 15q^-79 - 4q^-78 - 55q^-77 - 71q^-76 - 6q^-75 + 95q^-74 + 132q^-73 + 80q^-72 - 84q^-71 - 253q^-70 - 233q^-69 + 19q^-68 + 340q^-67 + 458q^-66 + 209q^-65 - 322q^-64 - 727q^-63 - 609q^-62 + 86q^-61 + 915q^-60 + 1126q^-59 + 446q^-58 - 786q^-57 - 1672q^-56 - 1310q^-55 + 287q^-54 + 1953q^-53 + 2302q^-52 + 801q^-51 - 1753q^-50 - 3300q^-49 - 2264q^-48 + 893q^-47 + 3853q^-46 + 4017q^-45 + 685q^-44 - 3832q^-43 - 5663q^-42 - 2803q^-41 + 2970q^-40 + 6977q^-39 + 5246q^-38 - 1427q^-37 - 7632q^-36 - 7696q^-35 - 758q^-34 + 7679q^-33 + 9899q^-32 + 3187q^-31 - 7050q^-30 - 11665q^-29 - 5764q^-28 + 6033q^-27 + 12989q^-26 + 8135q^-25 - 4733q^-24 - 13866q^-23 - 10288q^-22 + 3341q^-21 + 14447q^-20 + 12146q^-19 - 1986q^-18 - 14762q^-17 - 13722q^-16 + 628q^-15 + 14863q^-14 + 15120q^-13 + 700q^-12 - 14734q^-11 - 16230q^-10 - 2146q^-9 + 14227q^-8 + 17126q^-7 + 3673q^-6 - 13272q^-5 - 17577q^-4 - 5299q^-3 + 11752q^-2 + 17481q^-1 + 6860 - 9688q - 16662q² - 8168q³ + 7226q⁴ + 15092q⁵ + 8938q⁶ - 4599q⁷ - 12831q⁸ - 9044q⁹ + 2142q¹⁰ + 10157q¹¹ + 8431q¹² - 202q¹³ - 7358q¹⁴ - 7200q¹⁵ - 1109q¹⁶ + 4795q¹⁷ + 5681q¹⁸ + 1707q¹⁹ - 2804q²⁰ - 4006q²¹ - 1766q²² + 1327q²³ + 2637q²⁴ + 1488q²⁵ - 542q²⁶ - 1507q²⁷ - 1048q²⁸ + 63q²⁹ + 826q³⁰ + 668q³¹ + 36q³² - 366q³³ - 365q³⁴ - 96q³⁵ + 181q³⁶ + 184q³⁷ + 34q³⁸ - 45q³⁹ - 82q⁴⁰ - 47q⁴¹ + 41q⁴² + 36q⁴³ - 4q⁴⁴ + 3q⁴⁵ - 9q⁴⁶ - 21q⁴⁷ + 14q⁴⁸ + 10q⁴⁹ - 8q⁵⁰ + 3q⁵¹ + q⁵² - 8q⁵³ + 4q⁵⁴ + 3q⁵⁵ - 2q⁵⁶ + q⁵⁸ - 2q⁵⁹ + q⁶⁰

6 q^-126 - 3q^-125 + 5q^-123 - 7q^-120 + 6q^-119 - 13q^-118 - 10q^-117 + 29q^-116 + 15q^-115 + 15q^-114 - 27q^-113 + 3q^-112 - 66q^-111 - 71q^-110 + 59q^-109 + 86q^-108 + 132q^-107 + 12q^-106 + 62q^-105 - 228q^-104 - 346q^-103 - 106q^-102 + 81q^-101 + 416q^-100 + 350q^-99 + 595q^-98 - 167q^-97 - 809q^-96 - 878q^-95 - 683q^-94 + 179q^-93 + 732q^-92 + 2087q^-91 + 1214q^-90 - 153q^-89 - 1575q^-88 - 2599q^-87 - 2170q^-86 - 1002q^-85 + 3001q^-84 + 4012q^-83 + 3675q^-82 + 1026q^-81 - 2838q^-80 - 5993q^-79 - 7148q^-78 - 1233q^-77 + 3734q^-76 + 8804q^-75 + 9148q^-74 + 4437q^-73 - 4760q^-72 - 14007q^-71 - 12371q^-70 - 6498q^-69 + 6488q^-68 + 16939q^-67 + 19913q^-66 + 9074q^-65 - 10742q^-64 - 22234q^-63 - 25847q^-62 - 10965q^-61 + 12006q^-60 + 33080q^-59 + 32877q^-58 + 9876q^-57 - 17472q^-56 - 41933q^-55 - 39075q^-54 - 11733q^-53 + 30341q^-52 + 52988q^-51 + 41592q^-50 + 6538q^-49 - 41731q^-48 - 63587q^-47 - 46453q^-46 + 8709q^-45 + 57488q^-44 + 70135q^-43 + 40923q^-42 - 23928q^-41 - 73965q^-40 - 78314q^-39 - 22402q^-38 + 46392q^-37 + 86347q^-36 + 72749q^-35 + 2037q^-34 - 71019q^-33 - 99432q^-32 - 51354q^-31 + 28507q^-30 + 91347q^-29 + 95418q^-28 + 26139q^-27 - 62305q^-26 - 110909q^-25 - 72997q^-24 + 11750q^-23 + 91149q^-22 + 110170q^-21 + 44956q^-20 - 53331q^-19 - 117304q^-18 - 89116q^-17 - 2616q^-16 + 88906q^-15 + 120653q^-14 + 61269q^-13 - 43052q^-12 - 119790q^-11 - 102975q^-10 - 18714q^-9 + 81477q^-8 + 126576q^-7 + 78166q^-6 - 26119q^-5 - 113494q^-4 - 112917q^-3 - 39334q^-2 + 62685q^-1 + 121569 + 92403q - 565q² - 91890q³ - 110932q⁴ - 59204q⁵ + 31972q⁶ + 98984q⁷ + 94225q⁸ + 25849q⁹ - 56332q¹⁰ - 90348q¹¹ - 66747q¹² - 176q¹³ + 62162q¹⁴ + 77149q¹⁵ + 39828q¹⁶ - 19856q¹⁷ - 56250q¹⁸ - 55795q¹⁹ - 19296q²⁰ + 25842q²¹ + 47742q²² + 35737q²³ + 2797q²⁴ - 23968q²⁵ - 33790q²⁶ - 20763q²⁷ + 3711q²⁸ + 20913q²⁹ + 21464q³⁰ + 8535q³¹ - 5126q³² - 14275q³³ - 12736q³⁴ - 3087q³⁵ + 5840q³⁶ + 8774q³⁷ + 5519q³⁸ + 896q³⁹ - 3931q⁴⁰ - 5191q⁴¹ - 2440q⁴² + 770q⁴³ + 2404q⁴⁴ + 1977q⁴⁵ + 1143q⁴⁶ - 590q⁴⁷ - 1518q⁴⁸ - 866q⁴⁹ - 56q⁵⁰ + 433q⁵¹ + 386q⁵² + 473q⁵³ + 11q⁵⁴ - 376q⁵⁵ - 166q⁵⁶ - 24q⁵⁷ + 56q⁵⁸ + q⁵⁹ + 134q⁶⁰ + 35q⁶¹ - 104q⁶² - 7q⁶³ + 8q⁶⁴ + 14q⁶⁵ - 31q⁶⁶ + 33q⁶⁷ + 14q⁶⁸ - 35q⁶⁹ + 8q⁷⁰ + 4q⁷¹ + 9q⁷² - 13q⁷³ + 6q⁷⁴ + 5q⁷⁵ - 11q⁷⁶ + 4q⁷⁷ + 3q⁷⁹ - 2q⁸⁰ + q⁸² - 2q⁸³ + q⁸⁴

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, 58]]

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

In[3]:=
GaussCode[Knot[10, 58]]

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

In[4]:=
DTCode[Knot[10, 58]]

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

In[5]:=
br = BR[Knot[10, 58]]

Out[5]=
BR[6, {1, -2, 1, 3, -2, -4, -3, -3, 5, -4, 5}]

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

Out[6]=
{6, 11}

In[7]:=
BraidIndex[Knot[10, 58]]

Out[7]=
6

In[8]:=
Show[DrawMorseLink[Knot[10, 58]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 58]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 58]][t]

Out[10]=
3 16 2 27 + -- - -- - 16 t + 3 t 2 t t

In[11]:=
Conway[Knot[10, 58]][z]

Out[11]=
2 4 1 - 4 z + 3 z

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

Out[12]=
{Knot[10, 58]}

In[13]:=
{KnotDet[Knot[10, 58]], KnotSignature[Knot[10, 58]]}

Out[13]=
{65, 0}

In[14]:=
Jones[Knot[10, 58]][q]

Out[14]=
-6 3 6 8 10 11 2 3 4 10 + q - -- + -- - -- + -- - -- - 8 q + 5 q - 2 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, 58]}

In[16]:=
A2Invariant[Knot[10, 58]][q]

Out[16]=
-20 -18 2 -14 2 3 -4 2 4 6 8 10 -2 + q + q - --- + q - --- + -- + q + q - 3 q + q + 2 q - q + 16 10 8 q q q 12 14 > q + q

In[17]:=
HOMFLYPT[Knot[10, 58]][a, z]

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

In[18]:=
Kauffman[Knot[10, 58]][a, z]

Out[18]=
2 -4 2 4 6 4 z 3 5 2 2 z -2 + a - 3 a - 2 a - a - --- - 6 a z - 4 a z - 2 a z + 8 z - ---- + a 4 a 3 3 2 2 4 2 6 2 2 z 8 z 3 3 3 5 3 > 10 a z + 7 a z + 3 a z - ---- + ---- + 21 a z + 18 a z + 7 a z - 3 a a 4 4 5 5 4 z 2 z 2 4 4 4 6 4 2 z 6 z 5 > 5 z + -- - ---- - 4 a z - 5 a z - 3 a z + ---- - ---- - 22 a z - 4 2 3 a a a a 6 7 3 5 5 5 6 3 z 2 6 4 6 6 6 4 z > 23 a z - 9 a z - z + ---- - 10 a z - 5 a z + a z + ---- + 2 a a 7 3 7 5 7 8 2 8 4 8 9 3 9 > 7 a z + 6 a z + 3 a z + 3 z + 6 a z + 3 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 58]], Vassiliev[3][Knot[10, 58]]}

Out[19]=
{-4, 1}

In[20]:=
Kh[Knot[10, 58]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 58], 2][q]

Out[21]=
-18 3 11 13 10 36 20 36 63 12 68 78 6 57 + q - --- + --- - --- - --- + --- - --- - --- + -- - -- - -- + -- + -- - 17 15 14 13 12 11 10 9 8 7 6 5 q q q q q q q q q q q q 91 76 23 91 2 3 4 5 6 7 > -- + -- + -- - -- + 29 q - 66 q + 30 q + 20 q - 32 q + 12 q + 7 q - 4 3 2 q q q q 8 9 10 11 12 > 10 q + 4 q + q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₅₈

10₅₇
10₅₉

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-18 - 3q^-17 + 11q^-15 - 13q^-14 - 10q^-13 + 36q^-12 - 20q^-11 - 36q^-10 + 63q^-9 - 12q^-8 - 68q^-7 + 78q^-6 + 6q^-5 - 91q^-4 + 76q^-3 + 23q^-2 - 91q^-1 + 57 + 29q - 66q² + 30q³ + 20q⁴ - 32q⁵ + 12q⁶ + 7q⁷ - 10q⁸ + 4q⁹ + q¹⁰ - 2q¹¹ + q¹²
3	q^-36 - 3q^-35 + 5q^-33 + 6q^-32 - 13q^-31 - 17q^-30 + 20q^-29 + 38q^-28 - 20q^-27 - 69q^-26 + 5q^-25 + 109q^-24 + 22q^-23 - 136q^-22 - 77q^-21 + 157q^-20 + 138q^-19 - 151q^-18 - 210q^-17 + 130q^-16 + 269q^-15 - 81q^-14 - 328q^-13 + 31q^-12 + 366q^-11 + 31q^-10 - 392q^-9 - 92q^-8 + 403q^-7 + 146q^-6 - 393q^-5 - 198q^-4 + 370q^-3 + 229q^-2 - 320q^-1 - 247 + 259q + 242q² - 190q³ - 215q⁴ + 125q⁵ + 171q⁶ - 72q⁷ - 120q⁸ + 33q⁹ + 80q¹⁰ - 21q¹¹ - 37q¹² + 5q¹³ + 22q¹⁴ - 8q¹⁵ - 5q¹⁶ + 2q¹⁷ + 4q¹⁸ - 5q¹⁹ + 2q²⁰ + q²² - 2q²³ + q²⁴
4	q^-60 - 3q^-59 + 5q^-57 + 6q^-55 - 20q^-54 - 10q^-53 + 20q^-52 + 15q^-51 + 47q^-50 - 62q^-49 - 71q^-48 + 5q^-47 + 41q^-46 + 198q^-45 - 53q^-44 - 175q^-43 - 143q^-42 - 56q^-41 + 446q^-40 + 148q^-39 - 135q^-38 - 383q^-37 - 456q^-36 + 529q^-35 + 482q^-34 + 277q^-33 - 412q^-32 - 1071q^-31 + 188q^-30 + 608q^-29 + 963q^-28 + 26q^-27 - 1520q^-26 - 490q^-25 + 276q^-24 + 1571q^-23 + 812q^-22 - 1561q^-21 - 1172q^-20 - 392q^-19 + 1873q^-18 + 1629q^-17 - 1274q^-16 - 1659q^-15 - 1119q^-14 + 1905q^-13 + 2279q^-12 - 840q^-11 - 1928q^-10 - 1742q^-9 + 1722q^-8 + 2691q^-7 - 311q^-6 - 1939q^-5 - 2192q^-4 + 1280q^-3 + 2743q^-2 + 273q^-1 - 1568 - 2318q + 600q² + 2296q³ + 712q⁴ - 867q⁵ - 1959q⁶ - 29q⁷ + 1456q⁸ + 755q⁹ - 182q¹⁰ - 1234q¹¹ - 300q¹² + 639q¹³ + 464q¹⁴ + 148q¹⁵ - 550q¹⁶ - 229q¹⁷ + 179q¹⁸ + 152q¹⁹ + 155q²⁰ - 173q²¹ - 85q²² + 36q²³ + 10q²⁴ + 73q²⁵ - 45q²⁶ - 15q²⁷ + 11q²⁸ - 15q²⁹ + 25q³⁰ - 11q³¹ - q³² + 4q³³ - 8q³⁴ + 7q³⁵ - 2q³⁶ + q³⁸ - 2q³⁹ + q⁴⁰
5	q^-90 - 3q^-89 + 5q^-87 - q^-84 - 13q^-83 - 10q^-82 + 20q^-81 + 24q^-80 + 15q^-79 - 4q^-78 - 55q^-77 - 71q^-76 - 6q^-75 + 95q^-74 + 132q^-73 + 80q^-72 - 84q^-71 - 253q^-70 - 233q^-69 + 19q^-68 + 340q^-67 + 458q^-66 + 209q^-65 - 322q^-64 - 727q^-63 - 609q^-62 + 86q^-61 + 915q^-60 + 1126q^-59 + 446q^-58 - 786q^-57 - 1672q^-56 - 1310q^-55 + 287q^-54 + 1953q^-53 + 2302q^-52 + 801q^-51 - 1753q^-50 - 3300q^-49 - 2264q^-48 + 893q^-47 + 3853q^-46 + 4017q^-45 + 685q^-44 - 3832q^-43 - 5663q^-42 - 2803q^-41 + 2970q^-40 + 6977q^-39 + 5246q^-38 - 1427q^-37 - 7632q^-36 - 7696q^-35 - 758q^-34 + 7679q^-33 + 9899q^-32 + 3187q^-31 - 7050q^-30 - 11665q^-29 - 5764q^-28 + 6033q^-27 + 12989q^-26 + 8135q^-25 - 4733q^-24 - 13866q^-23 - 10288q^-22 + 3341q^-21 + 14447q^-20 + 12146q^-19 - 1986q^-18 - 14762q^-17 - 13722q^-16 + 628q^-15 + 14863q^-14 + 15120q^-13 + 700q^-12 - 14734q^-11 - 16230q^-10 - 2146q^-9 + 14227q^-8 + 17126q^-7 + 3673q^-6 - 13272q^-5 - 17577q^-4 - 5299q^-3 + 11752q^-2 + 17481q^-1 + 6860 - 9688q - 16662q² - 8168q³ + 7226q⁴ + 15092q⁵ + 8938q⁶ - 4599q⁷ - 12831q⁸ - 9044q⁹ + 2142q¹⁰ + 10157q¹¹ + 8431q¹² - 202q¹³ - 7358q¹⁴ - 7200q¹⁵ - 1109q¹⁶ + 4795q¹⁷ + 5681q¹⁸ + 1707q¹⁹ - 2804q²⁰ - 4006q²¹ - 1766q²² + 1327q²³ + 2637q²⁴ + 1488q²⁵ - 542q²⁶ - 1507q²⁷ - 1048q²⁸ + 63q²⁹ + 826q³⁰ + 668q³¹ + 36q³² - 366q³³ - 365q³⁴ - 96q³⁵ + 181q³⁶ + 184q³⁷ + 34q³⁸ - 45q³⁹ - 82q⁴⁰ - 47q⁴¹ + 41q⁴² + 36q⁴³ - 4q⁴⁴ + 3q⁴⁵ - 9q⁴⁶ - 21q⁴⁷ + 14q⁴⁸ + 10q⁴⁹ - 8q⁵⁰ + 3q⁵¹ + q⁵² - 8q⁵³ + 4q⁵⁴ + 3q⁵⁵ - 2q⁵⁶ + q⁵⁸ - 2q⁵⁹ + q⁶⁰
6	q^-126 - 3q^-125 + 5q^-123 - 7q^-120 + 6q^-119 - 13q^-118 - 10q^-117 + 29q^-116 + 15q^-115 + 15q^-114 - 27q^-113 + 3q^-112 - 66q^-111 - 71q^-110 + 59q^-109 + 86q^-108 + 132q^-107 + 12q^-106 + 62q^-105 - 228q^-104 - 346q^-103 - 106q^-102 + 81q^-101 + 416q^-100 + 350q^-99 + 595q^-98 - 167q^-97 - 809q^-96 - 878q^-95 - 683q^-94 + 179q^-93 + 732q^-92 + 2087q^-91 + 1214q^-90 - 153q^-89 - 1575q^-88 - 2599q^-87 - 2170q^-86 - 1002q^-85 + 3001q^-84 + 4012q^-83 + 3675q^-82 + 1026q^-81 - 2838q^-80 - 5993q^-79 - 7148q^-78 - 1233q^-77 + 3734q^-76 + 8804q^-75 + 9148q^-74 + 4437q^-73 - 4760q^-72 - 14007q^-71 - 12371q^-70 - 6498q^-69 + 6488q^-68 + 16939q^-67 + 19913q^-66 + 9074q^-65 - 10742q^-64 - 22234q^-63 - 25847q^-62 - 10965q^-61 + 12006q^-60 + 33080q^-59 + 32877q^-58 + 9876q^-57 - 17472q^-56 - 41933q^-55 - 39075q^-54 - 11733q^-53 + 30341q^-52 + 52988q^-51 + 41592q^-50 + 6538q^-49 - 41731q^-48 - 63587q^-47 - 46453q^-46 + 8709q^-45 + 57488q^-44 + 70135q^-43 + 40923q^-42 - 23928q^-41 - 73965q^-40 - 78314q^-39 - 22402q^-38 + 46392q^-37 + 86347q^-36 + 72749q^-35 + 2037q^-34 - 71019q^-33 - 99432q^-32 - 51354q^-31 + 28507q^-30 + 91347q^-29 + 95418q^-28 + 26139q^-27 - 62305q^-26 - 110909q^-25 - 72997q^-24 + 11750q^-23 + 91149q^-22 + 110170q^-21 + 44956q^-20 - 53331q^-19 - 117304q^-18 - 89116q^-17 - 2616q^-16 + 88906q^-15 + 120653q^-14 + 61269q^-13 - 43052q^-12 - 119790q^-11 - 102975q^-10 - 18714q^-9 + 81477q^-8 + 126576q^-7 + 78166q^-6 - 26119q^-5 - 113494q^-4 - 112917q^-3 - 39334q^-2 + 62685q^-1 + 121569 + 92403q - 565q² - 91890q³ - 110932q⁴ - 59204q⁵ + 31972q⁶ + 98984q⁷ + 94225q⁸ + 25849q⁹ - 56332q¹⁰ - 90348q¹¹ - 66747q¹² - 176q¹³ + 62162q¹⁴ + 77149q¹⁵ + 39828q¹⁶ - 19856q¹⁷ - 56250q¹⁸ - 55795q¹⁹ - 19296q²⁰ + 25842q²¹ + 47742q²² + 35737q²³ + 2797q²⁴ - 23968q²⁵ - 33790q²⁶ - 20763q²⁷ + 3711q²⁸ + 20913q²⁹ + 21464q³⁰ + 8535q³¹ - 5126q³² - 14275q³³ - 12736q³⁴ - 3087q³⁵ + 5840q³⁶ + 8774q³⁷ + 5519q³⁸ + 896q³⁹ - 3931q⁴⁰ - 5191q⁴¹ - 2440q⁴² + 770q⁴³ + 2404q⁴⁴ + 1977q⁴⁵ + 1143q⁴⁶ - 590q⁴⁷ - 1518q⁴⁸ - 866q⁴⁹ - 56q⁵⁰ + 433q⁵¹ + 386q⁵² + 473q⁵³ + 11q⁵⁴ - 376q⁵⁵ - 166q⁵⁶ - 24q⁵⁷ + 56q⁵⁸ + q⁵⁹ + 134q⁶⁰ + 35q⁶¹ - 104q⁶² - 7q⁶³ + 8q⁶⁴ + 14q⁶⁵ - 31q⁶⁶ + 33q⁶⁷ + 14q⁶⁸ - 35q⁶⁹ + 8q⁷⁰ + 4q⁷¹ + 9q⁷² - 13q⁷³ + 6q⁷⁴ + 5q⁷⁵ - 11q⁷⁶ + 4q⁷⁷ + 3q⁷⁹ - 2q⁸⁰ + q⁸² - 2q⁸³ + q⁸⁴

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 58]]
Out[2]=	PD[X[1, 4, 2, 5], X[7, 10, 8, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], > X[5, 14, 6, 15], X[11, 19, 12, 18], X[15, 20, 16, 1], X[19, 16, 20, 17], > X[17, 13, 18, 12], X[13, 6, 14, 7]]
In[3]:=	GaussCode[Knot[10, 58]]
Out[3]=	GaussCode[-1, 4, -3, 1, -5, 10, -2, 3, -4, 2, -6, 9, -10, 5, -7, 8, -9, 6, -8, > 7]
In[4]:=	DTCode[Knot[10, 58]]
Out[4]=	DTCode[4, 8, 14, 10, 2, 18, 6, 20, 12, 16]
In[5]:=	br = BR[Knot[10, 58]]
Out[5]=	BR[6, {1, -2, 1, 3, -2, -4, -3, -3, 5, -4, 5}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{6, 11}
In[7]:=	BraidIndex[Knot[10, 58]]
Out[7]=	6
In[8]:=	Show[DrawMorseLink[Knot[10, 58]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 58]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 58]][t]
Out[10]=	3 16 2 27 + -- - -- - 16 t + 3 t 2 t t
In[11]:=	Conway[Knot[10, 58]][z]
Out[11]=	2 4 1 - 4 z + 3 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 58]}
In[13]:=	{KnotDet[Knot[10, 58]], KnotSignature[Knot[10, 58]]}
Out[13]=	{65, 0}
In[14]:=	Jones[Knot[10, 58]][q]
Out[14]=	-6 3 6 8 10 11 2 3 4 10 + q - -- + -- - -- + -- - -- - 8 q + 5 q - 2 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, 58]}
In[16]:=	A2Invariant[Knot[10, 58]][q]
Out[16]=	-20 -18 2 -14 2 3 -4 2 4 6 8 10 -2 + q + q - --- + q - --- + -- + q + q - 3 q + q + 2 q - q + 16 10 8 q q q 12 14 > q + q
In[17]:=	HOMFLYPT[Knot[10, 58]][a, z]
Out[17]=	2 -4 2 4 6 2 2 z 2 2 4 2 4 2 4 -2 + a + 3 a - 2 a + a - 2 z - ---- + 3 a z - 3 a z + z + 2 a z 2 a
In[18]:=	Kauffman[Knot[10, 58]][a, z]
Out[18]=	2 -4 2 4 6 4 z 3 5 2 2 z -2 + a - 3 a - 2 a - a - --- - 6 a z - 4 a z - 2 a z + 8 z - ---- + a 4 a 3 3 2 2 4 2 6 2 2 z 8 z 3 3 3 5 3 > 10 a z + 7 a z + 3 a z - ---- + ---- + 21 a z + 18 a z + 7 a z - 3 a a 4 4 5 5 4 z 2 z 2 4 4 4 6 4 2 z 6 z 5 > 5 z + -- - ---- - 4 a z - 5 a z - 3 a z + ---- - ---- - 22 a z - 4 2 3 a a a a 6 7 3 5 5 5 6 3 z 2 6 4 6 6 6 4 z > 23 a z - 9 a z - z + ---- - 10 a z - 5 a z + a z + ---- + 2 a a 7 3 7 5 7 8 2 8 4 8 9 3 9 > 7 a z + 6 a z + 3 a z + 3 z + 6 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 58]], Vassiliev[3][Knot[10, 58]]}
Out[19]=	{-4, 1}
In[20]:=	Kh[Knot[10, 58]][q, t]
Out[20]=	5 1 2 1 4 2 4 4 6 - + 6 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 4 5 6 3 3 2 5 2 5 3 7 3 > ----- + ---- + --- + 4 q t + 4 q t + q t + 4 q t + q t + q t + 3 2 3 q t q t q t 9 4 > q t
In[21]:=	ColouredJones[Knot[10, 58], 2][q]
Out[21]=	-18 3 11 13 10 36 20 36 63 12 68 78 6 57 + q - --- + --- - --- - --- + --- - --- - --- + -- - -- - -- + -- + -- - 17 15 14 13 12 11 10 9 8 7 6 5 q q q q q q q q q q q q 91 76 23 91 2 3 4 5 6 7 > -- + -- + -- - -- + 29 q - 66 q + 30 q + 20 q - 32 q + 12 q + 7 q - 4 3 2 q q q q 8 9 10 11 12 > 10 q + 4 q + q - 2 q + q

The Alternating Knot 1058

The Alternating Knot 10₅₈