The Alternating Knot 10₂₆

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X_16,8,17,7 X_12,3,13,4 X_2,15,3,16 X_14,5,15,6 X_4,13,5,14 X_20,12,1,11 X_8,20,9,19 X_18,10,19,9 X_10,18,11,17

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

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

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 + 7t^-2 - 13t^-1 + 17 - 13t + 7t² - 2t³

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

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

Determinant and Signature: {61, 0}

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

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

A2 (sl(3)) Invariant: q^-12 - q^-10 + q^-8 + q^-6 - q^-4 + 3q^-2 - 1 + q² - q⁴ - 2q⁶ + q⁸ - 2q¹⁰ + q¹² + q¹⁴ + q¹⁸

HOMFLY-PT Polynomial: 2a^-4 + 3a^-4z² + a^-4z⁴ - 3a^-2 - 6a^-2z² - 4a^-2z⁴ - a^-2z⁶ + 1 - 2z² - 3z⁴ - z⁶ + a² + 2a²z² + a²z⁴

Kauffman Polynomial: 4a^-6z² - 4a^-6z⁴ + a^-6z⁶ - 2a^-5z + 7a^-5z³ - 7a^-5z⁵ + 2a^-5z⁷ + 2a^-4 - 4a^-4z² + 4a^-4z⁴ - 5a^-4z⁶ + 2a^-4z⁸ - 2a^-3z + 4a^-3z³ - 6a^-3z⁵ + a^-3z⁷ + a^-3z⁹ + 3a^-2 - 12a^-2z² + 14a^-2z⁴ - 12a^-2z⁶ + 5a^-2z⁸ - a^-1z + 5a^-1z³ - 9a^-1z⁵ + 4a^-1z⁷ + a^-1z⁹ + 1 + z² - 2z⁴ - z⁶ + 3z⁸ - az + 5az³ - 7az⁵ + 5az⁷ - a² + 4a²z² - 7a²z⁴ + 5a²z⁶ - 3a³z³ + 3a³z⁵ - a⁴z² + a⁴z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-3, -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=0 is the signature of 10₂₆. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

j = 13 1

j = 11 1

j = 9 3 1

j = 7 4 1

j = 5 5 3

j = 3 5 4

j = 1 5 5

j = -1 4 6

j = -3 2 4

j = -5 1 4

j = -7 2

j = -9 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-12 - 3q^-11 + 2q^-10 + 7q^-9 - 16q^-8 + 6q^-7 + 25q^-6 - 41q^-5 + 7q^-4 + 53q^-3 - 67q^-2 + 2q^-1 + 78 - 78q - 8q² + 85q³ - 67q⁴ - 19q⁵ + 73q⁶ - 42q⁷ - 23q⁸ + 47q⁹ - 18q¹⁰ - 17q¹¹ + 21q¹² - 4q¹³ - 8q¹⁴ + 6q¹⁵ - 2q¹⁷ + q¹⁸

3 q^-24 - 3q^-23 + 2q^-22 + 3q^-21 - q^-20 - 10q^-19 + 4q^-18 + 20q^-17 - 7q^-16 - 37q^-15 + 11q^-14 + 60q^-13 - 8q^-12 - 100q^-11 + 9q^-10 + 141q^-9 + 5q^-8 - 197q^-7 - 19q^-6 + 244q^-5 + 51q^-4 - 295q^-3 - 77q^-2 + 322q^-1 + 117 - 343q - 144q² + 335q³ + 178q⁴ - 318q⁵ - 199q⁶ + 284q⁷ + 214q⁸ - 237q⁹ - 224q¹⁰ + 187q¹¹ + 219q¹² - 133q¹³ - 207q¹⁴ + 85q¹⁵ + 180q¹⁶ - 39q¹⁷ - 150q¹⁸ + 9q¹⁹ + 113q²⁰ + 11q²¹ - 76q²² - 22q²³ + 49q²⁴ + 19q²⁵ - 24q²⁶ - 18q²⁷ + 14q²⁸ + 9q²⁹ - 4q³⁰ - 7q³¹ + 3q³² + 2q³³ - 2q³⁵ + q³⁶

4 q^-40 - 3q^-39 + 2q^-38 + 3q^-37 - 5q^-36 + 5q^-35 - 12q^-34 + 10q^-33 + 14q^-32 - 22q^-31 + 14q^-30 - 39q^-29 + 33q^-28 + 51q^-27 - 61q^-26 + 14q^-25 - 101q^-24 + 93q^-23 + 147q^-22 - 122q^-21 - 32q^-20 - 246q^-19 + 207q^-18 + 373q^-17 - 159q^-16 - 172q^-15 - 551q^-14 + 334q^-13 + 764q^-12 - 68q^-11 - 353q^-10 - 1042q^-9 + 351q^-8 + 1220q^-7 + 198q^-6 - 436q^-5 - 1579q^-4 + 196q^-3 + 1532q^-2 + 537q^-1 - 330 - 1949q - 66q² + 1577q³ + 798q⁴ - 84q⁵ - 2038q⁶ - 328q⁷ + 1372q⁸ + 921q⁹ + 220q¹⁰ - 1877q¹¹ - 543q¹² + 1002q¹³ + 920q¹⁴ + 520q¹⁵ - 1526q¹⁶ - 689q¹⁷ + 541q¹⁸ + 801q¹⁹ + 758q²⁰ - 1046q²¹ - 714q²² + 95q²³ + 552q²⁴ + 834q²⁵ - 530q²⁶ - 564q²⁷ - 206q²⁸ + 231q²⁹ + 695q³⁰ - 135q³¹ - 296q³² - 273q³³ - 18q³⁴ + 417q³⁵ + 37q³⁶ - 65q³⁷ - 175q³⁸ - 103q³⁹ + 173q⁴⁰ + 41q⁴¹ + 29q⁴² - 62q⁴³ - 74q⁴⁴ + 51q⁴⁵ + 8q⁴⁶ + 28q⁴⁷ - 11q⁴⁸ - 30q⁴⁹ + 14q⁵⁰ - 3q⁵¹ + 10q⁵² - 9q⁵⁴ + 4q⁵⁵ - q⁵⁶ + 2q⁵⁷ - 2q⁵⁹ + q⁶⁰

5 q^-60 - 3q^-59 + 2q^-58 + 3q^-57 - 5q^-56 + q^-55 + 3q^-54 - 6q^-53 + 4q^-52 + 10q^-51 - 11q^-50 - 8q^-49 + 8q^-48 - 4q^-47 + 10q^-46 + 15q^-45 - 16q^-44 - 29q^-43 - q^-42 + 26q^-41 + 41q^-40 + 20q^-39 - 60q^-38 - 100q^-37 - 27q^-36 + 117q^-35 + 206q^-34 + 72q^-33 - 222q^-32 - 393q^-31 - 175q^-30 + 336q^-29 + 716q^-28 + 427q^-27 - 500q^-26 - 1179q^-25 - 806q^-24 + 540q^-23 + 1786q^-22 + 1489q^-21 - 525q^-20 - 2484q^-19 - 2326q^-18 + 231q^-17 + 3154q^-16 + 3452q^-15 + 244q^-14 - 3727q^-13 - 4588q^-12 - 1029q^-11 + 4070q^-10 + 5771q^-9 + 1925q^-8 - 4157q^-7 - 6708q^-6 - 2970q^-5 + 3966q^-4 + 7477q^-3 + 3883q^-2 - 3559q^-1 - 7836 - 4759q + 3001q² + 7995q³ + 5351q⁴ - 2376q⁵ - 7803q⁶ - 5807q⁷ + 1702q⁸ + 7485q⁹ + 6039q¹⁰ - 1057q¹¹ - 6949q¹² - 6152q¹³ + 387q¹⁴ + 6316q¹⁵ + 6150q¹⁶ + 287q¹⁷ - 5572q¹⁸ - 6040q¹⁹ - 974q²⁰ + 4696q²¹ + 5837q²² + 1671q²³ - 3739q²⁴ - 5484q²⁵ - 2307q²⁶ + 2659q²⁷ + 4993q²⁸ + 2828q²⁹ - 1568q³⁰ - 4281q³¹ - 3180q³² + 496q³³ + 3447q³⁴ + 3257q³⁵ + 417q³⁶ - 2462q³⁷ - 3071q³⁸ - 1137q³⁹ + 1511q⁴⁰ + 2634q⁴¹ + 1527q⁴² - 631q⁴³ - 2015q⁴⁴ - 1643q⁴⁵ - 45q⁴⁶ + 1360q⁴⁷ + 1490q⁴⁸ + 452q⁴⁹ - 739q⁵⁰ - 1158q⁵¹ - 652q⁵² + 272q⁵³ + 806q⁵⁴ + 606q⁵⁵ + 25q⁵⁶ - 437q⁵⁷ - 502q⁵⁸ - 163q⁵⁹ + 213q⁶⁰ + 314q⁶¹ + 181q⁶² - 38q⁶³ - 191q⁶⁴ - 147q⁶⁵ - 7q⁶⁶ + 73q⁶⁷ + 93q⁶⁸ + 45q⁶⁹ - 39q⁷⁰ - 51q⁷¹ - 18q⁷² - 3q⁷³ + 22q⁷⁴ + 26q⁷⁵ - 5q⁷⁶ - 13q⁷⁷ - 6q⁷⁹ + q⁸⁰ + 9q⁸¹ - q⁸² - 5q⁸³ + 2q⁸⁴ - q⁸⁶ + 2q⁸⁷ - 2q⁸⁹ + q⁹⁰

6 q^-84 - 3q^-83 + 2q^-82 + 3q^-81 - 5q^-80 + q^-79 - q^-78 + 9q^-77 - 12q^-76 + 21q^-74 - 22q^-73 - q^-72 - q^-71 + 28q^-70 - 30q^-69 - 11q^-68 + 64q^-67 - 53q^-66 - 9q^-65 + 2q^-64 + 79q^-63 - 77q^-62 - 48q^-61 + 138q^-60 - 126q^-59 - 15q^-58 + 35q^-57 + 237q^-56 - 117q^-55 - 152q^-54 + 160q^-53 - 391q^-52 - 108q^-51 + 175q^-50 + 762q^-49 + 130q^-48 - 183q^-47 - 61q^-46 - 1349q^-45 - 809q^-44 + 232q^-43 + 2098q^-42 + 1516q^-41 + 669q^-40 - 347q^-39 - 3749q^-38 - 3431q^-37 - 931q^-36 + 4105q^-35 + 5165q^-34 + 4225q^-33 + 724q^-32 - 7460q^-31 - 9403q^-30 - 5564q^-29 + 4982q^-28 + 10841q^-27 + 12094q^-26 + 5736q^-25 - 10219q^-24 - 18130q^-23 - 15209q^-22 + 1745q^-21 + 15693q^-20 + 23124q^-19 + 16005q^-18 - 8665q^-17 - 25997q^-16 - 27995q^-15 - 6794q^-14 + 16016q^-13 + 32992q^-12 + 29034q^-11 - 1762q^-10 - 28980q^-9 - 39064q^-8 - 17871q^-7 + 10926q^-6 + 37603q^-5 + 39858q^-4 + 7636q^-3 - 26330q^-2 - 44552q^-1 - 26904 + 3228q + 36470q² + 45193q³ + 15511q⁴ - 20643q⁵ - 44351q⁶ - 31467q⁷ - 3705q⁸ + 32020q⁹ + 45409q¹⁰ + 20206q¹¹ - 14621q¹² - 40746q¹³ - 32380q¹⁴ - 8864q¹⁵ + 26403q¹⁶ + 42667q¹⁷ + 22785q¹⁸ - 8775q¹⁹ - 35451q²⁰ - 31571q²¹ - 13356q²² + 19818q²³ + 38251q²⁴ + 24733q²⁵ - 2122q²⁶ - 28415q²⁷ - 29698q²⁸ - 18097q²⁹ + 11443q³⁰ + 31729q³¹ + 25911q³² + 5635q³³ - 18883q³⁴ - 25658q³⁵ - 22100q³⁶ + 1482q³⁷ + 22182q³⁸ + 24440q³⁹ + 12715q⁴⁰ - 7413q⁴¹ - 18115q⁴² - 22779q⁴³ - 7659q⁴⁴ + 10345q⁴⁵ + 18619q⁴⁶ + 15938q⁴⁷ + 3090q⁴⁸ - 7858q⁴⁹ - 18276q⁵⁰ - 12397q⁵¹ - 478q⁵² + 9446q⁵³ + 13402q⁵⁴ + 8808q⁵⁵ + 1541q⁵⁶ - 9989q⁵⁷ - 11004q⁵⁸ - 6364q⁵⁹ + 849q⁶⁰ + 6902q⁶¹ + 8317q⁶² + 6247q⁶³ - 2183q⁶⁴ - 5716q⁶⁵ - 6282q⁶⁶ - 3447q⁶⁷ + 819q⁶⁸ + 4221q⁶⁹ + 5699q⁷⁰ + 1671q⁷¹ - 875q⁷² - 3091q⁷³ - 3253q⁷⁴ - 1846q⁷⁵ + 603q⁷⁶ + 2839q⁷⁷ + 1762q⁷⁸ + 1073q⁷⁹ - 445q⁸⁰ - 1354q⁸¹ - 1620q⁸² - 701q⁸³ + 707q⁸⁴ + 617q⁸⁵ + 886q⁸⁶ + 400q⁸⁷ - 105q⁸⁸ - 663q⁸⁹ - 535q⁹⁰ + 8q⁹¹ - 40q⁹² + 297q⁹³ + 276q⁹⁴ + 182q⁹⁵ - 146q⁹⁶ - 180q⁹⁷ - 21q⁹⁸ - 125q⁹⁹ + 32q¹⁰⁰ + 74q¹⁰¹ + 111q¹⁰² - 19q¹⁰³ - 38q¹⁰⁴ + 19q¹⁰⁵ - 56q¹⁰⁶ - 10q¹⁰⁷ + 5q¹⁰⁸ + 39q¹⁰⁹ - 5q¹¹⁰ - 11q¹¹¹ + 17q¹¹² - 14q¹¹³ - 4q¹¹⁴ - 3q¹¹⁵ + 11q¹¹⁶ - 2q¹¹⁷ - 6q¹¹⁸ + 6q¹¹⁹ - 2q¹²⁰ - q¹²² + 2q¹²³ - 2q¹²⁵ + q¹²⁶

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, 26]]

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

In[3]:=
GaussCode[Knot[10, 26]]

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

In[4]:=
DTCode[Knot[10, 26]]

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

In[5]:=
br = BR[Knot[10, 26]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 26]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 26]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 26]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 26]][t]

Out[10]=
2 7 13 2 3 17 - -- + -- - -- - 13 t + 7 t - 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 26]][z]

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

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

Out[12]=
{Knot[10, 26]}

In[13]:=
{KnotDet[Knot[10, 26]], KnotSignature[Knot[10, 26]]}

Out[13]=
{61, 0}

In[14]:=
Jones[Knot[10, 26]][q]

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

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

Out[15]=
{Knot[10, 26]}

In[16]:=
A2Invariant[Knot[10, 26]][q]

Out[16]=
-12 -10 -8 -6 -4 3 2 4 6 8 10 12 -1 + q - q + q + q - q + -- + q - q - 2 q + q - 2 q + q + 2 q 14 18 > q + q

In[17]:=
HOMFLYPT[Knot[10, 26]][a, z]

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

In[18]:=
Kauffman[Knot[10, 26]][a, z]

Out[18]=
2 2 2 2 3 2 2 z 2 z z 2 4 z 4 z 12 z 2 2 1 + -- + -- - a - --- - --- - - - a z + z + ---- - ---- - ----- + 4 a z - 4 2 5 3 a 6 4 2 a a a a a a a 3 3 3 4 4 4 2 7 z 4 z 5 z 3 3 3 4 4 z 4 z > a z + ---- + ---- + ---- + 5 a z - 3 a z - 2 z - ---- + ---- + 5 3 a 6 4 a a a a 4 5 5 5 6 14 z 2 4 4 4 7 z 6 z 9 z 5 3 5 6 z > ----- - 7 a z + a z - ---- - ---- - ---- - 7 a z + 3 a z - z + -- - 2 5 3 a 6 a a a a 6 6 7 7 7 8 8 5 z 12 z 2 6 2 z z 4 z 7 8 2 z 5 z > ---- - ----- + 5 a z + ---- + -- + ---- + 5 a z + 3 z + ---- + ---- + 4 2 5 3 a 4 2 a a a a a a 9 9 z z > -- + -- 3 a a

In[19]:=
{Vassiliev[2][Knot[10, 26]], Vassiliev[3][Knot[10, 26]]}

Out[19]=
{-3, -2}

In[20]:=
Kh[Knot[10, 26]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 26], 2][q]

Out[21]=
-12 3 2 7 16 6 25 41 7 53 67 2 78 + q - --- + --- + -- - -- + -- + -- - -- + -- + -- - -- + - - 78 q - 11 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 10 > 8 q + 85 q - 67 q - 19 q + 73 q - 42 q - 23 q + 47 q - 18 q - 11 12 13 14 15 17 18 > 17 q + 21 q - 4 q - 8 q + 6 q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₂₆

10₂₅
10₂₇

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-12 - 3q^-11 + 2q^-10 + 7q^-9 - 16q^-8 + 6q^-7 + 25q^-6 - 41q^-5 + 7q^-4 + 53q^-3 - 67q^-2 + 2q^-1 + 78 - 78q - 8q² + 85q³ - 67q⁴ - 19q⁵ + 73q⁶ - 42q⁷ - 23q⁸ + 47q⁹ - 18q¹⁰ - 17q¹¹ + 21q¹² - 4q¹³ - 8q¹⁴ + 6q¹⁵ - 2q¹⁷ + q¹⁸
3	q^-24 - 3q^-23 + 2q^-22 + 3q^-21 - q^-20 - 10q^-19 + 4q^-18 + 20q^-17 - 7q^-16 - 37q^-15 + 11q^-14 + 60q^-13 - 8q^-12 - 100q^-11 + 9q^-10 + 141q^-9 + 5q^-8 - 197q^-7 - 19q^-6 + 244q^-5 + 51q^-4 - 295q^-3 - 77q^-2 + 322q^-1 + 117 - 343q - 144q² + 335q³ + 178q⁴ - 318q⁵ - 199q⁶ + 284q⁷ + 214q⁸ - 237q⁹ - 224q¹⁰ + 187q¹¹ + 219q¹² - 133q¹³ - 207q¹⁴ + 85q¹⁵ + 180q¹⁶ - 39q¹⁷ - 150q¹⁸ + 9q¹⁹ + 113q²⁰ + 11q²¹ - 76q²² - 22q²³ + 49q²⁴ + 19q²⁵ - 24q²⁶ - 18q²⁷ + 14q²⁸ + 9q²⁹ - 4q³⁰ - 7q³¹ + 3q³² + 2q³³ - 2q³⁵ + q³⁶
4	q^-40 - 3q^-39 + 2q^-38 + 3q^-37 - 5q^-36 + 5q^-35 - 12q^-34 + 10q^-33 + 14q^-32 - 22q^-31 + 14q^-30 - 39q^-29 + 33q^-28 + 51q^-27 - 61q^-26 + 14q^-25 - 101q^-24 + 93q^-23 + 147q^-22 - 122q^-21 - 32q^-20 - 246q^-19 + 207q^-18 + 373q^-17 - 159q^-16 - 172q^-15 - 551q^-14 + 334q^-13 + 764q^-12 - 68q^-11 - 353q^-10 - 1042q^-9 + 351q^-8 + 1220q^-7 + 198q^-6 - 436q^-5 - 1579q^-4 + 196q^-3 + 1532q^-2 + 537q^-1 - 330 - 1949q - 66q² + 1577q³ + 798q⁴ - 84q⁵ - 2038q⁶ - 328q⁷ + 1372q⁸ + 921q⁹ + 220q¹⁰ - 1877q¹¹ - 543q¹² + 1002q¹³ + 920q¹⁴ + 520q¹⁵ - 1526q¹⁶ - 689q¹⁷ + 541q¹⁸ + 801q¹⁹ + 758q²⁰ - 1046q²¹ - 714q²² + 95q²³ + 552q²⁴ + 834q²⁵ - 530q²⁶ - 564q²⁷ - 206q²⁸ + 231q²⁹ + 695q³⁰ - 135q³¹ - 296q³² - 273q³³ - 18q³⁴ + 417q³⁵ + 37q³⁶ - 65q³⁷ - 175q³⁸ - 103q³⁹ + 173q⁴⁰ + 41q⁴¹ + 29q⁴² - 62q⁴³ - 74q⁴⁴ + 51q⁴⁵ + 8q⁴⁶ + 28q⁴⁷ - 11q⁴⁸ - 30q⁴⁹ + 14q⁵⁰ - 3q⁵¹ + 10q⁵² - 9q⁵⁴ + 4q⁵⁵ - q⁵⁶ + 2q⁵⁷ - 2q⁵⁹ + q⁶⁰
5	q^-60 - 3q^-59 + 2q^-58 + 3q^-57 - 5q^-56 + q^-55 + 3q^-54 - 6q^-53 + 4q^-52 + 10q^-51 - 11q^-50 - 8q^-49 + 8q^-48 - 4q^-47 + 10q^-46 + 15q^-45 - 16q^-44 - 29q^-43 - q^-42 + 26q^-41 + 41q^-40 + 20q^-39 - 60q^-38 - 100q^-37 - 27q^-36 + 117q^-35 + 206q^-34 + 72q^-33 - 222q^-32 - 393q^-31 - 175q^-30 + 336q^-29 + 716q^-28 + 427q^-27 - 500q^-26 - 1179q^-25 - 806q^-24 + 540q^-23 + 1786q^-22 + 1489q^-21 - 525q^-20 - 2484q^-19 - 2326q^-18 + 231q^-17 + 3154q^-16 + 3452q^-15 + 244q^-14 - 3727q^-13 - 4588q^-12 - 1029q^-11 + 4070q^-10 + 5771q^-9 + 1925q^-8 - 4157q^-7 - 6708q^-6 - 2970q^-5 + 3966q^-4 + 7477q^-3 + 3883q^-2 - 3559q^-1 - 7836 - 4759q + 3001q² + 7995q³ + 5351q⁴ - 2376q⁵ - 7803q⁶ - 5807q⁷ + 1702q⁸ + 7485q⁹ + 6039q¹⁰ - 1057q¹¹ - 6949q¹² - 6152q¹³ + 387q¹⁴ + 6316q¹⁵ + 6150q¹⁶ + 287q¹⁷ - 5572q¹⁸ - 6040q¹⁹ - 974q²⁰ + 4696q²¹ + 5837q²² + 1671q²³ - 3739q²⁴ - 5484q²⁵ - 2307q²⁶ + 2659q²⁷ + 4993q²⁸ + 2828q²⁹ - 1568q³⁰ - 4281q³¹ - 3180q³² + 496q³³ + 3447q³⁴ + 3257q³⁵ + 417q³⁶ - 2462q³⁷ - 3071q³⁸ - 1137q³⁹ + 1511q⁴⁰ + 2634q⁴¹ + 1527q⁴² - 631q⁴³ - 2015q⁴⁴ - 1643q⁴⁵ - 45q⁴⁶ + 1360q⁴⁷ + 1490q⁴⁸ + 452q⁴⁹ - 739q⁵⁰ - 1158q⁵¹ - 652q⁵² + 272q⁵³ + 806q⁵⁴ + 606q⁵⁵ + 25q⁵⁶ - 437q⁵⁷ - 502q⁵⁸ - 163q⁵⁹ + 213q⁶⁰ + 314q⁶¹ + 181q⁶² - 38q⁶³ - 191q⁶⁴ - 147q⁶⁵ - 7q⁶⁶ + 73q⁶⁷ + 93q⁶⁸ + 45q⁶⁹ - 39q⁷⁰ - 51q⁷¹ - 18q⁷² - 3q⁷³ + 22q⁷⁴ + 26q⁷⁵ - 5q⁷⁶ - 13q⁷⁷ - 6q⁷⁹ + q⁸⁰ + 9q⁸¹ - q⁸² - 5q⁸³ + 2q⁸⁴ - q⁸⁶ + 2q⁸⁷ - 2q⁸⁹ + q⁹⁰
6	q^-84 - 3q^-83 + 2q^-82 + 3q^-81 - 5q^-80 + q^-79 - q^-78 + 9q^-77 - 12q^-76 + 21q^-74 - 22q^-73 - q^-72 - q^-71 + 28q^-70 - 30q^-69 - 11q^-68 + 64q^-67 - 53q^-66 - 9q^-65 + 2q^-64 + 79q^-63 - 77q^-62 - 48q^-61 + 138q^-60 - 126q^-59 - 15q^-58 + 35q^-57 + 237q^-56 - 117q^-55 - 152q^-54 + 160q^-53 - 391q^-52 - 108q^-51 + 175q^-50 + 762q^-49 + 130q^-48 - 183q^-47 - 61q^-46 - 1349q^-45 - 809q^-44 + 232q^-43 + 2098q^-42 + 1516q^-41 + 669q^-40 - 347q^-39 - 3749q^-38 - 3431q^-37 - 931q^-36 + 4105q^-35 + 5165q^-34 + 4225q^-33 + 724q^-32 - 7460q^-31 - 9403q^-30 - 5564q^-29 + 4982q^-28 + 10841q^-27 + 12094q^-26 + 5736q^-25 - 10219q^-24 - 18130q^-23 - 15209q^-22 + 1745q^-21 + 15693q^-20 + 23124q^-19 + 16005q^-18 - 8665q^-17 - 25997q^-16 - 27995q^-15 - 6794q^-14 + 16016q^-13 + 32992q^-12 + 29034q^-11 - 1762q^-10 - 28980q^-9 - 39064q^-8 - 17871q^-7 + 10926q^-6 + 37603q^-5 + 39858q^-4 + 7636q^-3 - 26330q^-2 - 44552q^-1 - 26904 + 3228q + 36470q² + 45193q³ + 15511q⁴ - 20643q⁵ - 44351q⁶ - 31467q⁷ - 3705q⁸ + 32020q⁹ + 45409q¹⁰ + 20206q¹¹ - 14621q¹² - 40746q¹³ - 32380q¹⁴ - 8864q¹⁵ + 26403q¹⁶ + 42667q¹⁷ + 22785q¹⁸ - 8775q¹⁹ - 35451q²⁰ - 31571q²¹ - 13356q²² + 19818q²³ + 38251q²⁴ + 24733q²⁵ - 2122q²⁶ - 28415q²⁷ - 29698q²⁸ - 18097q²⁹ + 11443q³⁰ + 31729q³¹ + 25911q³² + 5635q³³ - 18883q³⁴ - 25658q³⁵ - 22100q³⁶ + 1482q³⁷ + 22182q³⁸ + 24440q³⁹ + 12715q⁴⁰ - 7413q⁴¹ - 18115q⁴² - 22779q⁴³ - 7659q⁴⁴ + 10345q⁴⁵ + 18619q⁴⁶ + 15938q⁴⁷ + 3090q⁴⁸ - 7858q⁴⁹ - 18276q⁵⁰ - 12397q⁵¹ - 478q⁵² + 9446q⁵³ + 13402q⁵⁴ + 8808q⁵⁵ + 1541q⁵⁶ - 9989q⁵⁷ - 11004q⁵⁸ - 6364q⁵⁹ + 849q⁶⁰ + 6902q⁶¹ + 8317q⁶² + 6247q⁶³ - 2183q⁶⁴ - 5716q⁶⁵ - 6282q⁶⁶ - 3447q⁶⁷ + 819q⁶⁸ + 4221q⁶⁹ + 5699q⁷⁰ + 1671q⁷¹ - 875q⁷² - 3091q⁷³ - 3253q⁷⁴ - 1846q⁷⁵ + 603q⁷⁶ + 2839q⁷⁷ + 1762q⁷⁸ + 1073q⁷⁹ - 445q⁸⁰ - 1354q⁸¹ - 1620q⁸² - 701q⁸³ + 707q⁸⁴ + 617q⁸⁵ + 886q⁸⁶ + 400q⁸⁷ - 105q⁸⁸ - 663q⁸⁹ - 535q⁹⁰ + 8q⁹¹ - 40q⁹² + 297q⁹³ + 276q⁹⁴ + 182q⁹⁵ - 146q⁹⁶ - 180q⁹⁷ - 21q⁹⁸ - 125q⁹⁹ + 32q¹⁰⁰ + 74q¹⁰¹ + 111q¹⁰² - 19q¹⁰³ - 38q¹⁰⁴ + 19q¹⁰⁵ - 56q¹⁰⁶ - 10q¹⁰⁷ + 5q¹⁰⁸ + 39q¹⁰⁹ - 5q¹¹⁰ - 11q¹¹¹ + 17q¹¹² - 14q¹¹³ - 4q¹¹⁴ - 3q¹¹⁵ + 11q¹¹⁶ - 2q¹¹⁷ - 6q¹¹⁸ + 6q¹¹⁹ - 2q¹²⁰ - q¹²² + 2q¹²³ - 2q¹²⁵ + q¹²⁶

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 26]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 8, 17, 7], X[12, 3, 13, 4], X[2, 15, 3, 16], > X[14, 5, 15, 6], X[4, 13, 5, 14], X[20, 12, 1, 11], X[8, 20, 9, 19], > X[18, 10, 19, 9], X[10, 18, 11, 17]]
In[3]:=	GaussCode[Knot[10, 26]]
Out[3]=	GaussCode[1, -4, 3, -6, 5, -1, 2, -8, 9, -10, 7, -3, 6, -5, 4, -2, 10, -9, 8, > -7]
In[4]:=	DTCode[Knot[10, 26]]
Out[4]=	DTCode[6, 12, 14, 16, 18, 20, 4, 2, 10, 8]
In[5]:=	br = BR[Knot[10, 26]]
Out[5]=	BR[4, {-1, -1, -1, 2, -1, 2, 2, 2, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 26]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 26]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 26]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 26]][t]
Out[10]=	2 7 13 2 3 17 - -- + -- - -- - 13 t + 7 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 26]][z]
Out[11]=	2 4 6 1 - 3 z - 5 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 26]}
In[13]:=	{KnotDet[Knot[10, 26]], KnotSignature[Knot[10, 26]]}
Out[13]=	{61, 0}
In[14]:=	Jones[Knot[10, 26]][q]
Out[14]=	-4 3 6 8 2 3 4 5 6 10 + q - -- + -- - - - 10 q + 9 q - 7 q + 4 q - 2 q + q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 26]}
In[16]:=	A2Invariant[Knot[10, 26]][q]
Out[16]=	-12 -10 -8 -6 -4 3 2 4 6 8 10 12 -1 + q - q + q + q - q + -- + q - q - 2 q + q - 2 q + q + 2 q 14 18 > q + q
In[17]:=	HOMFLYPT[Knot[10, 26]][a, z]
Out[17]=	2 2 4 4 2 3 2 2 3 z 6 z 2 2 4 z 4 z 2 4 1 + -- - -- + a - 2 z + ---- - ---- + 2 a z - 3 z + -- - ---- + a z - 4 2 4 2 4 2 a a a a a a 6 6 z > z - -- 2 a
In[18]:=	Kauffman[Knot[10, 26]][a, z]
Out[18]=	2 2 2 2 3 2 2 z 2 z z 2 4 z 4 z 12 z 2 2 1 + -- + -- - a - --- - --- - - - a z + z + ---- - ---- - ----- + 4 a z - 4 2 5 3 a 6 4 2 a a a a a a a 3 3 3 4 4 4 2 7 z 4 z 5 z 3 3 3 4 4 z 4 z > a z + ---- + ---- + ---- + 5 a z - 3 a z - 2 z - ---- + ---- + 5 3 a 6 4 a a a a 4 5 5 5 6 14 z 2 4 4 4 7 z 6 z 9 z 5 3 5 6 z > ----- - 7 a z + a z - ---- - ---- - ---- - 7 a z + 3 a z - z + -- - 2 5 3 a 6 a a a a 6 6 7 7 7 8 8 5 z 12 z 2 6 2 z z 4 z 7 8 2 z 5 z > ---- - ----- + 5 a z + ---- + -- + ---- + 5 a z + 3 z + ---- + ---- + 4 2 5 3 a 4 2 a a a a a a 9 9 z z > -- + -- 3 a a
In[19]:=	{Vassiliev[2][Knot[10, 26]], Vassiliev[3][Knot[10, 26]]}
Out[19]=	{-3, -2}
In[20]:=	Kh[Knot[10, 26]][q, t]
Out[20]=	6 1 2 1 4 2 4 4 3 - + 5 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 5 q t + 5 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 11 5 > 4 q t + 5 q t + 3 q t + 4 q t + q t + 3 q t + q t + q t + 13 6 > q t
In[21]:=	ColouredJones[Knot[10, 26], 2][q]
Out[21]=	-12 3 2 7 16 6 25 41 7 53 67 2 78 + q - --- + --- + -- - -- + -- + -- - -- + -- + -- - -- + - - 78 q - 11 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 10 > 8 q + 85 q - 67 q - 19 q + 73 q - 42 q - 23 q + 47 q - 18 q - 11 12 13 14 15 17 18 > 17 q + 21 q - 4 q - 8 q + 6 q - 2 q + q

The Alternating Knot 1026

The Alternating Knot 10₂₆