The Alternating Knot 10₄₉

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X₃₈₄₉ X_5,14,6,15 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_13,6,14,7 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 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 3 3 3 / NotAvailable 1

Alexander Polynomial: 3t^-3 - 8t^-2 + 12t^-1 - 13 + 12t - 8t² + 3t³

Conway Polynomial: 1 + 7z² + 10z⁴ + 3z⁶

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

Determinant and Signature: {59, -6}

Jones Polynomial: q^-13 - 3q^-12 + 5q^-11 - 8q^-10 + 9q^-9 - 10q^-8 + 9q^-7 - 6q^-6 + 5q^-5 - 2q^-4 + q^-3

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

A2 (sl(3)) Invariant: q^-40 + q^-38 - q^-36 - 3q^-32 - 2q^-30 - q^-28 - 2q^-26 + 3q^-24 + 3q^-20 + 2q^-18 + 2q^-14 - q^-12 + q^-10

HOMFLY-PT Polynomial: a⁶ + 4a⁶z² + 4a⁶z⁴ + a⁶z⁶ + 5a⁸ + 12a⁸z² + 9a⁸z⁴ + 2a⁸z⁶ - 7a¹⁰ - 10a¹⁰z² - 3a¹⁰z⁴ + 2a¹² + a¹²z²

Kauffman Polynomial: - a⁶ + 4a⁶z² - 4a⁶z⁴ + a⁶z⁶ + 3a⁷z³ - 6a⁷z⁵ + 2a⁷z⁷ + 5a⁸ - 13a⁸z² + 15a⁸z⁴ - 11a⁸z⁶ + 3a⁸z⁸ - 9a⁹z + 22a⁹z³ - 18a⁹z⁵ + 3a⁹z⁷ + a⁹z⁹ + 7a¹⁰ - 20a¹⁰z² + 26a¹⁰z⁴ - 19a¹⁰z⁶ + 6a¹⁰z⁸ - 10a¹¹z + 24a¹¹z³ - 19a¹¹z⁵ + 5a¹¹z⁷ + a¹¹z⁹ + 2a¹² - 2a¹²z² + 2a¹²z⁴ - 3a¹²z⁶ + 3a¹²z⁸ + a¹³z³ - 4a¹³z⁵ + 4a¹³z⁷ - 4a¹⁴z⁴ + 4a¹⁴z⁶ + a¹⁵z - 4a¹⁵z³ + 3a¹⁵z⁵ - a¹⁶z² + a¹⁶z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {7, -16}

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=-6 is the signature of 10₄₉. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0

j = -5 1

j = -7 2 1

j = -9 3

j = -11 3 2

j = -13 6 3

j = -15 4 3

j = -17 5 6

j = -19 3 4

j = -21 2 5

j = -23 1 3

j = -25 2

j = -27 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-36 - 3q^-35 + q^-34 + 7q^-33 - 13q^-32 + 4q^-31 + 20q^-30 - 32q^-29 + 7q^-28 + 41q^-27 - 56q^-26 + 6q^-25 + 63q^-24 - 69q^-23 - 3q^-22 + 72q^-21 - 62q^-20 - 14q^-19 + 65q^-18 - 41q^-17 - 20q^-16 + 46q^-15 - 18q^-14 - 19q^-13 + 24q^-12 - 3q^-11 - 10q^-10 + 7q^-9 + q^-8 - 2q^-7 + q^-6

3 q^-69 - 3q^-68 + q^-67 + 3q^-66 + 2q^-65 - 8q^-64 - 2q^-63 + 13q^-62 + q^-61 - 18q^-60 - q^-59 + 30q^-58 - 7q^-57 - 43q^-56 + 11q^-55 + 72q^-54 - 25q^-53 - 100q^-52 + 23q^-51 + 150q^-50 - 30q^-49 - 183q^-48 + 10q^-47 + 228q^-46 + 3q^-45 - 246q^-44 - 33q^-43 + 259q^-42 + 57q^-41 - 250q^-40 - 85q^-39 + 232q^-38 + 111q^-37 - 209q^-36 - 125q^-35 + 165q^-34 + 150q^-33 - 138q^-32 - 146q^-31 + 83q^-30 + 158q^-29 - 55q^-28 - 133q^-27 + 6q^-26 + 121q^-25 + 11q^-24 - 83q^-23 - 35q^-22 + 62q^-21 + 29q^-20 - 28q^-19 - 30q^-18 + 16q^-17 + 17q^-16 - 3q^-15 - 10q^-14 + 2q^-13 + 3q^-12 + q^-11 - 2q^-10 + q^-9

4 q^-112 - 3q^-111 + q^-110 + 3q^-109 - 2q^-108 + 7q^-107 - 14q^-106 + 2q^-105 + 7q^-104 - 8q^-103 + 31q^-102 - 34q^-101 + 3q^-100 + 5q^-99 - 35q^-98 + 76q^-97 - 42q^-96 + 31q^-95 - 8q^-94 - 125q^-93 + 105q^-92 - 27q^-91 + 151q^-90 + 20q^-89 - 304q^-88 + 22q^-87 - 44q^-86 + 407q^-85 + 195q^-84 - 503q^-83 - 223q^-82 - 210q^-81 + 707q^-80 + 553q^-79 - 569q^-78 - 516q^-77 - 548q^-76 + 864q^-75 + 937q^-74 - 437q^-73 - 658q^-72 - 909q^-71 + 802q^-70 + 1146q^-69 - 216q^-68 - 589q^-67 - 1120q^-66 + 604q^-65 + 1125q^-64 - 12q^-63 - 385q^-62 - 1163q^-61 + 347q^-60 + 957q^-59 + 165q^-58 - 123q^-57 - 1086q^-56 + 57q^-55 + 683q^-54 + 312q^-53 + 178q^-52 - 892q^-51 - 215q^-50 + 322q^-49 + 352q^-48 + 449q^-47 - 565q^-46 - 346q^-45 - 41q^-44 + 219q^-43 + 558q^-42 - 195q^-41 - 266q^-40 - 246q^-39 - 3q^-38 + 439q^-37 + 50q^-36 - 75q^-35 - 229q^-34 - 137q^-33 + 212q^-32 + 92q^-31 + 50q^-30 - 100q^-29 - 121q^-28 + 52q^-27 + 36q^-26 + 55q^-25 - 15q^-24 - 50q^-23 + 6q^-22 + 19q^-20 + 3q^-19 - 11q^-18 + 2q^-17 - 2q^-16 + 3q^-15 + q^-14 - 2q^-13 + q^-12

5 q^-165 - 3q^-164 + q^-163 + 3q^-162 - 2q^-161 + 3q^-160 + q^-159 - 10q^-158 - 4q^-157 + 9q^-156 + 2q^-155 + 13q^-154 + 9q^-153 - 24q^-152 - 30q^-151 - 6q^-150 + 18q^-149 + 50q^-148 + 51q^-147 - 17q^-146 - 95q^-145 - 100q^-144 - 12q^-143 + 124q^-142 + 196q^-141 + 105q^-140 - 151q^-139 - 353q^-138 - 245q^-137 + 127q^-136 + 512q^-135 + 535q^-134 - 3q^-133 - 739q^-132 - 919q^-131 - 256q^-130 + 874q^-129 + 1485q^-128 + 736q^-127 - 967q^-126 - 2102q^-125 - 1446q^-124 + 813q^-123 + 2807q^-122 + 2381q^-121 - 493q^-120 - 3336q^-119 - 3461q^-118 - 202q^-117 + 3790q^-116 + 4554q^-115 + 993q^-114 - 3845q^-113 - 5513q^-112 - 2038q^-111 + 3736q^-110 + 6252q^-109 + 2924q^-108 - 3288q^-107 - 6656q^-106 - 3794q^-105 + 2788q^-104 + 6783q^-103 + 4356q^-102 - 2184q^-101 - 6647q^-100 - 4761q^-99 + 1666q^-98 + 6344q^-97 + 4912q^-96 - 1145q^-95 - 5942q^-94 - 4953q^-93 + 696q^-92 + 5453q^-91 + 4898q^-90 - 246q^-89 - 4888q^-88 - 4766q^-87 - 263q^-86 + 4245q^-85 + 4614q^-84 + 726q^-83 - 3476q^-82 - 4281q^-81 - 1305q^-80 + 2621q^-79 + 3936q^-78 + 1673q^-77 - 1679q^-76 - 3268q^-75 - 2083q^-74 + 735q^-73 + 2624q^-72 + 2105q^-71 + 111q^-70 - 1669q^-69 - 2058q^-68 - 792q^-67 + 881q^-66 + 1603q^-65 + 1184q^-64 + 5q^-63 - 1129q^-62 - 1290q^-61 - 541q^-60 + 444q^-59 + 1110q^-58 + 967q^-57 + 44q^-56 - 759q^-55 - 962q^-54 - 515q^-53 + 332q^-52 + 884q^-51 + 648q^-50 + 13q^-49 - 541q^-48 - 682q^-47 - 272q^-46 + 313q^-45 + 505q^-44 + 336q^-43 - 45q^-42 - 338q^-41 - 318q^-40 - 51q^-39 + 154q^-38 + 220q^-37 + 111q^-36 - 62q^-35 - 128q^-34 - 75q^-33 - 9q^-32 + 60q^-31 + 59q^-30 + 9q^-29 - 26q^-28 - 15q^-27 - 16q^-26 + 3q^-25 + 16q^-24 + 4q^-23 - 5q^-22 + q^-21 - 2q^-20 - 2q^-19 + 3q^-18 + q^-17 - 2q^-16 + q^-15

6 q^-228 - 3q^-227 + q^-226 + 3q^-225 - 2q^-224 + 3q^-223 - 3q^-222 + 5q^-221 - 16q^-220 - 2q^-219 + 19q^-218 - 5q^-217 + 17q^-216 - 3q^-215 + 10q^-214 - 62q^-213 - 24q^-212 + 48q^-211 + 2q^-210 + 68q^-209 + 30q^-208 + 37q^-207 - 184q^-206 - 122q^-205 + 49q^-204 + 6q^-203 + 211q^-202 + 195q^-201 + 171q^-200 - 396q^-199 - 408q^-198 - 125q^-197 - 95q^-196 + 490q^-195 + 695q^-194 + 648q^-193 - 604q^-192 - 1033q^-191 - 813q^-190 - 626q^-189 + 854q^-188 + 1869q^-187 + 2030q^-186 - 390q^-185 - 2075q^-184 - 2655q^-183 - 2451q^-182 + 812q^-181 + 4009q^-180 + 5371q^-179 + 1467q^-178 - 2976q^-177 - 6233q^-176 - 7100q^-175 - 1231q^-174 + 6459q^-173 + 11498q^-172 + 6957q^-171 - 1703q^-170 - 10696q^-169 - 15495q^-168 - 7621q^-167 + 6667q^-166 + 19110q^-165 + 16862q^-164 + 4383q^-163 - 12863q^-162 - 25605q^-161 - 18758q^-160 + 1736q^-159 + 24224q^-158 + 28304q^-157 + 15236q^-156 - 9664q^-155 - 32690q^-154 - 30942q^-153 - 7840q^-152 + 23614q^-151 + 36104q^-150 + 26646q^-149 - 1981q^-148 - 33663q^-147 - 38986q^-146 - 17672q^-145 + 18451q^-144 + 37709q^-143 + 33877q^-142 + 5916q^-141 - 30043q^-140 - 41087q^-139 - 23783q^-138 + 12603q^-137 + 35014q^-136 + 35995q^-135 + 10985q^-134 - 25252q^-133 - 39398q^-132 - 26057q^-131 + 8172q^-130 + 30977q^-129 + 35282q^-128 + 13767q^-127 - 20636q^-126 - 36359q^-125 - 26760q^-124 + 4206q^-123 + 26376q^-122 + 33658q^-121 + 16299q^-120 - 15125q^-119 - 32251q^-118 - 27214q^-117 - 894q^-116 + 20112q^-115 + 30886q^-114 + 19197q^-113 - 7635q^-112 - 25864q^-111 - 26535q^-110 - 6958q^-109 + 11417q^-108 + 25452q^-107 + 20875q^-106 + 992q^-105 - 16559q^-104 - 22733q^-103 - 11654q^-102 + 1555q^-101 + 16640q^-100 + 18862q^-99 + 7818q^-98 - 5916q^-97 - 14969q^-96 - 12041q^-95 - 6117q^-94 + 6273q^-93 + 12321q^-92 + 9640q^-91 + 2308q^-90 - 5412q^-89 - 7378q^-88 - 8292q^-87 - 1567q^-86 + 3853q^-85 + 6038q^-84 + 4841q^-83 + 1534q^-82 - 741q^-81 - 5031q^-80 - 3765q^-79 - 1923q^-78 + 569q^-77 + 2287q^-76 + 3080q^-75 + 3247q^-74 - 319q^-73 - 1479q^-72 - 2724q^-71 - 2337q^-70 - 1169q^-69 + 931q^-68 + 2971q^-67 + 1823q^-66 + 1146q^-65 - 709q^-64 - 1764q^-63 - 2196q^-62 - 1033q^-61 + 883q^-60 + 1172q^-59 + 1596q^-58 + 726q^-57 - 192q^-56 - 1213q^-55 - 1139q^-54 - 289q^-53 + 56q^-52 + 722q^-51 + 679q^-50 + 426q^-49 - 242q^-48 - 451q^-47 - 282q^-46 - 255q^-45 + 92q^-44 + 215q^-43 + 273q^-42 + 31q^-41 - 70q^-40 - 53q^-39 - 124q^-38 - 35q^-37 + 18q^-36 + 80q^-35 + 16q^-34 - 3q^-33 + 11q^-32 - 26q^-31 - 14q^-30 - 6q^-29 + 18q^-28 + q^-27 - 4q^-26 + 7q^-25 - 3q^-24 - 2q^-23 - 2q^-22 + 3q^-21 + q^-20 - 2q^-19 + q^-18

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, 49]]

Out[2]=
PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 14, 6, 15], 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[13, 6, 14, 7], X[7, 2, 8, 3]]

In[3]:=
GaussCode[Knot[10, 49]]

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, 49]]

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

In[5]:=
br = BR[Knot[10, 49]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 49]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 49]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 49]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 49]][t]

Out[10]=
3 8 12 2 3 -13 + -- - -- + -- + 12 t - 8 t + 3 t 3 2 t t t

In[11]:=
Conway[Knot[10, 49]][z]

Out[11]=
2 4 6 1 + 7 z + 10 z + 3 z

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

Out[12]=
{Knot[10, 49]}

In[13]:=
{KnotDet[Knot[10, 49]], KnotSignature[Knot[10, 49]]}

Out[13]=
{59, -6}

In[14]:=
Jones[Knot[10, 49]][q]

Out[14]=
-13 3 5 8 9 10 9 6 5 2 -3 q - --- + --- - --- + -- - -- + -- - -- + -- - -- + q 12 11 10 9 8 7 6 5 4 q 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, 49]}

In[16]:=
A2Invariant[Knot[10, 49]][q]

Out[16]=
-40 -38 -36 3 2 -28 2 3 3 2 2 -12 q + q - q - --- - --- - q - --- + --- + --- + --- + --- - q + 32 30 26 24 20 18 14 q q q q q q q -10 > q

In[17]:=
HOMFLYPT[Knot[10, 49]][a, z]

Out[17]=
6 8 10 12 6 2 8 2 10 2 12 2 6 4 a + 5 a - 7 a + 2 a + 4 a z + 12 a z - 10 a z + a z + 4 a z + 8 4 10 4 6 6 8 6 > 9 a z - 3 a z + a z + 2 a z

In[18]:=
Kauffman[Knot[10, 49]][a, z]

Out[18]=
6 8 10 12 9 11 15 6 2 8 2 -a + 5 a + 7 a + 2 a - 9 a z - 10 a z + a z + 4 a z - 13 a z - 10 2 12 2 16 2 7 3 9 3 11 3 13 3 > 20 a z - 2 a z - a z + 3 a z + 22 a z + 24 a z + a z - 15 3 6 4 8 4 10 4 12 4 14 4 16 4 > 4 a z - 4 a z + 15 a z + 26 a z + 2 a z - 4 a z + a z - 7 5 9 5 11 5 13 5 15 5 6 6 8 6 > 6 a z - 18 a z - 19 a z - 4 a z + 3 a z + a z - 11 a z - 10 6 12 6 14 6 7 7 9 7 11 7 13 7 > 19 a z - 3 a z + 4 a z + 2 a z + 3 a z + 5 a z + 4 a z + 8 8 10 8 12 8 9 9 11 9 > 3 a z + 6 a z + 3 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 49]], Vassiliev[3][Knot[10, 49]]}

Out[19]=
{7, -16}

In[20]:=
Kh[Knot[10, 49]][q, t]

Out[20]=
-7 -5 1 2 1 3 2 5 3 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 27 10 25 9 23 9 23 8 21 8 21 7 19 7 q t q t q t q t q t q t q t 4 5 6 4 3 6 3 3 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 19 6 17 6 17 5 15 5 15 4 13 4 13 3 11 3 q t q t q t q t q t q t q t q t 2 3 2 > ------ + ----- + ---- 11 2 9 2 7 q t q t q t

In[21]:=
ColouredJones[Knot[10, 49], 2][q]

Out[21]=
-36 3 -34 7 13 4 20 32 7 41 56 6 63 q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- + --- + --- - 35 33 32 31 30 29 28 27 26 25 24 q q q q q q q q q q q 69 3 72 62 14 65 41 20 46 18 19 24 > --- - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 23 22 21 20 19 18 17 16 15 14 13 12 q q q q q q q q q q q q 3 10 7 -8 2 -6 > --- - --- + -- + q - -- + q 11 10 9 7 q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₄₉

10₄₈
10₅₀

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-36 - 3q^-35 + q^-34 + 7q^-33 - 13q^-32 + 4q^-31 + 20q^-30 - 32q^-29 + 7q^-28 + 41q^-27 - 56q^-26 + 6q^-25 + 63q^-24 - 69q^-23 - 3q^-22 + 72q^-21 - 62q^-20 - 14q^-19 + 65q^-18 - 41q^-17 - 20q^-16 + 46q^-15 - 18q^-14 - 19q^-13 + 24q^-12 - 3q^-11 - 10q^-10 + 7q^-9 + q^-8 - 2q^-7 + q^-6
3	q^-69 - 3q^-68 + q^-67 + 3q^-66 + 2q^-65 - 8q^-64 - 2q^-63 + 13q^-62 + q^-61 - 18q^-60 - q^-59 + 30q^-58 - 7q^-57 - 43q^-56 + 11q^-55 + 72q^-54 - 25q^-53 - 100q^-52 + 23q^-51 + 150q^-50 - 30q^-49 - 183q^-48 + 10q^-47 + 228q^-46 + 3q^-45 - 246q^-44 - 33q^-43 + 259q^-42 + 57q^-41 - 250q^-40 - 85q^-39 + 232q^-38 + 111q^-37 - 209q^-36 - 125q^-35 + 165q^-34 + 150q^-33 - 138q^-32 - 146q^-31 + 83q^-30 + 158q^-29 - 55q^-28 - 133q^-27 + 6q^-26 + 121q^-25 + 11q^-24 - 83q^-23 - 35q^-22 + 62q^-21 + 29q^-20 - 28q^-19 - 30q^-18 + 16q^-17 + 17q^-16 - 3q^-15 - 10q^-14 + 2q^-13 + 3q^-12 + q^-11 - 2q^-10 + q^-9
4	q^-112 - 3q^-111 + q^-110 + 3q^-109 - 2q^-108 + 7q^-107 - 14q^-106 + 2q^-105 + 7q^-104 - 8q^-103 + 31q^-102 - 34q^-101 + 3q^-100 + 5q^-99 - 35q^-98 + 76q^-97 - 42q^-96 + 31q^-95 - 8q^-94 - 125q^-93 + 105q^-92 - 27q^-91 + 151q^-90 + 20q^-89 - 304q^-88 + 22q^-87 - 44q^-86 + 407q^-85 + 195q^-84 - 503q^-83 - 223q^-82 - 210q^-81 + 707q^-80 + 553q^-79 - 569q^-78 - 516q^-77 - 548q^-76 + 864q^-75 + 937q^-74 - 437q^-73 - 658q^-72 - 909q^-71 + 802q^-70 + 1146q^-69 - 216q^-68 - 589q^-67 - 1120q^-66 + 604q^-65 + 1125q^-64 - 12q^-63 - 385q^-62 - 1163q^-61 + 347q^-60 + 957q^-59 + 165q^-58 - 123q^-57 - 1086q^-56 + 57q^-55 + 683q^-54 + 312q^-53 + 178q^-52 - 892q^-51 - 215q^-50 + 322q^-49 + 352q^-48 + 449q^-47 - 565q^-46 - 346q^-45 - 41q^-44 + 219q^-43 + 558q^-42 - 195q^-41 - 266q^-40 - 246q^-39 - 3q^-38 + 439q^-37 + 50q^-36 - 75q^-35 - 229q^-34 - 137q^-33 + 212q^-32 + 92q^-31 + 50q^-30 - 100q^-29 - 121q^-28 + 52q^-27 + 36q^-26 + 55q^-25 - 15q^-24 - 50q^-23 + 6q^-22 + 19q^-20 + 3q^-19 - 11q^-18 + 2q^-17 - 2q^-16 + 3q^-15 + q^-14 - 2q^-13 + q^-12
5	q^-165 - 3q^-164 + q^-163 + 3q^-162 - 2q^-161 + 3q^-160 + q^-159 - 10q^-158 - 4q^-157 + 9q^-156 + 2q^-155 + 13q^-154 + 9q^-153 - 24q^-152 - 30q^-151 - 6q^-150 + 18q^-149 + 50q^-148 + 51q^-147 - 17q^-146 - 95q^-145 - 100q^-144 - 12q^-143 + 124q^-142 + 196q^-141 + 105q^-140 - 151q^-139 - 353q^-138 - 245q^-137 + 127q^-136 + 512q^-135 + 535q^-134 - 3q^-133 - 739q^-132 - 919q^-131 - 256q^-130 + 874q^-129 + 1485q^-128 + 736q^-127 - 967q^-126 - 2102q^-125 - 1446q^-124 + 813q^-123 + 2807q^-122 + 2381q^-121 - 493q^-120 - 3336q^-119 - 3461q^-118 - 202q^-117 + 3790q^-116 + 4554q^-115 + 993q^-114 - 3845q^-113 - 5513q^-112 - 2038q^-111 + 3736q^-110 + 6252q^-109 + 2924q^-108 - 3288q^-107 - 6656q^-106 - 3794q^-105 + 2788q^-104 + 6783q^-103 + 4356q^-102 - 2184q^-101 - 6647q^-100 - 4761q^-99 + 1666q^-98 + 6344q^-97 + 4912q^-96 - 1145q^-95 - 5942q^-94 - 4953q^-93 + 696q^-92 + 5453q^-91 + 4898q^-90 - 246q^-89 - 4888q^-88 - 4766q^-87 - 263q^-86 + 4245q^-85 + 4614q^-84 + 726q^-83 - 3476q^-82 - 4281q^-81 - 1305q^-80 + 2621q^-79 + 3936q^-78 + 1673q^-77 - 1679q^-76 - 3268q^-75 - 2083q^-74 + 735q^-73 + 2624q^-72 + 2105q^-71 + 111q^-70 - 1669q^-69 - 2058q^-68 - 792q^-67 + 881q^-66 + 1603q^-65 + 1184q^-64 + 5q^-63 - 1129q^-62 - 1290q^-61 - 541q^-60 + 444q^-59 + 1110q^-58 + 967q^-57 + 44q^-56 - 759q^-55 - 962q^-54 - 515q^-53 + 332q^-52 + 884q^-51 + 648q^-50 + 13q^-49 - 541q^-48 - 682q^-47 - 272q^-46 + 313q^-45 + 505q^-44 + 336q^-43 - 45q^-42 - 338q^-41 - 318q^-40 - 51q^-39 + 154q^-38 + 220q^-37 + 111q^-36 - 62q^-35 - 128q^-34 - 75q^-33 - 9q^-32 + 60q^-31 + 59q^-30 + 9q^-29 - 26q^-28 - 15q^-27 - 16q^-26 + 3q^-25 + 16q^-24 + 4q^-23 - 5q^-22 + q^-21 - 2q^-20 - 2q^-19 + 3q^-18 + q^-17 - 2q^-16 + q^-15
6	q^-228 - 3q^-227 + q^-226 + 3q^-225 - 2q^-224 + 3q^-223 - 3q^-222 + 5q^-221 - 16q^-220 - 2q^-219 + 19q^-218 - 5q^-217 + 17q^-216 - 3q^-215 + 10q^-214 - 62q^-213 - 24q^-212 + 48q^-211 + 2q^-210 + 68q^-209 + 30q^-208 + 37q^-207 - 184q^-206 - 122q^-205 + 49q^-204 + 6q^-203 + 211q^-202 + 195q^-201 + 171q^-200 - 396q^-199 - 408q^-198 - 125q^-197 - 95q^-196 + 490q^-195 + 695q^-194 + 648q^-193 - 604q^-192 - 1033q^-191 - 813q^-190 - 626q^-189 + 854q^-188 + 1869q^-187 + 2030q^-186 - 390q^-185 - 2075q^-184 - 2655q^-183 - 2451q^-182 + 812q^-181 + 4009q^-180 + 5371q^-179 + 1467q^-178 - 2976q^-177 - 6233q^-176 - 7100q^-175 - 1231q^-174 + 6459q^-173 + 11498q^-172 + 6957q^-171 - 1703q^-170 - 10696q^-169 - 15495q^-168 - 7621q^-167 + 6667q^-166 + 19110q^-165 + 16862q^-164 + 4383q^-163 - 12863q^-162 - 25605q^-161 - 18758q^-160 + 1736q^-159 + 24224q^-158 + 28304q^-157 + 15236q^-156 - 9664q^-155 - 32690q^-154 - 30942q^-153 - 7840q^-152 + 23614q^-151 + 36104q^-150 + 26646q^-149 - 1981q^-148 - 33663q^-147 - 38986q^-146 - 17672q^-145 + 18451q^-144 + 37709q^-143 + 33877q^-142 + 5916q^-141 - 30043q^-140 - 41087q^-139 - 23783q^-138 + 12603q^-137 + 35014q^-136 + 35995q^-135 + 10985q^-134 - 25252q^-133 - 39398q^-132 - 26057q^-131 + 8172q^-130 + 30977q^-129 + 35282q^-128 + 13767q^-127 - 20636q^-126 - 36359q^-125 - 26760q^-124 + 4206q^-123 + 26376q^-122 + 33658q^-121 + 16299q^-120 - 15125q^-119 - 32251q^-118 - 27214q^-117 - 894q^-116 + 20112q^-115 + 30886q^-114 + 19197q^-113 - 7635q^-112 - 25864q^-111 - 26535q^-110 - 6958q^-109 + 11417q^-108 + 25452q^-107 + 20875q^-106 + 992q^-105 - 16559q^-104 - 22733q^-103 - 11654q^-102 + 1555q^-101 + 16640q^-100 + 18862q^-99 + 7818q^-98 - 5916q^-97 - 14969q^-96 - 12041q^-95 - 6117q^-94 + 6273q^-93 + 12321q^-92 + 9640q^-91 + 2308q^-90 - 5412q^-89 - 7378q^-88 - 8292q^-87 - 1567q^-86 + 3853q^-85 + 6038q^-84 + 4841q^-83 + 1534q^-82 - 741q^-81 - 5031q^-80 - 3765q^-79 - 1923q^-78 + 569q^-77 + 2287q^-76 + 3080q^-75 + 3247q^-74 - 319q^-73 - 1479q^-72 - 2724q^-71 - 2337q^-70 - 1169q^-69 + 931q^-68 + 2971q^-67 + 1823q^-66 + 1146q^-65 - 709q^-64 - 1764q^-63 - 2196q^-62 - 1033q^-61 + 883q^-60 + 1172q^-59 + 1596q^-58 + 726q^-57 - 192q^-56 - 1213q^-55 - 1139q^-54 - 289q^-53 + 56q^-52 + 722q^-51 + 679q^-50 + 426q^-49 - 242q^-48 - 451q^-47 - 282q^-46 - 255q^-45 + 92q^-44 + 215q^-43 + 273q^-42 + 31q^-41 - 70q^-40 - 53q^-39 - 124q^-38 - 35q^-37 + 18q^-36 + 80q^-35 + 16q^-34 - 3q^-33 + 11q^-32 - 26q^-31 - 14q^-30 - 6q^-29 + 18q^-28 + q^-27 - 4q^-26 + 7q^-25 - 3q^-24 - 2q^-23 - 2q^-22 + 3q^-21 + q^-20 - 2q^-19 + q^-18

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 49]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 14, 6, 15], 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[13, 6, 14, 7], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 49]]
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, 49]]
Out[4]=	DTCode[4, 8, 14, 2, 16, 18, 6, 20, 10, 12]
In[5]:=	br = BR[Knot[10, 49]]
Out[5]=	BR[4, {-1, -1, -1, -1, 2, -1, -3, -2, -2, -2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 49]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 49]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 49]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 3, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 49]][t]
Out[10]=	3 8 12 2 3 -13 + -- - -- + -- + 12 t - 8 t + 3 t 3 2 t t t
In[11]:=	Conway[Knot[10, 49]][z]
Out[11]=	2 4 6 1 + 7 z + 10 z + 3 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 49]}
In[13]:=	{KnotDet[Knot[10, 49]], KnotSignature[Knot[10, 49]]}
Out[13]=	{59, -6}
In[14]:=	Jones[Knot[10, 49]][q]
Out[14]=	-13 3 5 8 9 10 9 6 5 2 -3 q - --- + --- - --- + -- - -- + -- - -- + -- - -- + q 12 11 10 9 8 7 6 5 4 q 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, 49]}
In[16]:=	A2Invariant[Knot[10, 49]][q]
Out[16]=	-40 -38 -36 3 2 -28 2 3 3 2 2 -12 q + q - q - --- - --- - q - --- + --- + --- + --- + --- - q + 32 30 26 24 20 18 14 q q q q q q q -10 > q
In[17]:=	HOMFLYPT[Knot[10, 49]][a, z]
Out[17]=	6 8 10 12 6 2 8 2 10 2 12 2 6 4 a + 5 a - 7 a + 2 a + 4 a z + 12 a z - 10 a z + a z + 4 a z + 8 4 10 4 6 6 8 6 > 9 a z - 3 a z + a z + 2 a z
In[18]:=	Kauffman[Knot[10, 49]][a, z]
Out[18]=	6 8 10 12 9 11 15 6 2 8 2 -a + 5 a + 7 a + 2 a - 9 a z - 10 a z + a z + 4 a z - 13 a z - 10 2 12 2 16 2 7 3 9 3 11 3 13 3 > 20 a z - 2 a z - a z + 3 a z + 22 a z + 24 a z + a z - 15 3 6 4 8 4 10 4 12 4 14 4 16 4 > 4 a z - 4 a z + 15 a z + 26 a z + 2 a z - 4 a z + a z - 7 5 9 5 11 5 13 5 15 5 6 6 8 6 > 6 a z - 18 a z - 19 a z - 4 a z + 3 a z + a z - 11 a z - 10 6 12 6 14 6 7 7 9 7 11 7 13 7 > 19 a z - 3 a z + 4 a z + 2 a z + 3 a z + 5 a z + 4 a z + 8 8 10 8 12 8 9 9 11 9 > 3 a z + 6 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 49]], Vassiliev[3][Knot[10, 49]]}
Out[19]=	{7, -16}
In[20]:=	Kh[Knot[10, 49]][q, t]
Out[20]=	-7 -5 1 2 1 3 2 5 3 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 27 10 25 9 23 9 23 8 21 8 21 7 19 7 q t q t q t q t q t q t q t 4 5 6 4 3 6 3 3 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 19 6 17 6 17 5 15 5 15 4 13 4 13 3 11 3 q t q t q t q t q t q t q t q t 2 3 2 > ------ + ----- + ---- 11 2 9 2 7 q t q t q t
In[21]:=	ColouredJones[Knot[10, 49], 2][q]
Out[21]=	-36 3 -34 7 13 4 20 32 7 41 56 6 63 q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- + --- + --- - 35 33 32 31 30 29 28 27 26 25 24 q q q q q q q q q q q 69 3 72 62 14 65 41 20 46 18 19 24 > --- - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 23 22 21 20 19 18 17 16 15 14 13 12 q q q q q q q q q q q q 3 10 7 -8 2 -6 > --- - --- + -- + q - -- + q 11 10 9 7 q q q q

The Alternating Knot 1049

The Alternating Knot 10₄₉