The Alternating Knot 10₅₂

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X₈₄₉₃ X_14,6,15,5 X_20,15,1,16 X_16,9,17,10 X_10,19,11,20 X_18,11,19,12 X_12,17,13,18 X₂₈₃₇ X_4,14,5,13

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

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

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 2 3 3 / NotAvailable 1

Alexander Polynomial: 2t^-3 - 7t^-2 + 13t^-1 - 15 + 13t - 7t² + 2t³

Conway Polynomial: 1 + 3z² + 5z⁴ + 2z⁶

Other knots with the same Alexander/Conway Polynomial: {10₂₃, ...}

Determinant and Signature: {59, 2}

Jones Polynomial: - q^-4 + 2q^-3 - 4q^-2 + 7q^-1 - 8 + 10q - 9q² + 8q³ - 6q⁴ + 3q⁵ - q⁶

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

A2 (sl(3)) Invariant: - q^-12 - q^-8 - q^-6 + 2q^-4 + 3 + 2q² + 2q⁶ - 2q⁸ + q¹⁰ - q¹² - q¹⁴ + q¹⁶ - q¹⁸

HOMFLY-PT Polynomial: - a^-4 - 2a^-4z² - a^-4z⁴ + 2a^-2z² + 3a^-2z⁴ + a^-2z⁶ + 4 + 6z² + 4z⁴ + z⁶ - 2a² - 3a²z² - a²z⁴

Kauffman Polynomial: a^-7z³ + 3a^-6z⁴ + 2a^-5z - 5a^-5z³ + 6a^-5z⁵ - a^-4 + 6a^-4z² - 12a^-4z⁴ + 8a^-4z⁶ + 2a^-3z³ - 11a^-3z⁵ + 7a^-3z⁷ + 4a^-2z² - 9a^-2z⁴ - 3a^-2z⁶ + 4a^-2z⁸ - 7a^-1z + 24a^-1z³ - 28a^-1z⁵ + 7a^-1z⁷ + a^-1z⁹ + 4 - 9z² + 19z⁴ - 20z⁶ + 6z⁸ - 9az + 24az³ - 16az⁵ + az⁷ + az⁹ + 2a² - 7a²z² + 13a²z⁴ - 9a²z⁶ + 2a²z⁸ - 4a³z + 8a³z³ - 5a³z⁵ + a³z⁷

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {3, 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=2 is the signature of 10₅₂. 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 r = 4 r = 5

j = 13 1

j = 11 2

j = 9 4 1

j = 7 4 2

j = 5 5 4

j = 3 5 4

j = 1 4 6

j = -1 3 4

j = -3 1 4

j = -5 1 3

j = -7 1

j = -9 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-13 - 2q^-12 - q^-11 + 7q^-10 - 6q^-9 - 9q^-8 + 21q^-7 - 6q^-6 - 28q^-5 + 37q^-4 + 3q^-3 - 52q^-2 + 46q^-1 + 20 - 70q + 43q² + 35q³ - 73q⁴ + 30q⁵ + 41q⁶ - 60q⁷ + 17q⁸ + 31q⁹ - 37q¹⁰ + 9q¹¹ + 14q¹² - 15q¹³ + 4q¹⁴ + 3q¹⁵ - 3q¹⁶ + q¹⁷

3 - q^-27 + 2q^-26 + q^-25 - 2q^-24 - 6q^-23 + 5q^-22 + 11q^-21 - 2q^-20 - 24q^-19 - 2q^-18 + 34q^-17 + 20q^-16 - 48q^-15 - 41q^-14 + 49q^-13 + 73q^-12 - 43q^-11 - 105q^-10 + 26q^-9 + 130q^-8 + 6q^-7 - 155q^-6 - 35q^-5 + 160q^-4 + 80q^-3 - 169q^-2 - 108q^-1 + 150 + 155q - 145q² - 173q³ + 107q⁴ + 211q⁵ - 88q⁶ - 216q⁷ + 47q⁸ + 228q⁹ - 24q¹⁰ - 210q¹¹ - 6q¹² + 190q¹³ + 18q¹⁴ - 156q¹⁵ - 22q¹⁶ + 117q¹⁷ + 20q¹⁸ - 85q¹⁹ - 9q²⁰ + 53q²¹ + 5q²² - 35q²³ + 2q²⁴ + 19q²⁵ - 4q²⁶ - 9q²⁷ + 4q²⁸ + 3q²⁹ - q³⁰ - 3q³¹ + 3q³² - q³³

4 q^-46 - 2q^-45 - q^-44 + 2q^-43 + q^-42 + 7q^-41 - 9q^-40 - 9q^-39 + 2q^-38 + 3q^-37 + 35q^-36 - 11q^-35 - 32q^-34 - 23q^-33 - 17q^-32 + 97q^-31 + 31q^-30 - 30q^-29 - 83q^-28 - 125q^-27 + 134q^-26 + 123q^-25 + 84q^-24 - 88q^-23 - 321q^-22 + 39q^-21 + 141q^-20 + 300q^-19 + 88q^-18 - 454q^-17 - 163q^-16 - 49q^-15 + 442q^-14 + 404q^-13 - 378q^-12 - 294q^-11 - 396q^-10 + 375q^-9 + 672q^-8 - 134q^-7 - 233q^-6 - 728q^-5 + 137q^-4 + 778q^-3 + 149q^-2 - 23q^-1 - 955 - 163q + 749q² + 407q³ + 249q⁴ - 1085q⁵ - 478q⁶ + 630q⁷ + 643q⁸ + 543q⁹ - 1114q¹⁰ - 776q¹¹ + 416q¹² + 792q¹³ + 825q¹⁴ - 977q¹⁵ - 958q¹⁶ + 122q¹⁷ + 743q¹⁸ + 978q¹⁹ - 673q²⁰ - 888q²¹ - 120q²² + 481q²³ + 880q²⁴ - 343q²⁵ - 595q²⁶ - 178q²⁷ + 185q²⁸ + 581q²⁹ - 149q³⁰ - 277q³¹ - 92q³² + 19q³³ + 286q³⁴ - 82q³⁵ - 91q³⁶ - 8q³⁷ - 21q³⁸ + 111q³⁹ - 51q⁴⁰ - 20q⁴¹ + 15q⁴² - 19q⁴³ + 36q⁴⁴ - 22q⁴⁵ - 2q⁴⁶ + 10q⁴⁷ - 9q⁴⁸ + 8q⁴⁹ - 6q⁵⁰ + q⁵¹ + 3q⁵² - 3q⁵³ + q⁵⁴

5 - q^-70 + 2q^-69 + q^-68 - 2q^-67 - q^-66 - 2q^-65 - 3q^-64 + 7q^-63 + 11q^-62 - 2q^-61 - 8q^-60 - 13q^-59 - 19q^-58 + 8q^-57 + 40q^-56 + 32q^-55 + 2q^-54 - 39q^-53 - 81q^-52 - 50q^-51 + 50q^-50 + 120q^-49 + 120q^-48 + 16q^-47 - 155q^-46 - 234q^-45 - 121q^-44 + 110q^-43 + 328q^-42 + 326q^-41 + 21q^-40 - 362q^-39 - 520q^-38 - 302q^-37 + 239q^-36 + 702q^-35 + 644q^-34 + 44q^-33 - 681q^-32 - 995q^-31 - 529q^-30 + 471q^-29 + 1218q^-28 + 1052q^-27 + 13q^-26 - 1177q^-25 - 1559q^-24 - 681q^-23 + 874q^-22 + 1846q^-21 + 1386q^-20 - 246q^-19 - 1844q^-18 - 2055q^-17 - 529q^-16 + 1536q^-15 + 2448q^-14 + 1415q^-13 - 918q^-12 - 2648q^-11 - 2186q^-10 + 147q^-9 + 2475q^-8 + 2871q^-7 + 740q^-6 - 2185q^-5 - 3292q^-4 - 1573q^-3 + 1616q^-2 + 3633q^-1 + 2397 - 1143q - 3745q² - 3070q³ + 473q⁴ + 3906q⁵ + 3764q⁶ - 17q⁷ - 3928q⁸ - 4339q⁹ - 634q¹⁰ + 4030q¹¹ + 5005q¹² + 1130q¹³ - 3996q¹⁴ - 5565q¹⁵ - 1853q¹⁶ + 3925q¹⁷ + 6137q¹⁸ + 2501q¹⁹ - 3611q²⁰ - 6482q²¹ - 3295q²² + 3128q²³ + 6625q²⁴ + 3910q²⁵ - 2373q²⁶ - 6383q²⁷ - 4432q²⁸ + 1530q²⁹ + 5812q³⁰ + 4597q³¹ - 655q³² - 4915q³³ - 4443q³⁴ - 94q³⁵ + 3871q³⁶ + 3964q³⁷ + 583q³⁸ - 2808q³⁹ - 3231q⁴⁰ - 832q⁴¹ + 1860q⁴² + 2461q⁴³ + 821q⁴⁴ - 1161q⁴⁵ - 1678q⁴⁶ - 674q⁴⁷ + 647q⁴⁸ + 1080q⁴⁹ + 470q⁵⁰ - 366q⁵¹ - 619q⁵² - 276q⁵³ + 192q⁵⁴ + 334q⁵⁵ + 133q⁵⁶ - 103q⁵⁷ - 162q⁵⁸ - 59q⁵⁹ + 66q⁶⁰ + 75q⁶¹ + 7q⁶² - 34q⁶³ - 22q⁶⁴ - 5q⁶⁵ + 14q⁶⁶ + 21q⁶⁷ - 11q⁶⁸ - 12q⁶⁹ + 8q⁷⁰ - q⁷¹ - 4q⁷² + 8q⁷³ - 3q⁷⁴ - 5q⁷⁵ + 6q⁷⁶ - q⁷⁷ - 3q⁷⁸ + 3q⁷⁹ - q⁸⁰

6 q^-99 - 2q^-98 - q^-97 + 2q^-96 + q^-95 + 2q^-94 - 2q^-93 + 5q^-92 - 9q^-91 - 11q^-90 + 5q^-89 + 7q^-88 + 15q^-87 + 23q^-85 - 23q^-84 - 48q^-83 - 20q^-82 - q^-81 + 45q^-80 + 30q^-79 + 115q^-78 + 6q^-77 - 99q^-76 - 126q^-75 - 124q^-74 - 16q^-73 + 25q^-72 + 340q^-71 + 249q^-70 + 64q^-69 - 170q^-68 - 382q^-67 - 411q^-66 - 403q^-65 + 362q^-64 + 616q^-63 + 743q^-62 + 428q^-61 - 141q^-60 - 825q^-59 - 1500q^-58 - 615q^-57 + 85q^-56 + 1224q^-55 + 1705q^-54 + 1545q^-53 + 205q^-52 - 1967q^-51 - 2234q^-50 - 2254q^-49 - 388q^-48 + 1653q^-47 + 3671q^-46 + 3331q^-45 + 410q^-44 - 1751q^-43 - 4522q^-42 - 4233q^-41 - 2027q^-40 + 2795q^-39 + 5710q^-38 + 4996q^-37 + 2923q^-36 - 2737q^-35 - 6493q^-34 - 7734q^-33 - 2746q^-32 + 3105q^-31 + 7129q^-30 + 9167q^-29 + 4098q^-28 - 2946q^-27 - 10155q^-26 - 9434q^-25 - 4696q^-24 + 2872q^-23 + 11350q^-22 + 11713q^-21 + 5821q^-20 - 5804q^-19 - 11647q^-18 - 12944q^-17 - 6482q^-16 + 6619q^-15 + 14724q^-14 + 14845q^-13 + 3498q^-12 - 7209q^-11 - 16720q^-10 - 16031q^-9 - 2785q^-8 + 11592q^-7 + 19786q^-6 + 13095q^-5 + 1389q^-4 - 15065q^-3 - 22066q^-2 - 12517q^-1 + 4894 + 20150q + 19883q² + 10191q³ - 10547q⁴ - 24460q⁵ - 19974q⁶ - 1873q⁷ + 18392q⁸ + 24019q⁹ + 17092q¹⁰ - 6194q¹¹ - 25488q¹² - 25518q¹³ - 7174q¹⁴ + 16999q¹⁵ + 27575q¹⁶ + 22750q¹⁷ - 2896q¹⁸ - 27007q¹⁹ - 31018q²⁰ - 12149q²¹ + 16027q²² + 31710q²³ + 29006q²⁴ + 1142q²⁵ - 28088q²⁶ - 36938q²⁷ - 18752q²⁸ + 12861q²⁹ + 34436q³⁰ + 35758q³¹ + 8043q³² - 25325q³³ - 40357q³⁴ - 26325q³⁵ + 5350q³⁶ + 31754q³⁷ + 39319q³⁸ + 16488q³⁹ - 16824q⁴⁰ - 37060q⁴¹ - 30514q⁴² - 4323q⁴³ + 22409q⁴⁴ + 35577q⁴⁵ + 21506q⁴⁶ - 5805q⁴⁷ - 26739q⁴⁸ - 27467q⁴⁹ - 10654q⁵⁰ + 10645q⁵¹ + 25210q⁵² + 19644q⁵³ + 1913q⁵⁴ - 14467q⁵⁵ - 18675q⁵⁶ - 10692q⁵⁷ + 2283q⁵⁸ + 13693q⁵⁹ + 12961q⁶⁰ + 3925q⁶¹ - 5700q⁶² - 9469q⁶³ - 6779q⁶⁴ - 877q⁶⁵ + 5743q⁶⁶ + 6269q⁶⁷ + 2553q⁶⁸ - 1659q⁶⁹ - 3549q⁷⁰ - 2896q⁷¹ - 996q⁷² + 1962q⁷³ + 2241q⁷⁴ + 901q⁷⁵ - 422q⁷⁶ - 963q⁷⁷ - 798q⁷⁸ - 468q⁷⁹ + 612q⁸⁰ + 596q⁸¹ + 115q⁸² - 129q⁸³ - 170q⁸⁴ - 100q⁸⁵ - 164q⁸⁶ + 192q⁸⁷ + 123q⁸⁸ - 55q⁸⁹ - 42q⁹⁰ - 6q⁹¹ + 23q⁹² - 62q⁹³ + 59q⁹⁴ + 22q⁹⁵ - 37q⁹⁶ - 5q⁹⁷ + 5q⁹⁸ + 18q⁹⁹ - 26q¹⁰⁰ + 15q¹⁰¹ + 7q¹⁰² - 14q¹⁰³ + 4q¹⁰⁴ + 5q¹⁰⁶ - 6q¹⁰⁷ + q¹⁰⁸ + 3q¹⁰⁹ - 3q¹¹⁰ + q¹¹¹

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, 52]]

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

In[3]:=
GaussCode[Knot[10, 52]]

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

In[4]:=
DTCode[Knot[10, 52]]

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

In[5]:=
br = BR[Knot[10, 52]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 52]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 52]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 52]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 52]][t]

Out[10]=
2 7 13 2 3 -15 + -- - -- + -- + 13 t - 7 t + 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 52]][z]

Out[11]=
2 4 6 1 + 3 z + 5 z + 2 z

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

Out[12]=
{Knot[10, 23], Knot[10, 52]}

In[13]:=
{KnotDet[Knot[10, 52]], KnotSignature[Knot[10, 52]]}

Out[13]=
{59, 2}

In[14]:=
Jones[Knot[10, 52]][q]

Out[14]=
-4 2 4 7 2 3 4 5 6 -8 - q + -- - -- + - + 10 q - 9 q + 8 q - 6 q + 3 q - q 3 2 q q q

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

Out[15]=
{Knot[10, 52]}

In[16]:=
A2Invariant[Knot[10, 52]][q]

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

In[17]:=
HOMFLYPT[Knot[10, 52]][a, z]

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

In[18]:=
Kauffman[Knot[10, 52]][a, z]

Out[18]=
2 2 -4 2 2 z 7 z 3 2 6 z 4 z 2 2 4 - a + 2 a + --- - --- - 9 a z - 4 a z - 9 z + ---- + ---- - 7 a z + 5 a 4 2 a a a 3 3 3 3 4 4 z 5 z 2 z 24 z 3 3 3 4 3 z 12 z > -- - ---- + ---- + ----- + 24 a z + 8 a z + 19 z + ---- - ----- - 7 5 3 a 6 4 a a a a a 4 5 5 5 6 9 z 2 4 6 z 11 z 28 z 5 3 5 6 8 z > ---- + 13 a z + ---- - ----- - ----- - 16 a z - 5 a z - 20 z + ---- - 2 5 3 a 4 a a a a 6 7 7 8 9 3 z 2 6 7 z 7 z 7 3 7 8 4 z 2 8 z > ---- - 9 a z + ---- + ---- + a z + a z + 6 z + ---- + 2 a z + -- + 2 3 a 2 a a a a 9 > a z

In[19]:=
{Vassiliev[2][Knot[10, 52]], Vassiliev[3][Knot[10, 52]]}

Out[19]=
{3, 1}

In[20]:=
Kh[Knot[10, 52]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 52], 2][q]

Out[21]=
-13 2 -11 7 6 9 21 6 28 37 3 52 46 20 + q - --- - q + --- - -- - -- + -- - -- - -- + -- + -- - -- + -- - 12 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 2 3 4 5 6 7 8 9 > 70 q + 43 q + 35 q - 73 q + 30 q + 41 q - 60 q + 17 q + 31 q - 10 11 12 13 14 15 16 17 > 37 q + 9 q + 14 q - 15 q + 4 q + 3 q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₅₂

10₅₁
10₅₃

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-13 - 2q^-12 - q^-11 + 7q^-10 - 6q^-9 - 9q^-8 + 21q^-7 - 6q^-6 - 28q^-5 + 37q^-4 + 3q^-3 - 52q^-2 + 46q^-1 + 20 - 70q + 43q² + 35q³ - 73q⁴ + 30q⁵ + 41q⁶ - 60q⁷ + 17q⁸ + 31q⁹ - 37q¹⁰ + 9q¹¹ + 14q¹² - 15q¹³ + 4q¹⁴ + 3q¹⁵ - 3q¹⁶ + q¹⁷
3	- q^-27 + 2q^-26 + q^-25 - 2q^-24 - 6q^-23 + 5q^-22 + 11q^-21 - 2q^-20 - 24q^-19 - 2q^-18 + 34q^-17 + 20q^-16 - 48q^-15 - 41q^-14 + 49q^-13 + 73q^-12 - 43q^-11 - 105q^-10 + 26q^-9 + 130q^-8 + 6q^-7 - 155q^-6 - 35q^-5 + 160q^-4 + 80q^-3 - 169q^-2 - 108q^-1 + 150 + 155q - 145q² - 173q³ + 107q⁴ + 211q⁵ - 88q⁶ - 216q⁷ + 47q⁸ + 228q⁹ - 24q¹⁰ - 210q¹¹ - 6q¹² + 190q¹³ + 18q¹⁴ - 156q¹⁵ - 22q¹⁶ + 117q¹⁷ + 20q¹⁸ - 85q¹⁹ - 9q²⁰ + 53q²¹ + 5q²² - 35q²³ + 2q²⁴ + 19q²⁵ - 4q²⁶ - 9q²⁷ + 4q²⁸ + 3q²⁹ - q³⁰ - 3q³¹ + 3q³² - q³³
4	q^-46 - 2q^-45 - q^-44 + 2q^-43 + q^-42 + 7q^-41 - 9q^-40 - 9q^-39 + 2q^-38 + 3q^-37 + 35q^-36 - 11q^-35 - 32q^-34 - 23q^-33 - 17q^-32 + 97q^-31 + 31q^-30 - 30q^-29 - 83q^-28 - 125q^-27 + 134q^-26 + 123q^-25 + 84q^-24 - 88q^-23 - 321q^-22 + 39q^-21 + 141q^-20 + 300q^-19 + 88q^-18 - 454q^-17 - 163q^-16 - 49q^-15 + 442q^-14 + 404q^-13 - 378q^-12 - 294q^-11 - 396q^-10 + 375q^-9 + 672q^-8 - 134q^-7 - 233q^-6 - 728q^-5 + 137q^-4 + 778q^-3 + 149q^-2 - 23q^-1 - 955 - 163q + 749q² + 407q³ + 249q⁴ - 1085q⁵ - 478q⁶ + 630q⁷ + 643q⁸ + 543q⁹ - 1114q¹⁰ - 776q¹¹ + 416q¹² + 792q¹³ + 825q¹⁴ - 977q¹⁵ - 958q¹⁶ + 122q¹⁷ + 743q¹⁸ + 978q¹⁹ - 673q²⁰ - 888q²¹ - 120q²² + 481q²³ + 880q²⁴ - 343q²⁵ - 595q²⁶ - 178q²⁷ + 185q²⁸ + 581q²⁹ - 149q³⁰ - 277q³¹ - 92q³² + 19q³³ + 286q³⁴ - 82q³⁵ - 91q³⁶ - 8q³⁷ - 21q³⁸ + 111q³⁹ - 51q⁴⁰ - 20q⁴¹ + 15q⁴² - 19q⁴³ + 36q⁴⁴ - 22q⁴⁵ - 2q⁴⁶ + 10q⁴⁷ - 9q⁴⁸ + 8q⁴⁹ - 6q⁵⁰ + q⁵¹ + 3q⁵² - 3q⁵³ + q⁵⁴
5	- q^-70 + 2q^-69 + q^-68 - 2q^-67 - q^-66 - 2q^-65 - 3q^-64 + 7q^-63 + 11q^-62 - 2q^-61 - 8q^-60 - 13q^-59 - 19q^-58 + 8q^-57 + 40q^-56 + 32q^-55 + 2q^-54 - 39q^-53 - 81q^-52 - 50q^-51 + 50q^-50 + 120q^-49 + 120q^-48 + 16q^-47 - 155q^-46 - 234q^-45 - 121q^-44 + 110q^-43 + 328q^-42 + 326q^-41 + 21q^-40 - 362q^-39 - 520q^-38 - 302q^-37 + 239q^-36 + 702q^-35 + 644q^-34 + 44q^-33 - 681q^-32 - 995q^-31 - 529q^-30 + 471q^-29 + 1218q^-28 + 1052q^-27 + 13q^-26 - 1177q^-25 - 1559q^-24 - 681q^-23 + 874q^-22 + 1846q^-21 + 1386q^-20 - 246q^-19 - 1844q^-18 - 2055q^-17 - 529q^-16 + 1536q^-15 + 2448q^-14 + 1415q^-13 - 918q^-12 - 2648q^-11 - 2186q^-10 + 147q^-9 + 2475q^-8 + 2871q^-7 + 740q^-6 - 2185q^-5 - 3292q^-4 - 1573q^-3 + 1616q^-2 + 3633q^-1 + 2397 - 1143q - 3745q² - 3070q³ + 473q⁴ + 3906q⁵ + 3764q⁶ - 17q⁷ - 3928q⁸ - 4339q⁹ - 634q¹⁰ + 4030q¹¹ + 5005q¹² + 1130q¹³ - 3996q¹⁴ - 5565q¹⁵ - 1853q¹⁶ + 3925q¹⁷ + 6137q¹⁸ + 2501q¹⁹ - 3611q²⁰ - 6482q²¹ - 3295q²² + 3128q²³ + 6625q²⁴ + 3910q²⁵ - 2373q²⁶ - 6383q²⁷ - 4432q²⁸ + 1530q²⁹ + 5812q³⁰ + 4597q³¹ - 655q³² - 4915q³³ - 4443q³⁴ - 94q³⁵ + 3871q³⁶ + 3964q³⁷ + 583q³⁸ - 2808q³⁹ - 3231q⁴⁰ - 832q⁴¹ + 1860q⁴² + 2461q⁴³ + 821q⁴⁴ - 1161q⁴⁵ - 1678q⁴⁶ - 674q⁴⁷ + 647q⁴⁸ + 1080q⁴⁹ + 470q⁵⁰ - 366q⁵¹ - 619q⁵² - 276q⁵³ + 192q⁵⁴ + 334q⁵⁵ + 133q⁵⁶ - 103q⁵⁷ - 162q⁵⁸ - 59q⁵⁹ + 66q⁶⁰ + 75q⁶¹ + 7q⁶² - 34q⁶³ - 22q⁶⁴ - 5q⁶⁵ + 14q⁶⁶ + 21q⁶⁷ - 11q⁶⁸ - 12q⁶⁹ + 8q⁷⁰ - q⁷¹ - 4q⁷² + 8q⁷³ - 3q⁷⁴ - 5q⁷⁵ + 6q⁷⁶ - q⁷⁷ - 3q⁷⁸ + 3q⁷⁹ - q⁸⁰
6	q^-99 - 2q^-98 - q^-97 + 2q^-96 + q^-95 + 2q^-94 - 2q^-93 + 5q^-92 - 9q^-91 - 11q^-90 + 5q^-89 + 7q^-88 + 15q^-87 + 23q^-85 - 23q^-84 - 48q^-83 - 20q^-82 - q^-81 + 45q^-80 + 30q^-79 + 115q^-78 + 6q^-77 - 99q^-76 - 126q^-75 - 124q^-74 - 16q^-73 + 25q^-72 + 340q^-71 + 249q^-70 + 64q^-69 - 170q^-68 - 382q^-67 - 411q^-66 - 403q^-65 + 362q^-64 + 616q^-63 + 743q^-62 + 428q^-61 - 141q^-60 - 825q^-59 - 1500q^-58 - 615q^-57 + 85q^-56 + 1224q^-55 + 1705q^-54 + 1545q^-53 + 205q^-52 - 1967q^-51 - 2234q^-50 - 2254q^-49 - 388q^-48 + 1653q^-47 + 3671q^-46 + 3331q^-45 + 410q^-44 - 1751q^-43 - 4522q^-42 - 4233q^-41 - 2027q^-40 + 2795q^-39 + 5710q^-38 + 4996q^-37 + 2923q^-36 - 2737q^-35 - 6493q^-34 - 7734q^-33 - 2746q^-32 + 3105q^-31 + 7129q^-30 + 9167q^-29 + 4098q^-28 - 2946q^-27 - 10155q^-26 - 9434q^-25 - 4696q^-24 + 2872q^-23 + 11350q^-22 + 11713q^-21 + 5821q^-20 - 5804q^-19 - 11647q^-18 - 12944q^-17 - 6482q^-16 + 6619q^-15 + 14724q^-14 + 14845q^-13 + 3498q^-12 - 7209q^-11 - 16720q^-10 - 16031q^-9 - 2785q^-8 + 11592q^-7 + 19786q^-6 + 13095q^-5 + 1389q^-4 - 15065q^-3 - 22066q^-2 - 12517q^-1 + 4894 + 20150q + 19883q² + 10191q³ - 10547q⁴ - 24460q⁵ - 19974q⁶ - 1873q⁷ + 18392q⁸ + 24019q⁹ + 17092q¹⁰ - 6194q¹¹ - 25488q¹² - 25518q¹³ - 7174q¹⁴ + 16999q¹⁵ + 27575q¹⁶ + 22750q¹⁷ - 2896q¹⁸ - 27007q¹⁹ - 31018q²⁰ - 12149q²¹ + 16027q²² + 31710q²³ + 29006q²⁴ + 1142q²⁵ - 28088q²⁶ - 36938q²⁷ - 18752q²⁸ + 12861q²⁹ + 34436q³⁰ + 35758q³¹ + 8043q³² - 25325q³³ - 40357q³⁴ - 26325q³⁵ + 5350q³⁶ + 31754q³⁷ + 39319q³⁸ + 16488q³⁹ - 16824q⁴⁰ - 37060q⁴¹ - 30514q⁴² - 4323q⁴³ + 22409q⁴⁴ + 35577q⁴⁵ + 21506q⁴⁶ - 5805q⁴⁷ - 26739q⁴⁸ - 27467q⁴⁹ - 10654q⁵⁰ + 10645q⁵¹ + 25210q⁵² + 19644q⁵³ + 1913q⁵⁴ - 14467q⁵⁵ - 18675q⁵⁶ - 10692q⁵⁷ + 2283q⁵⁸ + 13693q⁵⁹ + 12961q⁶⁰ + 3925q⁶¹ - 5700q⁶² - 9469q⁶³ - 6779q⁶⁴ - 877q⁶⁵ + 5743q⁶⁶ + 6269q⁶⁷ + 2553q⁶⁸ - 1659q⁶⁹ - 3549q⁷⁰ - 2896q⁷¹ - 996q⁷² + 1962q⁷³ + 2241q⁷⁴ + 901q⁷⁵ - 422q⁷⁶ - 963q⁷⁷ - 798q⁷⁸ - 468q⁷⁹ + 612q⁸⁰ + 596q⁸¹ + 115q⁸² - 129q⁸³ - 170q⁸⁴ - 100q⁸⁵ - 164q⁸⁶ + 192q⁸⁷ + 123q⁸⁸ - 55q⁸⁹ - 42q⁹⁰ - 6q⁹¹ + 23q⁹² - 62q⁹³ + 59q⁹⁴ + 22q⁹⁵ - 37q⁹⁶ - 5q⁹⁷ + 5q⁹⁸ + 18q⁹⁹ - 26q¹⁰⁰ + 15q¹⁰¹ + 7q¹⁰² - 14q¹⁰³ + 4q¹⁰⁴ + 5q¹⁰⁶ - 6q¹⁰⁷ + q¹⁰⁸ + 3q¹⁰⁹ - 3q¹¹⁰ + q¹¹¹

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 52]]
Out[2]=	PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[20, 15, 1, 16], > X[16, 9, 17, 10], X[10, 19, 11, 20], X[18, 11, 19, 12], X[12, 17, 13, 18], > X[2, 8, 3, 7], X[4, 14, 5, 13]]
In[3]:=	GaussCode[Knot[10, 52]]
Out[3]=	GaussCode[1, -9, 2, -10, 3, -1, 9, -2, 5, -6, 7, -8, 10, -3, 4, -5, 8, -7, 6, > -4]
In[4]:=	DTCode[Knot[10, 52]]
Out[4]=	DTCode[6, 8, 14, 2, 16, 18, 4, 20, 12, 10]
In[5]:=	br = BR[Knot[10, 52]]
Out[5]=	BR[4, {1, 1, 1, -2, 1, 1, -2, -2, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 52]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 52]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 52]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 52]][t]
Out[10]=	2 7 13 2 3 -15 + -- - -- + -- + 13 t - 7 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 52]][z]
Out[11]=	2 4 6 1 + 3 z + 5 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 23], Knot[10, 52]}
In[13]:=	{KnotDet[Knot[10, 52]], KnotSignature[Knot[10, 52]]}
Out[13]=	{59, 2}
In[14]:=	Jones[Knot[10, 52]][q]
Out[14]=	-4 2 4 7 2 3 4 5 6 -8 - q + -- - -- + - + 10 q - 9 q + 8 q - 6 q + 3 q - q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 52]}
In[16]:=	A2Invariant[Knot[10, 52]][q]
Out[16]=	-12 -8 -6 2 2 6 8 10 12 14 16 18 3 - q - q - q + -- + 2 q + 2 q - 2 q + q - q - q + q - q 4 q
In[17]:=	HOMFLYPT[Knot[10, 52]][a, z]
Out[17]=	2 2 4 4 -4 2 2 2 z 2 z 2 2 4 z 3 z 2 4 6 4 - a - 2 a + 6 z - ---- + ---- - 3 a z + 4 z - -- + ---- - a z + z + 4 2 4 2 a a a a 6 z > -- 2 a
In[18]:=	Kauffman[Knot[10, 52]][a, z]
Out[18]=	2 2 -4 2 2 z 7 z 3 2 6 z 4 z 2 2 4 - a + 2 a + --- - --- - 9 a z - 4 a z - 9 z + ---- + ---- - 7 a z + 5 a 4 2 a a a 3 3 3 3 4 4 z 5 z 2 z 24 z 3 3 3 4 3 z 12 z > -- - ---- + ---- + ----- + 24 a z + 8 a z + 19 z + ---- - ----- - 7 5 3 a 6 4 a a a a a 4 5 5 5 6 9 z 2 4 6 z 11 z 28 z 5 3 5 6 8 z > ---- + 13 a z + ---- - ----- - ----- - 16 a z - 5 a z - 20 z + ---- - 2 5 3 a 4 a a a a 6 7 7 8 9 3 z 2 6 7 z 7 z 7 3 7 8 4 z 2 8 z > ---- - 9 a z + ---- + ---- + a z + a z + 6 z + ---- + 2 a z + -- + 2 3 a 2 a a a a 9 > a z
In[19]:=	{Vassiliev[2][Knot[10, 52]], Vassiliev[3][Knot[10, 52]]}
Out[19]=	{3, 1}
In[20]:=	Kh[Knot[10, 52]][q, t]
Out[20]=	3 1 1 1 3 1 4 3 4 4 q 6 q + 5 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 > 4 q t + 5 q t + 4 q t + 4 q t + 2 q t + 4 q t + q t + 11 4 13 5 > 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 52], 2][q]
Out[21]=	-13 2 -11 7 6 9 21 6 28 37 3 52 46 20 + q - --- - q + --- - -- - -- + -- - -- - -- + -- + -- - -- + -- - 12 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 2 3 4 5 6 7 8 9 > 70 q + 43 q + 35 q - 73 q + 30 q + 41 q - 60 q + 17 q + 31 q - 10 11 12 13 14 15 16 17 > 37 q + 9 q + 14 q - 15 q + 4 q + 3 q - 3 q + q

The Alternating Knot 1052

The Alternating Knot 10₅₂