The Non Alternating Knot 10₁₅₉

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Acknowledgement

KnotPlot

PD Presentation: X₁₆₂₇ X₃₉₄₈ X_18,11,19,12 X_20,13,1,14 X_15,2,16,3 X_17,5,18,4 X_12,19,13,20 X_5,10,6,11 X_7,15,8,14 X_9,16,10,17

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

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

Minimum Braid Representative:

Length is 10, width is 3
Braid index is 3

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 - 4t^-2 + 9t^-1 - 11 + 9t - 4t² + t³

Conway Polynomial: 1 + 2z² + 2z⁴ + z⁶

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

Determinant and Signature: {39, -2}

Jones Polynomial: - q^-8 + 3q^-7 - 5q^-6 + 6q^-5 - 7q^-4 + 7q^-3 - 5q^-2 + 4q^-1 - 1

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

A2 (sl(3)) Invariant: - q^-24 + q^-22 - q^-20 + q^-16 - 2q^-14 + q^-12 - q^-10 + 2q^-8 + 2q^-6 + 2q^-2 - 1

HOMFLY-PT Polynomial: a² - a²z² - a²z⁴ + a⁴ + 5a⁴z² + 4a⁴z⁴ + a⁴z⁶ - a⁶ - 2a⁶z² - a⁶z⁴

Kauffman Polynomial: az³ - a² - 2a²z² + 4a²z⁴ + a³z + a³z⁵ + a³z⁷ + a⁴ - 4a⁴z² + 3a⁴z⁴ + a⁴z⁸ + a⁵z - 5a⁵z⁵ + 4a⁵z⁷ + a⁶ + a⁶z² - 8a⁶z⁴ + 3a⁶z⁶ + a⁶z⁸ + a⁷z - a⁷z³ - 5a⁷z⁵ + 3a⁷z⁷ + 3a⁸z² - 7a⁸z⁴ + 3a⁸z⁶ + a⁹z - 2a⁹z³ + a⁹z⁵

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

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 10₁₅₉. 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

j = 1 1

j = -1 3

j = -3 3 2

j = -5 4 2

j = -7 3 3

j = -9 3 4

j = -11 2 3

j = -13 1 3

j = -15 2

j = -17 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-23 - 3q^-22 + 10q^-20 - 11q^-19 - 9q^-18 + 28q^-17 - 13q^-16 - 27q^-15 + 42q^-14 - 6q^-13 - 43q^-12 + 46q^-11 + 3q^-10 - 49q^-9 + 39q^-8 + 10q^-7 - 39q^-6 + 22q^-5 + 11q^-4 - 19q^-3 + 6q^-2 + 5q^-1 - 3

3 - q^-45 + 3q^-44 - 5q^-42 - 5q^-41 + 11q^-40 + 16q^-39 - 17q^-38 - 33q^-37 + 13q^-36 + 58q^-35 + 3q^-34 - 84q^-33 - 29q^-32 + 99q^-31 + 68q^-30 - 105q^-29 - 109q^-28 + 98q^-27 + 148q^-26 - 84q^-25 - 178q^-24 + 62q^-23 + 205q^-22 - 42q^-21 - 220q^-20 + 21q^-19 + 225q^-18 + 2q^-17 - 224q^-16 - 20q^-15 + 202q^-14 + 46q^-13 - 180q^-12 - 54q^-11 + 133q^-10 + 68q^-9 - 95q^-8 - 58q^-7 + 48q^-6 + 53q^-5 - 25q^-4 - 27q^-3 + q^-2 + 16q^-1 + 3 - 5q - q² - q³ + q⁴

4 q^-74 - 3q^-73 + 5q^-71 + 5q^-69 - 18q^-68 - 9q^-67 + 17q^-66 + 15q^-65 + 42q^-64 - 51q^-63 - 66q^-62 - 7q^-61 + 38q^-60 + 173q^-59 - 17q^-58 - 144q^-57 - 156q^-56 - 65q^-55 + 347q^-54 + 186q^-53 - 68q^-52 - 344q^-51 - 387q^-50 + 345q^-49 + 441q^-48 + 259q^-47 - 355q^-46 - 783q^-45 + 92q^-44 + 541q^-43 + 675q^-42 - 150q^-41 - 1046q^-40 - 255q^-39 + 463q^-38 + 996q^-37 + 121q^-36 - 1145q^-35 - 539q^-34 + 322q^-33 + 1177q^-32 + 347q^-31 - 1133q^-30 - 734q^-29 + 171q^-28 + 1234q^-27 + 526q^-26 - 1002q^-25 - 844q^-24 - 27q^-23 + 1123q^-22 + 668q^-21 - 700q^-20 - 814q^-19 - 263q^-18 + 801q^-17 + 688q^-16 - 288q^-15 - 575q^-14 - 396q^-13 + 356q^-12 + 499q^-11 + 26q^-10 - 232q^-9 - 311q^-8 + 36q^-7 + 209q^-6 + 94q^-5 - 11q^-4 - 119q^-3 - 43q^-2 + 31q^-1 + 33 + 23q - 13q² - 13q³ - 3q⁴ + 4q⁶ + q⁷ - q⁸

5 - q^-110 + 3q^-109 - 5q^-107 + 2q^-104 + 11q^-103 + 9q^-102 - 17q^-101 - 24q^-100 - 15q^-99 + 5q^-98 + 50q^-97 + 68q^-96 + 16q^-95 - 87q^-94 - 137q^-93 - 92q^-92 + 62q^-91 + 244q^-90 + 264q^-89 + 38q^-88 - 311q^-87 - 488q^-86 - 300q^-85 + 227q^-84 + 726q^-83 + 722q^-82 + 67q^-81 - 829q^-80 - 1201q^-79 - 632q^-78 + 638q^-77 + 1642q^-76 + 1400q^-75 - 128q^-74 - 1830q^-73 - 2221q^-72 - 730q^-71 + 1675q^-70 + 2958q^-69 + 1779q^-68 - 1159q^-67 - 3436q^-66 - 2885q^-65 + 343q^-64 + 3622q^-63 + 3895q^-62 + 619q^-61 - 3509q^-60 - 4729q^-59 - 1607q^-58 + 3223q^-57 + 5318q^-56 + 2517q^-55 - 2818q^-54 - 5736q^-53 - 3288q^-52 + 2420q^-51 + 5990q^-50 + 3900q^-49 - 2023q^-48 - 6154q^-47 - 4403q^-46 + 1674q^-45 + 6246q^-44 + 4800q^-43 - 1325q^-42 - 6232q^-41 - 5161q^-40 + 908q^-39 + 6153q^-38 + 5457q^-37 - 445q^-36 - 5842q^-35 - 5683q^-34 - 197q^-33 + 5383q^-32 + 5778q^-31 + 837q^-30 - 4590q^-29 - 5639q^-28 - 1581q^-27 + 3640q^-26 + 5219q^-25 + 2129q^-24 - 2455q^-23 - 4499q^-22 - 2518q^-21 + 1345q^-20 + 3528q^-19 + 2500q^-18 - 304q^-17 - 2453q^-16 - 2264q^-15 - 313q^-14 + 1430q^-13 + 1694q^-12 + 681q^-11 - 624q^-10 - 1148q^-9 - 657q^-8 + 138q^-7 + 581q^-6 + 512q^-5 + 95q^-4 - 238q^-3 - 295q^-2 - 124q^-1 + 52 + 120q + 86q² + 16q³ - 38q⁴ - 41q⁵ - 8q⁶ + q⁷ + 9q⁸ + 8q⁹ - 3q¹¹

6 q^-153 - 3q^-152 + 5q^-150 - 7q^-147 + 5q^-146 - 11q^-145 - 9q^-144 + 26q^-143 + 15q^-142 + 15q^-141 - 23q^-140 - 4q^-139 - 63q^-138 - 66q^-137 + 46q^-136 + 88q^-135 + 140q^-134 + 44q^-133 + 50q^-132 - 227q^-131 - 363q^-130 - 185q^-129 + 43q^-128 + 424q^-127 + 499q^-126 + 677q^-125 - 39q^-124 - 794q^-123 - 1126q^-122 - 990q^-121 - 93q^-120 + 876q^-119 + 2310q^-118 + 1802q^-117 + 344q^-116 - 1634q^-115 - 3146q^-114 - 3073q^-113 - 1394q^-112 + 2753q^-111 + 4796q^-110 + 4799q^-109 + 1709q^-108 - 2972q^-107 - 6979q^-106 - 7726q^-105 - 2066q^-104 + 4246q^-103 + 9724q^-102 + 9611q^-101 + 3758q^-100 - 6123q^-99 - 13892q^-98 - 11789q^-97 - 3732q^-96 + 8868q^-95 + 16740q^-94 + 15408q^-93 + 2574q^-92 - 13648q^-91 - 20316q^-90 - 16295q^-89 + 248q^-88 + 17400q^-87 + 25517q^-86 + 15291q^-85 - 6197q^-84 - 22755q^-83 - 27089q^-82 - 11680q^-81 + 11841q^-80 + 30067q^-79 + 26096q^-78 + 3792q^-77 - 20007q^-76 - 32940q^-75 - 21635q^-74 + 4490q^-73 + 30215q^-72 + 32552q^-71 + 11907q^-70 - 15869q^-69 - 35025q^-68 - 27854q^-67 - 1280q^-66 + 28945q^-65 + 35796q^-64 + 17120q^-63 - 12640q^-62 - 35634q^-61 - 31543q^-60 - 5207q^-59 + 27646q^-58 + 37694q^-57 + 20901q^-56 - 9898q^-55 - 35586q^-54 - 34412q^-53 - 9057q^-52 + 25382q^-51 + 38699q^-50 + 24928q^-49 - 5576q^-48 - 33554q^-47 - 36622q^-46 - 14457q^-45 + 19851q^-44 + 37062q^-43 + 28935q^-42 + 1815q^-41 - 26993q^-40 - 35841q^-39 - 20547q^-38 + 9877q^-37 + 29961q^-36 + 29799q^-35 + 10547q^-34 - 15195q^-33 - 28939q^-32 - 23264q^-31 - 1647q^-30 + 17301q^-29 + 24091q^-28 + 15477q^-27 - 2300q^-26 - 16515q^-25 - 18998q^-24 - 8584q^-23 + 4322q^-22 + 13151q^-21 + 13055q^-20 + 5066q^-19 - 4600q^-18 - 9913q^-17 - 7902q^-16 - 2521q^-15 + 3413q^-14 + 6313q^-13 + 4970q^-12 + 1087q^-11 - 2456q^-10 - 3456q^-9 - 2681q^-8 - 609q^-7 + 1294q^-6 + 1951q^-5 + 1300q^-4 + 198q^-3 - 523q^-2 - 856q^-1 - 630 - 134q + 253q² + 324q³ + 209q⁴ + 82q⁵ - 58q⁶ - 119q⁷ - 73q⁸ - 15q⁹ + 12q¹⁰ + 17q¹¹ + 18q¹² + 12q¹³ - 6q¹⁴ - 4q¹⁵ - q¹⁸ - q¹⁹ + q²⁰

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, 159]]

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

In[3]:=
GaussCode[Knot[10, 159]]

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

In[4]:=
DTCode[Knot[10, 159]]

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

In[5]:=
br = BR[Knot[10, 159]]

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

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

Out[6]=
{3, 10}

In[7]:=
BraidIndex[Knot[10, 159]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[10, 159]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 159]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 159]][t]

Out[10]=
-3 4 9 2 3 -11 + t - -- + - + 9 t - 4 t + t 2 t t

In[11]:=
Conway[Knot[10, 159]][z]

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

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

Out[12]=
{Knot[10, 159]}

In[13]:=
{KnotDet[Knot[10, 159]], KnotSignature[Knot[10, 159]]}

Out[13]=
{39, -2}

In[14]:=
Jones[Knot[10, 159]][q]

Out[14]=
-8 3 5 6 7 7 5 4 -1 - 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, 159]}

In[16]:=
A2Invariant[Knot[10, 159]][q]

Out[16]=
-24 -22 -20 -16 2 -12 -10 2 2 2 -1 - q + q - q + q - --- + q - q + -- + -- + -- 14 8 6 2 q q q q

In[17]:=
HOMFLYPT[Knot[10, 159]][a, z]

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

In[18]:=
Kauffman[Knot[10, 159]][a, z]

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

In[19]:=
{Vassiliev[2][Knot[10, 159]], Vassiliev[3][Knot[10, 159]]}

Out[19]=
{2, -3}

In[20]:=
Kh[Knot[10, 159]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 159], 2][q]

Out[21]=
-23 3 10 11 9 28 13 27 42 6 43 46 -3 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- + 22 20 19 18 17 16 15 14 13 12 11 q q q q q q q q q q q 3 49 39 10 39 22 11 19 6 5 > --- - -- + -- + -- - -- + -- + -- - -- + -- + - 10 9 8 7 6 5 4 3 2 q q q q q q q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₅₉

10₁₅₈
10₁₆₀

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-23 - 3q^-22 + 10q^-20 - 11q^-19 - 9q^-18 + 28q^-17 - 13q^-16 - 27q^-15 + 42q^-14 - 6q^-13 - 43q^-12 + 46q^-11 + 3q^-10 - 49q^-9 + 39q^-8 + 10q^-7 - 39q^-6 + 22q^-5 + 11q^-4 - 19q^-3 + 6q^-2 + 5q^-1 - 3
3	- q^-45 + 3q^-44 - 5q^-42 - 5q^-41 + 11q^-40 + 16q^-39 - 17q^-38 - 33q^-37 + 13q^-36 + 58q^-35 + 3q^-34 - 84q^-33 - 29q^-32 + 99q^-31 + 68q^-30 - 105q^-29 - 109q^-28 + 98q^-27 + 148q^-26 - 84q^-25 - 178q^-24 + 62q^-23 + 205q^-22 - 42q^-21 - 220q^-20 + 21q^-19 + 225q^-18 + 2q^-17 - 224q^-16 - 20q^-15 + 202q^-14 + 46q^-13 - 180q^-12 - 54q^-11 + 133q^-10 + 68q^-9 - 95q^-8 - 58q^-7 + 48q^-6 + 53q^-5 - 25q^-4 - 27q^-3 + q^-2 + 16q^-1 + 3 - 5q - q² - q³ + q⁴
4	q^-74 - 3q^-73 + 5q^-71 + 5q^-69 - 18q^-68 - 9q^-67 + 17q^-66 + 15q^-65 + 42q^-64 - 51q^-63 - 66q^-62 - 7q^-61 + 38q^-60 + 173q^-59 - 17q^-58 - 144q^-57 - 156q^-56 - 65q^-55 + 347q^-54 + 186q^-53 - 68q^-52 - 344q^-51 - 387q^-50 + 345q^-49 + 441q^-48 + 259q^-47 - 355q^-46 - 783q^-45 + 92q^-44 + 541q^-43 + 675q^-42 - 150q^-41 - 1046q^-40 - 255q^-39 + 463q^-38 + 996q^-37 + 121q^-36 - 1145q^-35 - 539q^-34 + 322q^-33 + 1177q^-32 + 347q^-31 - 1133q^-30 - 734q^-29 + 171q^-28 + 1234q^-27 + 526q^-26 - 1002q^-25 - 844q^-24 - 27q^-23 + 1123q^-22 + 668q^-21 - 700q^-20 - 814q^-19 - 263q^-18 + 801q^-17 + 688q^-16 - 288q^-15 - 575q^-14 - 396q^-13 + 356q^-12 + 499q^-11 + 26q^-10 - 232q^-9 - 311q^-8 + 36q^-7 + 209q^-6 + 94q^-5 - 11q^-4 - 119q^-3 - 43q^-2 + 31q^-1 + 33 + 23q - 13q² - 13q³ - 3q⁴ + 4q⁶ + q⁷ - q⁸
5	- q^-110 + 3q^-109 - 5q^-107 + 2q^-104 + 11q^-103 + 9q^-102 - 17q^-101 - 24q^-100 - 15q^-99 + 5q^-98 + 50q^-97 + 68q^-96 + 16q^-95 - 87q^-94 - 137q^-93 - 92q^-92 + 62q^-91 + 244q^-90 + 264q^-89 + 38q^-88 - 311q^-87 - 488q^-86 - 300q^-85 + 227q^-84 + 726q^-83 + 722q^-82 + 67q^-81 - 829q^-80 - 1201q^-79 - 632q^-78 + 638q^-77 + 1642q^-76 + 1400q^-75 - 128q^-74 - 1830q^-73 - 2221q^-72 - 730q^-71 + 1675q^-70 + 2958q^-69 + 1779q^-68 - 1159q^-67 - 3436q^-66 - 2885q^-65 + 343q^-64 + 3622q^-63 + 3895q^-62 + 619q^-61 - 3509q^-60 - 4729q^-59 - 1607q^-58 + 3223q^-57 + 5318q^-56 + 2517q^-55 - 2818q^-54 - 5736q^-53 - 3288q^-52 + 2420q^-51 + 5990q^-50 + 3900q^-49 - 2023q^-48 - 6154q^-47 - 4403q^-46 + 1674q^-45 + 6246q^-44 + 4800q^-43 - 1325q^-42 - 6232q^-41 - 5161q^-40 + 908q^-39 + 6153q^-38 + 5457q^-37 - 445q^-36 - 5842q^-35 - 5683q^-34 - 197q^-33 + 5383q^-32 + 5778q^-31 + 837q^-30 - 4590q^-29 - 5639q^-28 - 1581q^-27 + 3640q^-26 + 5219q^-25 + 2129q^-24 - 2455q^-23 - 4499q^-22 - 2518q^-21 + 1345q^-20 + 3528q^-19 + 2500q^-18 - 304q^-17 - 2453q^-16 - 2264q^-15 - 313q^-14 + 1430q^-13 + 1694q^-12 + 681q^-11 - 624q^-10 - 1148q^-9 - 657q^-8 + 138q^-7 + 581q^-6 + 512q^-5 + 95q^-4 - 238q^-3 - 295q^-2 - 124q^-1 + 52 + 120q + 86q² + 16q³ - 38q⁴ - 41q⁵ - 8q⁶ + q⁷ + 9q⁸ + 8q⁹ - 3q¹¹
6	q^-153 - 3q^-152 + 5q^-150 - 7q^-147 + 5q^-146 - 11q^-145 - 9q^-144 + 26q^-143 + 15q^-142 + 15q^-141 - 23q^-140 - 4q^-139 - 63q^-138 - 66q^-137 + 46q^-136 + 88q^-135 + 140q^-134 + 44q^-133 + 50q^-132 - 227q^-131 - 363q^-130 - 185q^-129 + 43q^-128 + 424q^-127 + 499q^-126 + 677q^-125 - 39q^-124 - 794q^-123 - 1126q^-122 - 990q^-121 - 93q^-120 + 876q^-119 + 2310q^-118 + 1802q^-117 + 344q^-116 - 1634q^-115 - 3146q^-114 - 3073q^-113 - 1394q^-112 + 2753q^-111 + 4796q^-110 + 4799q^-109 + 1709q^-108 - 2972q^-107 - 6979q^-106 - 7726q^-105 - 2066q^-104 + 4246q^-103 + 9724q^-102 + 9611q^-101 + 3758q^-100 - 6123q^-99 - 13892q^-98 - 11789q^-97 - 3732q^-96 + 8868q^-95 + 16740q^-94 + 15408q^-93 + 2574q^-92 - 13648q^-91 - 20316q^-90 - 16295q^-89 + 248q^-88 + 17400q^-87 + 25517q^-86 + 15291q^-85 - 6197q^-84 - 22755q^-83 - 27089q^-82 - 11680q^-81 + 11841q^-80 + 30067q^-79 + 26096q^-78 + 3792q^-77 - 20007q^-76 - 32940q^-75 - 21635q^-74 + 4490q^-73 + 30215q^-72 + 32552q^-71 + 11907q^-70 - 15869q^-69 - 35025q^-68 - 27854q^-67 - 1280q^-66 + 28945q^-65 + 35796q^-64 + 17120q^-63 - 12640q^-62 - 35634q^-61 - 31543q^-60 - 5207q^-59 + 27646q^-58 + 37694q^-57 + 20901q^-56 - 9898q^-55 - 35586q^-54 - 34412q^-53 - 9057q^-52 + 25382q^-51 + 38699q^-50 + 24928q^-49 - 5576q^-48 - 33554q^-47 - 36622q^-46 - 14457q^-45 + 19851q^-44 + 37062q^-43 + 28935q^-42 + 1815q^-41 - 26993q^-40 - 35841q^-39 - 20547q^-38 + 9877q^-37 + 29961q^-36 + 29799q^-35 + 10547q^-34 - 15195q^-33 - 28939q^-32 - 23264q^-31 - 1647q^-30 + 17301q^-29 + 24091q^-28 + 15477q^-27 - 2300q^-26 - 16515q^-25 - 18998q^-24 - 8584q^-23 + 4322q^-22 + 13151q^-21 + 13055q^-20 + 5066q^-19 - 4600q^-18 - 9913q^-17 - 7902q^-16 - 2521q^-15 + 3413q^-14 + 6313q^-13 + 4970q^-12 + 1087q^-11 - 2456q^-10 - 3456q^-9 - 2681q^-8 - 609q^-7 + 1294q^-6 + 1951q^-5 + 1300q^-4 + 198q^-3 - 523q^-2 - 856q^-1 - 630 - 134q + 253q² + 324q³ + 209q⁴ + 82q⁵ - 58q⁶ - 119q⁷ - 73q⁸ - 15q⁹ + 12q¹⁰ + 17q¹¹ + 18q¹² + 12q¹³ - 6q¹⁴ - 4q¹⁵ - q¹⁸ - q¹⁹ + q²⁰

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 159]]
Out[2]=	PD[X[1, 6, 2, 7], X[3, 9, 4, 8], X[18, 11, 19, 12], X[20, 13, 1, 14], > X[15, 2, 16, 3], X[17, 5, 18, 4], X[12, 19, 13, 20], X[5, 10, 6, 11], > X[7, 15, 8, 14], X[9, 16, 10, 17]]
In[3]:=	GaussCode[Knot[10, 159]]
Out[3]=	GaussCode[-1, 5, -2, 6, -8, 1, -9, 2, -10, 8, 3, -7, 4, 9, -5, 10, -6, -3, 7, > -4]
In[4]:=	DTCode[Knot[10, 159]]
Out[4]=	DTCode[6, 8, 10, 14, 16, -18, -20, 2, 4, -12]
In[5]:=	br = BR[Knot[10, 159]]
Out[5]=	BR[3, {-1, -1, -1, -2, 1, -2, 1, 1, -2, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 159]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 159]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 159]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 159]][t]
Out[10]=	-3 4 9 2 3 -11 + t - -- + - + 9 t - 4 t + t 2 t t
In[11]:=	Conway[Knot[10, 159]][z]
Out[11]=	2 4 6 1 + 2 z + 2 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 159]}
In[13]:=	{KnotDet[Knot[10, 159]], KnotSignature[Knot[10, 159]]}
Out[13]=	{39, -2}
In[14]:=	Jones[Knot[10, 159]][q]
Out[14]=	-8 3 5 6 7 7 5 4 -1 - 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, 159]}
In[16]:=	A2Invariant[Knot[10, 159]][q]
Out[16]=	-24 -22 -20 -16 2 -12 -10 2 2 2 -1 - q + q - q + q - --- + q - q + -- + -- + -- 14 8 6 2 q q q q
In[17]:=	HOMFLYPT[Knot[10, 159]][a, z]
Out[17]=	2 4 6 2 2 4 2 6 2 2 4 4 4 6 4 4 6 a + a - a - a z + 5 a z - 2 a z - a z + 4 a z - a z + a z
In[18]:=	Kauffman[Knot[10, 159]][a, z]
Out[18]=	2 4 6 3 5 7 9 2 2 4 2 6 2 -a + a + a + a z + a z + a z + a z - 2 a z - 4 a z + a z + 8 2 3 7 3 9 3 2 4 4 4 6 4 8 4 > 3 a z + a z - a z - 2 a z + 4 a z + 3 a z - 8 a z - 7 a z + 3 5 5 5 7 5 9 5 6 6 8 6 3 7 5 7 > a z - 5 a z - 5 a z + a z + 3 a z + 3 a z + a z + 4 a z + 7 7 4 8 6 8 > 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 159]], Vassiliev[3][Knot[10, 159]]}
Out[19]=	{2, -3}
In[20]:=	Kh[Knot[10, 159]][q, t]
Out[20]=	2 3 1 2 1 3 2 3 3 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 3 3 4 2 3 > ----- + ----- + ----- + ---- + ---- + q t 7 3 7 2 5 2 5 3 q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 159], 2][q]
Out[21]=	-23 3 10 11 9 28 13 27 42 6 43 46 -3 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- + 22 20 19 18 17 16 15 14 13 12 11 q q q q q q q q q q q 3 49 39 10 39 22 11 19 6 5 > --- - -- + -- + -- - -- + -- + -- - -- + -- + - 10 9 8 7 6 5 4 3 2 q q q q q q q q q q

The Non Alternating Knot 10159

The Non Alternating Knot 10₁₅₉