The Non Alternating Knot 10₁₂₇

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X₃₈₄₉ X_14,6,15,5 X_15,20,16,1 X_9,16,10,17 X_11,18,12,19 X_17,10,18,11 X_19,12,20,13 X_6,14,7,13 X₇₂₈₃

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

DT (Dowker-Thistlethwaite) Code: 4 8 -14 2 16 18 -6 20 10 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 2 3 3 / NotAvailable 1

Alexander Polynomial: - t^-3 + 4t^-2 - 6t^-1 + 7 - 6t + 4t² - t³

Conway Polynomial: 1 + z² - 2z⁴ - z⁶

Other knots with the same Alexander/Conway Polynomial: {10₁₅₀, K11n51, ...}

Determinant and Signature: {29, -4}

Jones Polynomial: q^-10 - 2q^-9 + 3q^-8 - 5q^-7 + 5q^-6 - 5q^-5 + 4q^-4 - 2q^-3 + 2q^-2

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

A2 (sl(3)) Invariant: q^-30 + q^-26 - 2q^-22 - q^-20 - 3q^-18 + q^-12 + 3q^-10 + q^-8 + 2q^-6

HOMFLY-PT Polynomial: 5a⁴ + 7a⁴z² + 2a⁴z⁴ - 6a⁶ - 9a⁶z² - 5a⁶z⁴ - a⁶z⁶ + 2a⁸ + 3a⁸z² + a⁸z⁴

Kauffman Polynomial: 5a⁴ - 9a⁴z² + 3a⁴z⁴ - 5a⁵z + 5a⁵z³ - 3a⁵z⁵ + a⁵z⁷ + 6a⁶ - 14a⁶z² + 11a⁶z⁴ - 4a⁶z⁶ + a⁶z⁸ - 8a⁷z + 16a⁷z³ - 10a⁷z⁵ + 3a⁷z⁷ + 2a⁸ - 2a⁸z² + 4a⁸z⁴ - 2a⁸z⁶ + a⁸z⁸ - 2a⁹z + 7a⁹z³ - 5a⁹z⁵ + 2a⁹z⁷ + a¹⁰z² - 3a¹⁰z⁴ + 2a¹⁰z⁶ + a¹¹z - 4a¹¹z³ + 2a¹¹z⁵ - 2a¹²z² + a¹²z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {1, 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=-4 is the signature of 10₁₂₇. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0

j = -3 2

j = -5 1 1

j = -7 3 1

j = -9 2 1

j = -11 3 3

j = -13 2 2

j = -15 1 3

j = -17 1 2

j = -19 1

j = -21 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-28 - 2q^-27 + 4q^-25 - 6q^-24 + 11q^-22 - 12q^-21 - 4q^-20 + 22q^-19 - 17q^-18 - 9q^-17 + 28q^-16 - 17q^-15 - 12q^-14 + 25q^-13 - 11q^-12 - 12q^-11 + 17q^-10 - 4q^-9 - 9q^-8 + 8q^-7 + q^-6 - 4q^-5 + 2q^-4 + q^-3

3 q^-54 - 2q^-53 + q^-51 + 2q^-50 - 3q^-49 - q^-48 + 4q^-47 + 2q^-46 - 8q^-45 - 3q^-44 + 13q^-43 + 7q^-42 - 19q^-41 - 16q^-40 + 27q^-39 + 25q^-38 - 29q^-37 - 42q^-36 + 36q^-35 + 49q^-34 - 31q^-33 - 62q^-32 + 32q^-31 + 65q^-30 - 26q^-29 - 68q^-28 + 22q^-27 + 65q^-26 - 14q^-25 - 61q^-24 + 6q^-23 + 56q^-22 - 45q^-20 - 12q^-19 + 40q^-18 + 12q^-17 - 23q^-16 - 22q^-15 + 19q^-14 + 14q^-13 - 4q^-12 - 16q^-11 + 3q^-10 + 7q^-9 + 4q^-8 - 6q^-7 + 2q^-4

4 q^-88 - 2q^-87 + q^-85 - q^-84 + 5q^-83 - 5q^-82 + q^-81 + q^-80 - 7q^-79 + 11q^-78 - 9q^-77 + 6q^-76 + 8q^-75 - 17q^-74 + 11q^-73 - 24q^-72 + 14q^-71 + 31q^-70 - 14q^-69 + 11q^-68 - 68q^-67 + q^-66 + 64q^-65 + 25q^-64 + 38q^-63 - 133q^-62 - 54q^-61 + 75q^-60 + 85q^-59 + 109q^-58 - 181q^-57 - 128q^-56 + 51q^-55 + 125q^-54 + 187q^-53 - 189q^-52 - 176q^-51 + 13q^-50 + 131q^-49 + 234q^-48 - 172q^-47 - 187q^-46 - 15q^-45 + 115q^-44 + 242q^-43 - 140q^-42 - 170q^-41 - 41q^-40 + 84q^-39 + 229q^-38 - 89q^-37 - 136q^-36 - 68q^-35 + 35q^-34 + 196q^-33 - 24q^-32 - 80q^-31 - 85q^-30 - 24q^-29 + 138q^-28 + 26q^-27 - 13q^-26 - 69q^-25 - 62q^-24 + 63q^-23 + 34q^-22 + 32q^-21 - 27q^-20 - 57q^-19 + 8q^-18 + 11q^-17 + 33q^-16 + 6q^-15 - 25q^-14 - 8q^-13 - 5q^-12 + 12q^-11 + 9q^-10 - 3q^-9 - 2q^-8 - 4q^-7 + 2q^-5 + q^-4

5 q^-130 - 2q^-129 + q^-127 - q^-126 + 2q^-125 + 3q^-124 - 3q^-123 - 3q^-122 + q^-121 - 4q^-120 + q^-119 + 9q^-118 + 2q^-117 - 3q^-116 - q^-115 - 10q^-114 - 9q^-113 + 7q^-112 + 14q^-111 + 13q^-110 + 9q^-109 - 16q^-108 - 33q^-107 - 25q^-106 + 8q^-105 + 47q^-104 + 59q^-103 + 19q^-102 - 54q^-101 - 105q^-100 - 73q^-99 + 42q^-98 + 152q^-97 + 157q^-96 + 6q^-95 - 195q^-94 - 258q^-93 - 93q^-92 + 201q^-91 + 377q^-90 + 218q^-89 - 188q^-88 - 469q^-87 - 353q^-86 + 107q^-85 + 557q^-84 + 502q^-83 - 47q^-82 - 579q^-81 - 611q^-80 - 70q^-79 + 598q^-78 + 709q^-77 + 131q^-76 - 570q^-75 - 750q^-74 - 217q^-73 + 549q^-72 + 779q^-71 + 251q^-70 - 511q^-69 - 775q^-68 - 289q^-67 + 478q^-66 + 767q^-65 + 301q^-64 - 433q^-63 - 741q^-62 - 326q^-61 + 388q^-60 + 707q^-59 + 344q^-58 - 321q^-57 - 660q^-56 - 365q^-55 + 237q^-54 + 594q^-53 + 393q^-52 - 149q^-51 - 509q^-50 - 389q^-49 + 31q^-48 + 399q^-47 + 400q^-46 + 51q^-45 - 284q^-44 - 335q^-43 - 152q^-42 + 146q^-41 + 304q^-40 + 178q^-39 - 43q^-38 - 183q^-37 - 206q^-36 - 64q^-35 + 124q^-34 + 158q^-33 + 102q^-32 - 5q^-31 - 125q^-30 - 125q^-29 - 26q^-28 + 45q^-27 + 97q^-26 + 79q^-25 - 11q^-24 - 70q^-23 - 52q^-22 - 34q^-21 + 23q^-20 + 56q^-19 + 30q^-18 - 8q^-17 - 15q^-16 - 28q^-15 - 14q^-14 + 12q^-13 + 15q^-12 + 4q^-11 + 6q^-10 - 6q^-9 - 6q^-8 - 2q^-7 + 2q^-6 + 2q^-4

6 q^-180 - 2q^-179 + q^-177 - q^-176 + 2q^-175 + 5q^-173 - 7q^-172 - 3q^-171 + 3q^-170 - 5q^-169 + 3q^-168 + 2q^-167 + 17q^-166 - 11q^-165 - 7q^-164 + 5q^-163 - 15q^-162 - 4q^-161 + 39q^-159 - 7q^-158 - 5q^-157 + 14q^-156 - 33q^-155 - 32q^-154 - 21q^-153 + 67q^-152 + 12q^-151 + 23q^-150 + 55q^-149 - 48q^-148 - 95q^-147 - 104q^-146 + 53q^-145 + 26q^-144 + 110q^-143 + 204q^-142 + 22q^-141 - 157q^-140 - 298q^-139 - 125q^-138 - 107q^-137 + 196q^-136 + 529q^-135 + 358q^-134 - 11q^-133 - 504q^-132 - 544q^-131 - 620q^-130 + 9q^-129 + 867q^-128 + 1022q^-127 + 604q^-126 - 381q^-125 - 971q^-124 - 1524q^-123 - 702q^-122 + 834q^-121 + 1699q^-120 + 1601q^-119 + 269q^-118 - 1001q^-117 - 2399q^-116 - 1740q^-115 + 297q^-114 + 1970q^-113 + 2490q^-112 + 1161q^-111 - 580q^-110 - 2826q^-109 - 2582q^-108 - 411q^-107 + 1814q^-106 + 2907q^-105 + 1822q^-104 - 46q^-103 - 2827q^-102 - 2958q^-101 - 905q^-100 + 1528q^-99 + 2937q^-98 + 2095q^-97 + 308q^-96 - 2667q^-95 - 3001q^-94 - 1121q^-93 + 1314q^-92 + 2817q^-91 + 2145q^-90 + 484q^-89 - 2479q^-88 - 2915q^-87 - 1231q^-86 + 1119q^-85 + 2629q^-84 + 2147q^-83 + 659q^-82 - 2189q^-81 - 2751q^-80 - 1381q^-79 + 783q^-78 + 2292q^-77 + 2121q^-76 + 952q^-75 - 1661q^-74 - 2423q^-73 - 1564q^-72 + 234q^-71 + 1696q^-70 + 1954q^-69 + 1303q^-68 - 878q^-67 - 1819q^-66 - 1602q^-65 - 395q^-64 + 844q^-63 + 1492q^-62 + 1470q^-61 - 38q^-60 - 950q^-59 - 1288q^-58 - 788q^-57 - 28q^-56 + 738q^-55 + 1213q^-54 + 489q^-53 - 86q^-52 - 629q^-51 - 691q^-50 - 527q^-49 - 9q^-48 + 590q^-47 + 475q^-46 + 368q^-45 + 16q^-44 - 225q^-43 - 470q^-42 - 348q^-41 + 14q^-40 + 117q^-39 + 298q^-38 + 258q^-37 + 165q^-36 - 128q^-35 - 237q^-34 - 168q^-33 - 140q^-32 + 30q^-31 + 133q^-30 + 207q^-29 + 79q^-28 - 20q^-27 - 66q^-26 - 122q^-25 - 83q^-24 - 19q^-23 + 73q^-22 + 62q^-21 + 44q^-20 + 21q^-19 - 22q^-18 - 41q^-17 - 37q^-16 - q^-15 + 8q^-14 + 13q^-13 + 16q^-12 + 9q^-11 - 3q^-10 - 8q^-9 - 4q^-8 - 2q^-7 + 2q^-4 + q^-3

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, 127]]

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

In[3]:=
GaussCode[Knot[10, 127]]

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

In[4]:=
DTCode[Knot[10, 127]]

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

In[5]:=
br = BR[Knot[10, 127]]

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

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

Out[6]=
{3, 10}

In[7]:=
BraidIndex[Knot[10, 127]]

Out[7]=
3

In[8]:=
Show[DrawMorseLink[Knot[10, 127]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 127]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 127]][t]

Out[10]=
-3 4 6 2 3 7 - t + -- - - - 6 t + 4 t - t 2 t t

In[11]:=
Conway[Knot[10, 127]][z]

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

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

Out[12]=
{Knot[10, 127], Knot[10, 150], Knot[11, NonAlternating, 51]}

In[13]:=
{KnotDet[Knot[10, 127]], KnotSignature[Knot[10, 127]]}

Out[13]=
{29, -4}

In[14]:=
Jones[Knot[10, 127]][q]

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

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

Out[15]=
{Knot[10, 127]}

In[16]:=
A2Invariant[Knot[10, 127]][q]

Out[16]=
-30 -26 2 -20 3 -12 3 -8 2 q + q - --- - q - --- + q + --- + q + -- 22 18 10 6 q q q q

In[17]:=
HOMFLYPT[Knot[10, 127]][a, z]

Out[17]=
4 6 8 4 2 6 2 8 2 4 4 6 4 8 4 5 a - 6 a + 2 a + 7 a z - 9 a z + 3 a z + 2 a z - 5 a z + a z - 6 6 > a z

In[18]:=
Kauffman[Knot[10, 127]][a, z]

Out[18]=
4 6 8 5 7 9 11 4 2 6 2 5 a + 6 a + 2 a - 5 a z - 8 a z - 2 a z + a z - 9 a z - 14 a z - 8 2 10 2 12 2 5 3 7 3 9 3 11 3 > 2 a z + a z - 2 a z + 5 a z + 16 a z + 7 a z - 4 a z + 4 4 6 4 8 4 10 4 12 4 5 5 7 5 > 3 a z + 11 a z + 4 a z - 3 a z + a z - 3 a z - 10 a z - 9 5 11 5 6 6 8 6 10 6 5 7 7 7 > 5 a z + 2 a z - 4 a z - 2 a z + 2 a z + a z + 3 a z + 9 7 6 8 8 8 > 2 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 127]], Vassiliev[3][Knot[10, 127]]}

Out[19]=
{1, 1}

In[20]:=
Kh[Knot[10, 127]][q, t]

Out[20]=
-5 2 1 1 1 2 1 3 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q t q t q t q t q t q t q t 2 3 3 2 1 3 1 1 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- 13 4 11 4 11 3 9 3 9 2 7 2 7 5 q t q t q t q t q t q t q t q t

In[21]:=
ColouredJones[Knot[10, 127], 2][q]

Out[21]=
-28 2 4 6 11 12 4 22 17 9 28 17 12 q - --- + --- - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + 27 25 24 22 21 20 19 18 17 16 15 14 q q q q q q q q q q q q 25 11 12 17 4 9 8 -6 4 2 -3 > --- - --- - --- + --- - -- - -- + -- + q - -- + -- + q 13 12 11 10 9 8 7 5 4 q q q q q q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₂₇

10₁₂₆
10₁₂₈

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-28 - 2q^-27 + 4q^-25 - 6q^-24 + 11q^-22 - 12q^-21 - 4q^-20 + 22q^-19 - 17q^-18 - 9q^-17 + 28q^-16 - 17q^-15 - 12q^-14 + 25q^-13 - 11q^-12 - 12q^-11 + 17q^-10 - 4q^-9 - 9q^-8 + 8q^-7 + q^-6 - 4q^-5 + 2q^-4 + q^-3
3	q^-54 - 2q^-53 + q^-51 + 2q^-50 - 3q^-49 - q^-48 + 4q^-47 + 2q^-46 - 8q^-45 - 3q^-44 + 13q^-43 + 7q^-42 - 19q^-41 - 16q^-40 + 27q^-39 + 25q^-38 - 29q^-37 - 42q^-36 + 36q^-35 + 49q^-34 - 31q^-33 - 62q^-32 + 32q^-31 + 65q^-30 - 26q^-29 - 68q^-28 + 22q^-27 + 65q^-26 - 14q^-25 - 61q^-24 + 6q^-23 + 56q^-22 - 45q^-20 - 12q^-19 + 40q^-18 + 12q^-17 - 23q^-16 - 22q^-15 + 19q^-14 + 14q^-13 - 4q^-12 - 16q^-11 + 3q^-10 + 7q^-9 + 4q^-8 - 6q^-7 + 2q^-4
4	q^-88 - 2q^-87 + q^-85 - q^-84 + 5q^-83 - 5q^-82 + q^-81 + q^-80 - 7q^-79 + 11q^-78 - 9q^-77 + 6q^-76 + 8q^-75 - 17q^-74 + 11q^-73 - 24q^-72 + 14q^-71 + 31q^-70 - 14q^-69 + 11q^-68 - 68q^-67 + q^-66 + 64q^-65 + 25q^-64 + 38q^-63 - 133q^-62 - 54q^-61 + 75q^-60 + 85q^-59 + 109q^-58 - 181q^-57 - 128q^-56 + 51q^-55 + 125q^-54 + 187q^-53 - 189q^-52 - 176q^-51 + 13q^-50 + 131q^-49 + 234q^-48 - 172q^-47 - 187q^-46 - 15q^-45 + 115q^-44 + 242q^-43 - 140q^-42 - 170q^-41 - 41q^-40 + 84q^-39 + 229q^-38 - 89q^-37 - 136q^-36 - 68q^-35 + 35q^-34 + 196q^-33 - 24q^-32 - 80q^-31 - 85q^-30 - 24q^-29 + 138q^-28 + 26q^-27 - 13q^-26 - 69q^-25 - 62q^-24 + 63q^-23 + 34q^-22 + 32q^-21 - 27q^-20 - 57q^-19 + 8q^-18 + 11q^-17 + 33q^-16 + 6q^-15 - 25q^-14 - 8q^-13 - 5q^-12 + 12q^-11 + 9q^-10 - 3q^-9 - 2q^-8 - 4q^-7 + 2q^-5 + q^-4
5	q^-130 - 2q^-129 + q^-127 - q^-126 + 2q^-125 + 3q^-124 - 3q^-123 - 3q^-122 + q^-121 - 4q^-120 + q^-119 + 9q^-118 + 2q^-117 - 3q^-116 - q^-115 - 10q^-114 - 9q^-113 + 7q^-112 + 14q^-111 + 13q^-110 + 9q^-109 - 16q^-108 - 33q^-107 - 25q^-106 + 8q^-105 + 47q^-104 + 59q^-103 + 19q^-102 - 54q^-101 - 105q^-100 - 73q^-99 + 42q^-98 + 152q^-97 + 157q^-96 + 6q^-95 - 195q^-94 - 258q^-93 - 93q^-92 + 201q^-91 + 377q^-90 + 218q^-89 - 188q^-88 - 469q^-87 - 353q^-86 + 107q^-85 + 557q^-84 + 502q^-83 - 47q^-82 - 579q^-81 - 611q^-80 - 70q^-79 + 598q^-78 + 709q^-77 + 131q^-76 - 570q^-75 - 750q^-74 - 217q^-73 + 549q^-72 + 779q^-71 + 251q^-70 - 511q^-69 - 775q^-68 - 289q^-67 + 478q^-66 + 767q^-65 + 301q^-64 - 433q^-63 - 741q^-62 - 326q^-61 + 388q^-60 + 707q^-59 + 344q^-58 - 321q^-57 - 660q^-56 - 365q^-55 + 237q^-54 + 594q^-53 + 393q^-52 - 149q^-51 - 509q^-50 - 389q^-49 + 31q^-48 + 399q^-47 + 400q^-46 + 51q^-45 - 284q^-44 - 335q^-43 - 152q^-42 + 146q^-41 + 304q^-40 + 178q^-39 - 43q^-38 - 183q^-37 - 206q^-36 - 64q^-35 + 124q^-34 + 158q^-33 + 102q^-32 - 5q^-31 - 125q^-30 - 125q^-29 - 26q^-28 + 45q^-27 + 97q^-26 + 79q^-25 - 11q^-24 - 70q^-23 - 52q^-22 - 34q^-21 + 23q^-20 + 56q^-19 + 30q^-18 - 8q^-17 - 15q^-16 - 28q^-15 - 14q^-14 + 12q^-13 + 15q^-12 + 4q^-11 + 6q^-10 - 6q^-9 - 6q^-8 - 2q^-7 + 2q^-6 + 2q^-4
6	q^-180 - 2q^-179 + q^-177 - q^-176 + 2q^-175 + 5q^-173 - 7q^-172 - 3q^-171 + 3q^-170 - 5q^-169 + 3q^-168 + 2q^-167 + 17q^-166 - 11q^-165 - 7q^-164 + 5q^-163 - 15q^-162 - 4q^-161 + 39q^-159 - 7q^-158 - 5q^-157 + 14q^-156 - 33q^-155 - 32q^-154 - 21q^-153 + 67q^-152 + 12q^-151 + 23q^-150 + 55q^-149 - 48q^-148 - 95q^-147 - 104q^-146 + 53q^-145 + 26q^-144 + 110q^-143 + 204q^-142 + 22q^-141 - 157q^-140 - 298q^-139 - 125q^-138 - 107q^-137 + 196q^-136 + 529q^-135 + 358q^-134 - 11q^-133 - 504q^-132 - 544q^-131 - 620q^-130 + 9q^-129 + 867q^-128 + 1022q^-127 + 604q^-126 - 381q^-125 - 971q^-124 - 1524q^-123 - 702q^-122 + 834q^-121 + 1699q^-120 + 1601q^-119 + 269q^-118 - 1001q^-117 - 2399q^-116 - 1740q^-115 + 297q^-114 + 1970q^-113 + 2490q^-112 + 1161q^-111 - 580q^-110 - 2826q^-109 - 2582q^-108 - 411q^-107 + 1814q^-106 + 2907q^-105 + 1822q^-104 - 46q^-103 - 2827q^-102 - 2958q^-101 - 905q^-100 + 1528q^-99 + 2937q^-98 + 2095q^-97 + 308q^-96 - 2667q^-95 - 3001q^-94 - 1121q^-93 + 1314q^-92 + 2817q^-91 + 2145q^-90 + 484q^-89 - 2479q^-88 - 2915q^-87 - 1231q^-86 + 1119q^-85 + 2629q^-84 + 2147q^-83 + 659q^-82 - 2189q^-81 - 2751q^-80 - 1381q^-79 + 783q^-78 + 2292q^-77 + 2121q^-76 + 952q^-75 - 1661q^-74 - 2423q^-73 - 1564q^-72 + 234q^-71 + 1696q^-70 + 1954q^-69 + 1303q^-68 - 878q^-67 - 1819q^-66 - 1602q^-65 - 395q^-64 + 844q^-63 + 1492q^-62 + 1470q^-61 - 38q^-60 - 950q^-59 - 1288q^-58 - 788q^-57 - 28q^-56 + 738q^-55 + 1213q^-54 + 489q^-53 - 86q^-52 - 629q^-51 - 691q^-50 - 527q^-49 - 9q^-48 + 590q^-47 + 475q^-46 + 368q^-45 + 16q^-44 - 225q^-43 - 470q^-42 - 348q^-41 + 14q^-40 + 117q^-39 + 298q^-38 + 258q^-37 + 165q^-36 - 128q^-35 - 237q^-34 - 168q^-33 - 140q^-32 + 30q^-31 + 133q^-30 + 207q^-29 + 79q^-28 - 20q^-27 - 66q^-26 - 122q^-25 - 83q^-24 - 19q^-23 + 73q^-22 + 62q^-21 + 44q^-20 + 21q^-19 - 22q^-18 - 41q^-17 - 37q^-16 - q^-15 + 8q^-14 + 13q^-13 + 16q^-12 + 9q^-11 - 3q^-10 - 8q^-9 - 4q^-8 - 2q^-7 + 2q^-4 + q^-3

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 127]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[14, 6, 15, 5], X[15, 20, 16, 1], > X[9, 16, 10, 17], X[11, 18, 12, 19], X[17, 10, 18, 11], X[19, 12, 20, 13], > X[6, 14, 7, 13], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 127]]
Out[3]=	GaussCode[-1, 10, -2, 1, 3, -9, -10, 2, -5, 7, -6, 8, 9, -3, -4, 5, -7, 6, -8, > 4]
In[4]:=	DTCode[Knot[10, 127]]
Out[4]=	DTCode[4, 8, -14, 2, 16, 18, -6, 20, 10, 12]
In[5]:=	br = BR[Knot[10, 127]]
Out[5]=	BR[3, {-1, -1, -1, -1, -1, -2, 1, 1, -2, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 127]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 127]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 127]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 127]][t]
Out[10]=	-3 4 6 2 3 7 - t + -- - - - 6 t + 4 t - t 2 t t
In[11]:=	Conway[Knot[10, 127]][z]
Out[11]=	2 4 6 1 + z - 2 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 127], Knot[10, 150], Knot[11, NonAlternating, 51]}
In[13]:=	{KnotDet[Knot[10, 127]], KnotSignature[Knot[10, 127]]}
Out[13]=	{29, -4}
In[14]:=	Jones[Knot[10, 127]][q]
Out[14]=	-10 2 3 5 5 5 4 2 2 q - -- + -- - -- + -- - -- + -- - -- + -- 9 8 7 6 5 4 3 2 q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 127]}
In[16]:=	A2Invariant[Knot[10, 127]][q]
Out[16]=	-30 -26 2 -20 3 -12 3 -8 2 q + q - --- - q - --- + q + --- + q + -- 22 18 10 6 q q q q
In[17]:=	HOMFLYPT[Knot[10, 127]][a, z]
Out[17]=	4 6 8 4 2 6 2 8 2 4 4 6 4 8 4 5 a - 6 a + 2 a + 7 a z - 9 a z + 3 a z + 2 a z - 5 a z + a z - 6 6 > a z
In[18]:=	Kauffman[Knot[10, 127]][a, z]
Out[18]=	4 6 8 5 7 9 11 4 2 6 2 5 a + 6 a + 2 a - 5 a z - 8 a z - 2 a z + a z - 9 a z - 14 a z - 8 2 10 2 12 2 5 3 7 3 9 3 11 3 > 2 a z + a z - 2 a z + 5 a z + 16 a z + 7 a z - 4 a z + 4 4 6 4 8 4 10 4 12 4 5 5 7 5 > 3 a z + 11 a z + 4 a z - 3 a z + a z - 3 a z - 10 a z - 9 5 11 5 6 6 8 6 10 6 5 7 7 7 > 5 a z + 2 a z - 4 a z - 2 a z + 2 a z + a z + 3 a z + 9 7 6 8 8 8 > 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 127]], Vassiliev[3][Knot[10, 127]]}
Out[19]=	{1, 1}
In[20]:=	Kh[Knot[10, 127]][q, t]
Out[20]=	-5 2 1 1 1 2 1 3 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q t q t q t q t q t q t q t 2 3 3 2 1 3 1 1 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- 13 4 11 4 11 3 9 3 9 2 7 2 7 5 q t q t q t q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 127], 2][q]
Out[21]=	-28 2 4 6 11 12 4 22 17 9 28 17 12 q - --- + --- - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + 27 25 24 22 21 20 19 18 17 16 15 14 q q q q q q q q q q q q 25 11 12 17 4 9 8 -6 4 2 -3 > --- - --- - --- + --- - -- - -- + -- + q - -- + -- + q 13 12 11 10 9 8 7 5 4 q q q q q q q q q

The Non Alternating Knot 10127

The Non Alternating Knot 10₁₂₇