The Alternating Knot 10₃₂

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_3,12,4,13 X_5,14,6,15 X_15,20,16,1 X_7,17,8,16 X_19,7,20,6 X_9,19,10,18 X_17,9,18,8 X_13,10,14,11 X_11,2,12,3

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

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

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 2 / NotAvailable 1

Alexander Polynomial: - 2t^-3 + 8t^-2 - 15t^-1 + 19 - 15t + 8t² - 2t³

Conway Polynomial: 1 - z² - 4z⁴ - 2z⁶

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

Determinant and Signature: {69, 0}

Jones Polynomial: q^-6 - 3q^-5 + 5q^-4 - 8q^-3 + 11q^-2 - 11q^-1 + 11 - 9q + 6q² - 3q³ + q⁴

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

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

HOMFLY-PT Polynomial: a^-2 + 2a^-2z² + a^-2z⁴ - 1 - 3z² - 3z⁴ - z⁶ + a² - 2a²z² - 3a²z⁴ - a²z⁶ + 2a⁴z² + a⁴z⁴

Kauffman Polynomial: - a^-4z² + a^-4z⁴ + a^-3z - 3a^-3z³ + 3a^-3z⁵ - a^-2 + 4a^-2z² - 6a^-2z⁴ + 5a^-2z⁶ + a^-1z - 4a^-1z⁵ + 5a^-1z⁷ - 1 + 7z² - 11z⁴ + 3z⁶ + 3z⁸ - az + 7az³ - 15az⁵ + 7az⁷ + az⁹ - a² + 2a²z⁴ - 10a²z⁶ + 6a²z⁸ - 2a³z + 13a³z³ - 18a³z⁵ + 5a³z⁷ + a³z⁹ + 3a⁴z⁴ - 7a⁴z⁶ + 3a⁴z⁸ - a⁵z + 9a⁵z³ - 10a⁵z⁵ + 3a⁵z⁷ + 2a⁶z² - 3a⁶z⁴ + a⁶z⁶

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-1, 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 10₃₂. 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 2

j = 5 4 1

j = 3 5 2

j = 1 6 4

j = -1 6 6

j = -3 5 5

j = -5 3 6

j = -7 2 5

j = -9 1 3

j = -11 2

j = -13 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-18 - 3q^-17 + 10q^-15 - 12q^-14 - 8q^-13 + 33q^-12 - 21q^-11 - 32q^-10 + 65q^-9 - 22q^-8 - 66q^-7 + 93q^-6 - 11q^-5 - 95q^-4 + 101q^-3 + 6q^-2 - 103q^-1 + 86 + 16q - 83q² + 55q³ + 15q⁴ - 47q⁵ + 25q⁶ + 8q⁷ - 17q⁸ + 7q⁹ + 2q¹⁰ - 3q¹¹ + q¹²

3 q^-36 - 3q^-35 + 5q^-33 + 5q^-32 - 12q^-31 - 14q^-30 + 19q^-29 + 32q^-28 - 23q^-27 - 61q^-26 + 18q^-25 + 99q^-24 + 3q^-23 - 144q^-22 - 37q^-21 + 181q^-20 + 94q^-19 - 215q^-18 - 157q^-17 + 230q^-16 + 231q^-15 - 231q^-14 - 308q^-13 + 222q^-12 + 370q^-11 - 187q^-10 - 439q^-9 + 158q^-8 + 476q^-7 - 104q^-6 - 511q^-5 + 67q^-4 + 504q^-3 - 11q^-2 - 489q^-1 - 22 + 437q + 54q² - 374q³ - 68q⁴ + 299q⁵ + 70q⁶ - 224q⁷ - 62q⁸ + 158q⁹ + 49q¹⁰ - 107q¹¹ - 32q¹² + 65q¹³ + 22q¹⁴ - 39q¹⁵ - 12q¹⁶ + 21q¹⁷ + 6q¹⁸ - 11q¹⁹ - q²⁰ + 3q²¹ + 2q²² - 3q²³ + q²⁴

4 q^-60 - 3q^-59 + 5q^-57 + 5q^-55 - 19q^-54 - 7q^-53 + 20q^-52 + 11q^-51 + 38q^-50 - 61q^-49 - 58q^-48 + 22q^-47 + 44q^-46 + 165q^-45 - 88q^-44 - 178q^-43 - 83q^-42 + 23q^-41 + 439q^-40 + 32q^-39 - 274q^-38 - 358q^-37 - 223q^-36 + 742q^-35 + 369q^-34 - 125q^-33 - 666q^-32 - 786q^-31 + 825q^-30 + 775q^-29 + 376q^-28 - 748q^-27 - 1508q^-26 + 561q^-25 + 1000q^-24 + 1091q^-23 - 495q^-22 - 2134q^-21 + 51q^-20 + 945q^-19 + 1791q^-18 - 9q^-17 - 2525q^-16 - 536q^-15 + 687q^-14 + 2346q^-13 + 544q^-12 - 2661q^-11 - 1070q^-10 + 313q^-9 + 2661q^-8 + 1061q^-7 - 2503q^-6 - 1448q^-5 - 138q^-4 + 2626q^-3 + 1438q^-2 - 2029q^-1 - 1519 - 569q + 2168q² + 1523q³ - 1341q⁴ - 1217q⁵ - 800q⁶ + 1438q⁷ + 1248q⁸ - 707q⁹ - 700q¹⁰ - 726q¹¹ + 754q¹² + 776q¹³ - 323q¹⁴ - 261q¹⁵ - 465q¹⁶ + 330q¹⁷ + 367q¹⁸ - 159q¹⁹ - 43q²⁰ - 224q²¹ + 133q²² + 141q²³ - 84q²⁴ + 12q²⁵ - 87q²⁶ + 51q²⁷ + 46q²⁸ - 39q²⁹ + 13q³⁰ - 26q³¹ + 15q³² + 12q³³ - 13q³⁴ + 5q³⁵ - 5q³⁶ + 3q³⁷ + 2q³⁸ - 3q³⁹ + q⁴⁰

5 q^-90 - 3q^-89 + 5q^-87 - 2q^-84 - 12q^-83 - 7q^-82 + 20q^-81 + 21q^-80 + 8q^-79 - 9q^-78 - 50q^-77 - 54q^-76 + 16q^-75 + 93q^-74 + 103q^-73 + 31q^-72 - 117q^-71 - 229q^-70 - 136q^-69 + 128q^-68 + 363q^-67 + 337q^-66 - 24q^-65 - 493q^-64 - 658q^-63 - 231q^-62 + 540q^-61 + 1030q^-60 + 695q^-59 - 375q^-58 - 1384q^-57 - 1357q^-56 - 74q^-55 + 1555q^-54 + 2125q^-53 + 880q^-52 - 1420q^-51 - 2839q^-50 - 1962q^-49 + 807q^-48 + 3324q^-47 + 3251q^-46 + 228q^-45 - 3394q^-44 - 4462q^-43 - 1725q^-42 + 2932q^-41 + 5526q^-40 + 3432q^-39 - 1972q^-38 - 6153q^-37 - 5229q^-36 + 526q^-35 + 6365q^-34 + 6950q^-33 + 1175q^-32 - 6148q^-31 - 8373q^-30 - 3062q^-29 + 5520q^-28 + 9620q^-27 + 4915q^-26 - 4740q^-25 - 10464q^-24 - 6674q^-23 + 3685q^-22 + 11195q^-21 + 8303q^-20 - 2728q^-19 - 11570q^-18 - 9750q^-17 + 1564q^-16 + 11898q^-15 + 11049q^-14 - 529q^-13 - 11847q^-12 - 12149q^-11 - 736q^-10 + 11683q^-9 + 13001q^-8 + 1901q^-7 - 10991q^-6 - 13532q^-5 - 3240q^-4 + 10042q^-3 + 13626q^-2 + 4379q^-1 - 8582 - 13168q - 5436q² + 6893q³ + 12178q⁴ + 6067q⁵ - 5016q⁶ - 10665q⁷ - 6282q⁸ + 3212q⁹ + 8808q¹⁰ + 6010q¹¹ - 1668q¹² - 6822q¹³ - 5306q¹⁴ + 521q¹⁵ + 4905q¹⁶ + 4337q¹⁷ + 208q¹⁸ - 3284q¹⁹ - 3280q²⁰ - 517q²¹ + 2023q²² + 2264q²³ + 587q²⁴ - 1147q²⁵ - 1471q²⁶ - 469q²⁷ + 614q²⁸ + 856q²⁹ + 327q³⁰ - 308q³¹ - 475q³² - 187q³³ + 157q³⁴ + 243q³⁵ + 94q³⁶ - 88q³⁷ - 122q³⁸ - 26q³⁹ + 43q⁴⁰ + 52q⁴¹ + 19q⁴² - 33q⁴³ - 34q⁴⁴ + 12q⁴⁵ + 17q⁴⁶ - 2q⁴⁷ + 9q⁴⁸ - 8q⁴⁹ - 15q⁵⁰ + 11q⁵¹ + 6q⁵² - 7q⁵³ + 3q⁵⁴ + q⁵⁵ - 5q⁵⁶ + 3q⁵⁷ + 2q⁵⁸ - 3q⁵⁹ + q⁶⁰

6 q^-126 - 3q^-125 + 5q^-123 - 7q^-120 + 5q^-119 - 12q^-118 - 7q^-117 + 29q^-116 + 12q^-115 + 9q^-114 - 30q^-113 + 2q^-112 - 57q^-111 - 50q^-110 + 76q^-109 + 80q^-108 + 98q^-107 - 33q^-106 + 15q^-105 - 223q^-104 - 277q^-103 + 27q^-102 + 192q^-101 + 416q^-100 + 219q^-99 + 313q^-98 - 437q^-97 - 904q^-96 - 593q^-95 - 111q^-94 + 771q^-93 + 988q^-92 + 1651q^-91 + 120q^-90 - 1464q^-89 - 2122q^-88 - 1887q^-87 - 183q^-86 + 1380q^-85 + 4282q^-84 + 2883q^-83 + 89q^-82 - 3062q^-81 - 5094q^-80 - 4247q^-79 - 1449q^-78 + 5783q^-77 + 7349q^-76 + 5815q^-75 + 306q^-74 - 6174q^-73 - 10236q^-72 - 9606q^-71 + 1510q^-70 + 8744q^-69 + 13583q^-68 + 9995q^-67 + 312q^-66 - 12162q^-65 - 19881q^-64 - 10285q^-63 + 1102q^-62 + 16265q^-61 + 21777q^-60 + 15615q^-59 - 3790q^-58 - 24292q^-57 - 24473q^-56 - 16177q^-55 + 7649q^-54 + 27250q^-53 + 33809q^-52 + 14804q^-51 - 16977q^-50 - 32612q^-49 - 36634q^-48 - 11463q^-47 + 21237q^-46 + 46584q^-45 + 36906q^-44 + 617q^-43 - 30301q^-42 - 52415q^-41 - 34274q^-40 + 5807q^-39 + 50146q^-38 + 55246q^-37 + 21879q^-36 - 19911q^-35 - 60363q^-34 - 54141q^-33 - 12762q^-32 + 46732q^-31 + 67133q^-30 + 40979q^-29 - 6934q^-28 - 62481q^-27 - 68790q^-26 - 29556q^-25 + 40667q^-24 + 74131q^-23 + 56198q^-22 + 5092q^-21 - 61872q^-20 - 79406q^-19 - 43669q^-18 + 33829q^-17 + 78120q^-16 + 68635q^-15 + 16521q^-14 - 58801q^-13 - 86850q^-12 - 56510q^-11 + 24454q^-10 + 77935q^-9 + 78463q^-8 + 29319q^-7 - 50302q^-6 - 88763q^-5 - 67824q^-4 + 10175q^-3 + 69733q^-2 + 82343q^-1 + 42712 - 34022q - 80674q² - 73368q³ - 7291q⁴ + 51540q⁵ + 75348q⁶ + 51339q⁷ - 12969q⁸ - 61209q⁹ - 67829q¹⁰ - 21191q¹¹ + 27752q¹² + 56779q¹³ + 49423q¹⁴ + 4784q¹⁵ - 36080q¹⁶ - 51118q¹⁷ - 25188q¹⁸ + 7408q¹⁹ + 33255q²⁰ + 37220q²¹ + 12727q²² - 14793q²³ - 30304q²⁴ - 19617q²⁵ - 3221q²⁶ + 14012q²⁷ + 21583q²⁸ + 11446q²⁹ - 2951q³⁰ - 13770q³¹ - 10652q³² - 5075q³³ + 3504q³⁴ + 9562q³⁵ + 6538q³⁶ + 780q³⁷ - 4755q³⁸ - 3921q³⁹ - 3161q⁴⁰ - 43q⁴¹ + 3271q⁴² + 2600q⁴³ + 833q⁴⁴ - 1333q⁴⁵ - 794q⁴⁶ - 1257q⁴⁷ - 479q⁴⁸ + 933q⁴⁹ + 731q⁵⁰ + 292q⁵¹ - 406q⁵² + 75q⁵³ - 344q⁵⁴ - 255q⁵⁵ + 276q⁵⁶ + 144q⁵⁷ + 39q⁵⁸ - 170q⁵⁹ + 137q⁶⁰ - 63q⁶¹ - 97q⁶² + 94q⁶³ + 22q⁶⁴ - 7q⁶⁵ - 74q⁶⁶ + 68q⁶⁷ - 7q⁶⁸ - 34q⁶⁹ + 31q⁷⁰ + q⁷¹ - q⁷² - 26q⁷³ + 22q⁷⁴ + 2q⁷⁵ - 13q⁷⁶ + 9q⁷⁷ - q⁷⁸ + q⁷⁹ - 5q⁸⁰ + 3q⁸¹ + 2q⁸² - 3q⁸³ + 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, 32]]

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

In[3]:=
GaussCode[Knot[10, 32]]

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

In[4]:=
DTCode[Knot[10, 32]]

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

In[5]:=
br = BR[Knot[10, 32]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 32]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 32]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 32]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 32]][t]

Out[10]=
2 8 15 2 3 19 - -- + -- - -- - 15 t + 8 t - 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 32]][z]

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

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

Out[12]=
{Knot[10, 32]}

In[13]:=
{KnotDet[Knot[10, 32]], KnotSignature[Knot[10, 32]]}

Out[13]=
{69, 0}

In[14]:=
Jones[Knot[10, 32]][q]

Out[14]=
-6 3 5 8 11 11 2 3 4 11 + q - -- + -- - -- + -- - -- - 9 q + 6 q - 3 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, 32]}

In[16]:=
A2Invariant[Knot[10, 32]][q]

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

In[17]:=
HOMFLYPT[Knot[10, 32]][a, z]

Out[17]=
2 4 -2 2 2 2 z 2 2 4 2 4 z 2 4 4 4 -1 + a + a - 3 z + ---- - 2 a z + 2 a z - 3 z + -- - 3 a z + a z - 2 2 a a 6 2 6 > z - a z

In[18]:=
Kauffman[Knot[10, 32]][a, z]

Out[18]=
2 2 -2 2 z z 3 5 2 z 4 z 6 2 -1 - a - a + -- + - - a z - 2 a z - a z + 7 z - -- + ---- + 2 a z - 3 a 4 2 a a a 3 4 4 3 z 3 3 3 5 3 4 z 6 z 2 4 > ---- + 7 a z + 13 a z + 9 a z - 11 z + -- - ---- + 2 a z + 3 4 2 a a a 5 5 4 4 6 4 3 z 4 z 5 3 5 5 5 6 > 3 a z - 3 a z + ---- - ---- - 15 a z - 18 a z - 10 a z + 3 z + 3 a a 6 7 5 z 2 6 4 6 6 6 5 z 7 3 7 5 7 > ---- - 10 a z - 7 a z + a z + ---- + 7 a z + 5 a z + 3 a z + 2 a a 8 2 8 4 8 9 3 9 > 3 z + 6 a z + 3 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 32]], Vassiliev[3][Knot[10, 32]]}

Out[19]=
{-1, 0}

In[20]:=
Kh[Knot[10, 32]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 32], 2][q]

Out[21]=
-18 3 10 12 8 33 21 32 65 22 66 93 11 86 + 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 95 101 6 103 2 3 4 5 6 7 > -- + --- + -- - --- + 16 q - 83 q + 55 q + 15 q - 47 q + 25 q + 8 q - 4 3 2 q q q q 8 9 10 11 12 > 17 q + 7 q + 2 q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₃₂

10₃₁
10₃₃

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-18 - 3q^-17 + 10q^-15 - 12q^-14 - 8q^-13 + 33q^-12 - 21q^-11 - 32q^-10 + 65q^-9 - 22q^-8 - 66q^-7 + 93q^-6 - 11q^-5 - 95q^-4 + 101q^-3 + 6q^-2 - 103q^-1 + 86 + 16q - 83q² + 55q³ + 15q⁴ - 47q⁵ + 25q⁶ + 8q⁷ - 17q⁸ + 7q⁹ + 2q¹⁰ - 3q¹¹ + q¹²
3	q^-36 - 3q^-35 + 5q^-33 + 5q^-32 - 12q^-31 - 14q^-30 + 19q^-29 + 32q^-28 - 23q^-27 - 61q^-26 + 18q^-25 + 99q^-24 + 3q^-23 - 144q^-22 - 37q^-21 + 181q^-20 + 94q^-19 - 215q^-18 - 157q^-17 + 230q^-16 + 231q^-15 - 231q^-14 - 308q^-13 + 222q^-12 + 370q^-11 - 187q^-10 - 439q^-9 + 158q^-8 + 476q^-7 - 104q^-6 - 511q^-5 + 67q^-4 + 504q^-3 - 11q^-2 - 489q^-1 - 22 + 437q + 54q² - 374q³ - 68q⁴ + 299q⁵ + 70q⁶ - 224q⁷ - 62q⁸ + 158q⁹ + 49q¹⁰ - 107q¹¹ - 32q¹² + 65q¹³ + 22q¹⁴ - 39q¹⁵ - 12q¹⁶ + 21q¹⁷ + 6q¹⁸ - 11q¹⁹ - q²⁰ + 3q²¹ + 2q²² - 3q²³ + q²⁴
4	q^-60 - 3q^-59 + 5q^-57 + 5q^-55 - 19q^-54 - 7q^-53 + 20q^-52 + 11q^-51 + 38q^-50 - 61q^-49 - 58q^-48 + 22q^-47 + 44q^-46 + 165q^-45 - 88q^-44 - 178q^-43 - 83q^-42 + 23q^-41 + 439q^-40 + 32q^-39 - 274q^-38 - 358q^-37 - 223q^-36 + 742q^-35 + 369q^-34 - 125q^-33 - 666q^-32 - 786q^-31 + 825q^-30 + 775q^-29 + 376q^-28 - 748q^-27 - 1508q^-26 + 561q^-25 + 1000q^-24 + 1091q^-23 - 495q^-22 - 2134q^-21 + 51q^-20 + 945q^-19 + 1791q^-18 - 9q^-17 - 2525q^-16 - 536q^-15 + 687q^-14 + 2346q^-13 + 544q^-12 - 2661q^-11 - 1070q^-10 + 313q^-9 + 2661q^-8 + 1061q^-7 - 2503q^-6 - 1448q^-5 - 138q^-4 + 2626q^-3 + 1438q^-2 - 2029q^-1 - 1519 - 569q + 2168q² + 1523q³ - 1341q⁴ - 1217q⁵ - 800q⁶ + 1438q⁷ + 1248q⁸ - 707q⁹ - 700q¹⁰ - 726q¹¹ + 754q¹² + 776q¹³ - 323q¹⁴ - 261q¹⁵ - 465q¹⁶ + 330q¹⁷ + 367q¹⁸ - 159q¹⁹ - 43q²⁰ - 224q²¹ + 133q²² + 141q²³ - 84q²⁴ + 12q²⁵ - 87q²⁶ + 51q²⁷ + 46q²⁸ - 39q²⁹ + 13q³⁰ - 26q³¹ + 15q³² + 12q³³ - 13q³⁴ + 5q³⁵ - 5q³⁶ + 3q³⁷ + 2q³⁸ - 3q³⁹ + q⁴⁰
5	q^-90 - 3q^-89 + 5q^-87 - 2q^-84 - 12q^-83 - 7q^-82 + 20q^-81 + 21q^-80 + 8q^-79 - 9q^-78 - 50q^-77 - 54q^-76 + 16q^-75 + 93q^-74 + 103q^-73 + 31q^-72 - 117q^-71 - 229q^-70 - 136q^-69 + 128q^-68 + 363q^-67 + 337q^-66 - 24q^-65 - 493q^-64 - 658q^-63 - 231q^-62 + 540q^-61 + 1030q^-60 + 695q^-59 - 375q^-58 - 1384q^-57 - 1357q^-56 - 74q^-55 + 1555q^-54 + 2125q^-53 + 880q^-52 - 1420q^-51 - 2839q^-50 - 1962q^-49 + 807q^-48 + 3324q^-47 + 3251q^-46 + 228q^-45 - 3394q^-44 - 4462q^-43 - 1725q^-42 + 2932q^-41 + 5526q^-40 + 3432q^-39 - 1972q^-38 - 6153q^-37 - 5229q^-36 + 526q^-35 + 6365q^-34 + 6950q^-33 + 1175q^-32 - 6148q^-31 - 8373q^-30 - 3062q^-29 + 5520q^-28 + 9620q^-27 + 4915q^-26 - 4740q^-25 - 10464q^-24 - 6674q^-23 + 3685q^-22 + 11195q^-21 + 8303q^-20 - 2728q^-19 - 11570q^-18 - 9750q^-17 + 1564q^-16 + 11898q^-15 + 11049q^-14 - 529q^-13 - 11847q^-12 - 12149q^-11 - 736q^-10 + 11683q^-9 + 13001q^-8 + 1901q^-7 - 10991q^-6 - 13532q^-5 - 3240q^-4 + 10042q^-3 + 13626q^-2 + 4379q^-1 - 8582 - 13168q - 5436q² + 6893q³ + 12178q⁴ + 6067q⁵ - 5016q⁶ - 10665q⁷ - 6282q⁸ + 3212q⁹ + 8808q¹⁰ + 6010q¹¹ - 1668q¹² - 6822q¹³ - 5306q¹⁴ + 521q¹⁵ + 4905q¹⁶ + 4337q¹⁷ + 208q¹⁸ - 3284q¹⁹ - 3280q²⁰ - 517q²¹ + 2023q²² + 2264q²³ + 587q²⁴ - 1147q²⁵ - 1471q²⁶ - 469q²⁷ + 614q²⁸ + 856q²⁹ + 327q³⁰ - 308q³¹ - 475q³² - 187q³³ + 157q³⁴ + 243q³⁵ + 94q³⁶ - 88q³⁷ - 122q³⁸ - 26q³⁹ + 43q⁴⁰ + 52q⁴¹ + 19q⁴² - 33q⁴³ - 34q⁴⁴ + 12q⁴⁵ + 17q⁴⁶ - 2q⁴⁷ + 9q⁴⁸ - 8q⁴⁹ - 15q⁵⁰ + 11q⁵¹ + 6q⁵² - 7q⁵³ + 3q⁵⁴ + q⁵⁵ - 5q⁵⁶ + 3q⁵⁷ + 2q⁵⁸ - 3q⁵⁹ + q⁶⁰
6	q^-126 - 3q^-125 + 5q^-123 - 7q^-120 + 5q^-119 - 12q^-118 - 7q^-117 + 29q^-116 + 12q^-115 + 9q^-114 - 30q^-113 + 2q^-112 - 57q^-111 - 50q^-110 + 76q^-109 + 80q^-108 + 98q^-107 - 33q^-106 + 15q^-105 - 223q^-104 - 277q^-103 + 27q^-102 + 192q^-101 + 416q^-100 + 219q^-99 + 313q^-98 - 437q^-97 - 904q^-96 - 593q^-95 - 111q^-94 + 771q^-93 + 988q^-92 + 1651q^-91 + 120q^-90 - 1464q^-89 - 2122q^-88 - 1887q^-87 - 183q^-86 + 1380q^-85 + 4282q^-84 + 2883q^-83 + 89q^-82 - 3062q^-81 - 5094q^-80 - 4247q^-79 - 1449q^-78 + 5783q^-77 + 7349q^-76 + 5815q^-75 + 306q^-74 - 6174q^-73 - 10236q^-72 - 9606q^-71 + 1510q^-70 + 8744q^-69 + 13583q^-68 + 9995q^-67 + 312q^-66 - 12162q^-65 - 19881q^-64 - 10285q^-63 + 1102q^-62 + 16265q^-61 + 21777q^-60 + 15615q^-59 - 3790q^-58 - 24292q^-57 - 24473q^-56 - 16177q^-55 + 7649q^-54 + 27250q^-53 + 33809q^-52 + 14804q^-51 - 16977q^-50 - 32612q^-49 - 36634q^-48 - 11463q^-47 + 21237q^-46 + 46584q^-45 + 36906q^-44 + 617q^-43 - 30301q^-42 - 52415q^-41 - 34274q^-40 + 5807q^-39 + 50146q^-38 + 55246q^-37 + 21879q^-36 - 19911q^-35 - 60363q^-34 - 54141q^-33 - 12762q^-32 + 46732q^-31 + 67133q^-30 + 40979q^-29 - 6934q^-28 - 62481q^-27 - 68790q^-26 - 29556q^-25 + 40667q^-24 + 74131q^-23 + 56198q^-22 + 5092q^-21 - 61872q^-20 - 79406q^-19 - 43669q^-18 + 33829q^-17 + 78120q^-16 + 68635q^-15 + 16521q^-14 - 58801q^-13 - 86850q^-12 - 56510q^-11 + 24454q^-10 + 77935q^-9 + 78463q^-8 + 29319q^-7 - 50302q^-6 - 88763q^-5 - 67824q^-4 + 10175q^-3 + 69733q^-2 + 82343q^-1 + 42712 - 34022q - 80674q² - 73368q³ - 7291q⁴ + 51540q⁵ + 75348q⁶ + 51339q⁷ - 12969q⁸ - 61209q⁹ - 67829q¹⁰ - 21191q¹¹ + 27752q¹² + 56779q¹³ + 49423q¹⁴ + 4784q¹⁵ - 36080q¹⁶ - 51118q¹⁷ - 25188q¹⁸ + 7408q¹⁹ + 33255q²⁰ + 37220q²¹ + 12727q²² - 14793q²³ - 30304q²⁴ - 19617q²⁵ - 3221q²⁶ + 14012q²⁷ + 21583q²⁸ + 11446q²⁹ - 2951q³⁰ - 13770q³¹ - 10652q³² - 5075q³³ + 3504q³⁴ + 9562q³⁵ + 6538q³⁶ + 780q³⁷ - 4755q³⁸ - 3921q³⁹ - 3161q⁴⁰ - 43q⁴¹ + 3271q⁴² + 2600q⁴³ + 833q⁴⁴ - 1333q⁴⁵ - 794q⁴⁶ - 1257q⁴⁷ - 479q⁴⁸ + 933q⁴⁹ + 731q⁵⁰ + 292q⁵¹ - 406q⁵² + 75q⁵³ - 344q⁵⁴ - 255q⁵⁵ + 276q⁵⁶ + 144q⁵⁷ + 39q⁵⁸ - 170q⁵⁹ + 137q⁶⁰ - 63q⁶¹ - 97q⁶² + 94q⁶³ + 22q⁶⁴ - 7q⁶⁵ - 74q⁶⁶ + 68q⁶⁷ - 7q⁶⁸ - 34q⁶⁹ + 31q⁷⁰ + q⁷¹ - q⁷² - 26q⁷³ + 22q⁷⁴ + 2q⁷⁵ - 13q⁷⁶ + 9q⁷⁷ - q⁷⁸ + q⁷⁹ - 5q⁸⁰ + 3q⁸¹ + 2q⁸² - 3q⁸³ + q⁸⁴

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 32]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[5, 14, 6, 15], X[15, 20, 16, 1], > X[7, 17, 8, 16], X[19, 7, 20, 6], X[9, 19, 10, 18], X[17, 9, 18, 8], > X[13, 10, 14, 11], X[11, 2, 12, 3]]
In[3]:=	GaussCode[Knot[10, 32]]
Out[3]=	GaussCode[-1, 10, -2, 1, -3, 6, -5, 8, -7, 9, -10, 2, -9, 3, -4, 5, -8, 7, -6, > 4]
In[4]:=	DTCode[Knot[10, 32]]
Out[4]=	DTCode[4, 12, 14, 16, 18, 2, 10, 20, 8, 6]
In[5]:=	br = BR[Knot[10, 32]]
Out[5]=	BR[4, {1, 1, 1, -2, 1, -2, -2, -3, 2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 32]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 32]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 32]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 32]][t]
Out[10]=	2 8 15 2 3 19 - -- + -- - -- - 15 t + 8 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 32]][z]
Out[11]=	2 4 6 1 - z - 4 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 32]}
In[13]:=	{KnotDet[Knot[10, 32]], KnotSignature[Knot[10, 32]]}
Out[13]=	{69, 0}
In[14]:=	Jones[Knot[10, 32]][q]
Out[14]=	-6 3 5 8 11 11 2 3 4 11 + q - -- + -- - -- + -- - -- - 9 q + 6 q - 3 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, 32]}
In[16]:=	A2Invariant[Knot[10, 32]][q]
Out[16]=	-18 -16 2 3 -4 -2 2 4 6 8 10 12 -2 + q - q - --- + -- + q + q + 2 q - 2 q + q + q - q + q 10 8 q q
In[17]:=	HOMFLYPT[Knot[10, 32]][a, z]
Out[17]=	2 4 -2 2 2 2 z 2 2 4 2 4 z 2 4 4 4 -1 + a + a - 3 z + ---- - 2 a z + 2 a z - 3 z + -- - 3 a z + a z - 2 2 a a 6 2 6 > z - a z
In[18]:=	Kauffman[Knot[10, 32]][a, z]
Out[18]=	2 2 -2 2 z z 3 5 2 z 4 z 6 2 -1 - a - a + -- + - - a z - 2 a z - a z + 7 z - -- + ---- + 2 a z - 3 a 4 2 a a a 3 4 4 3 z 3 3 3 5 3 4 z 6 z 2 4 > ---- + 7 a z + 13 a z + 9 a z - 11 z + -- - ---- + 2 a z + 3 4 2 a a a 5 5 4 4 6 4 3 z 4 z 5 3 5 5 5 6 > 3 a z - 3 a z + ---- - ---- - 15 a z - 18 a z - 10 a z + 3 z + 3 a a 6 7 5 z 2 6 4 6 6 6 5 z 7 3 7 5 7 > ---- - 10 a z - 7 a z + a z + ---- + 7 a z + 5 a z + 3 a z + 2 a a 8 2 8 4 8 9 3 9 > 3 z + 6 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 32]], Vassiliev[3][Knot[10, 32]]}
Out[19]=	{-1, 0}
In[20]:=	Kh[Knot[10, 32]][q, t]
Out[20]=	6 1 2 1 3 2 5 3 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 5 5 6 3 3 2 5 2 5 3 7 3 > ----- + ---- + --- + 4 q t + 5 q t + 2 q t + 4 q t + q t + 2 q t + 3 2 3 q t q t q t 9 4 > q t
In[21]:=	ColouredJones[Knot[10, 32], 2][q]
Out[21]=	-18 3 10 12 8 33 21 32 65 22 66 93 11 86 + 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 95 101 6 103 2 3 4 5 6 7 > -- + --- + -- - --- + 16 q - 83 q + 55 q + 15 q - 47 q + 25 q + 8 q - 4 3 2 q q q q 8 9 10 11 12 > 17 q + 7 q + 2 q - 3 q + q

The Alternating Knot 1032

The Alternating Knot 10₃₂