The Alternating Knot 10₁₀₃

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Acknowledgement

KnotPlot

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

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

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

Alexander Polynomial: 2t^-3 - 8t^-2 + 17t^-1 - 21 + 17t - 8t² + 2t³

Conway Polynomial: 1 + 3z² + 4z⁴ + 2z⁶

Other knots with the same Alexander/Conway Polynomial: {10₄₀, ...}

Determinant and Signature: {75, -2}

Jones Polynomial: - q^-8 + 3q^-7 - 6q^-6 + 9q^-5 - 12q^-4 + 13q^-3 - 11q^-2 + 10q^-1 - 6 + 3q - q²

Other knots (up to mirrors) with the same Jones Polynomial: {10₄₀, ...}

A2 (sl(3)) Invariant: - q^-24 + q^-22 - q^-20 - q^-18 + 2q^-16 - 3q^-14 + q^-12 + q^-8 + 4q^-6 - q^-4 + 3q^-2 - 1 - q² + q⁴ - q⁶

HOMFLY-PT Polynomial: - 1 - 2z² - z⁴ + 3a² + 4a²z² + 3a²z⁴ + a²z⁶ + 3a⁴z² + 3a⁴z⁴ + a⁴z⁶ - a⁶ - 2a⁶z² - a⁶z⁴

Kauffman Polynomial: a^-1z - 2a^-1z³ + a^-1z⁵ - 1 + 3z² - 6z⁴ + 3z⁶ + az - 2az³ - 5az⁵ + 4az⁷ - 3a² + 2a²z² - 5a²z⁶ + 4a²z⁸ - 2a³z + 9a³z³ - 9a³z⁵ + 2a³z⁷ + 2a³z⁹ - 8a⁴z² + 25a⁴z⁴ - 23a⁴z⁶ + 9a⁴z⁸ - 6a⁵z + 21a⁵z³ - 16a⁵z⁵ + 3a⁵z⁷ + 2a⁵z⁹ + a⁶ - 6a⁶z² + 13a⁶z⁴ - 12a⁶z⁶ + 5a⁶z⁸ - 4a⁷z + 10a⁷z³ - 12a⁷z⁵ + 5a⁷z⁷ + a⁸z² - 6a⁸z⁴ + 3a⁸z⁶ - 2a⁹z³ + a⁹z⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {3, -4}

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 10₁₀₃. 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 r = 2 r = 3

j = 5 1

j = 3 2

j = 1 4 1

j = -1 6 2

j = -3 6 5

j = -5 7 5

j = -7 5 6

j = -9 4 7

j = -11 2 5

j = -13 1 4

j = -15 2

j = -17 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-23 - 3q^-22 + 2q^-21 + 7q^-20 - 17q^-19 + 6q^-18 + 30q^-17 - 46q^-16 - 3q^-15 + 75q^-14 - 72q^-13 - 30q^-12 + 122q^-11 - 81q^-10 - 61q^-9 + 141q^-8 - 67q^-7 - 75q^-6 + 123q^-5 - 35q^-4 - 69q^-3 + 78q^-2 - 7q^-1 - 44 + 32q + 3q² - 16q³ + 8q⁴ + q⁵ - 3q⁶ + q⁷

3 - q^-45 + 3q^-44 - 2q^-43 - 3q^-42 + 2q^-41 + 10q^-40 - 8q^-39 - 22q^-38 + 15q^-37 + 49q^-36 - 20q^-35 - 91q^-34 + 4q^-33 + 155q^-32 + 35q^-31 - 218q^-30 - 115q^-29 + 273q^-28 + 225q^-27 - 301q^-26 - 355q^-25 + 299q^-24 + 482q^-23 - 262q^-22 - 602q^-21 + 212q^-20 + 688q^-19 - 145q^-18 - 739q^-17 + 65q^-16 + 769q^-15 - q^-14 - 742q^-13 - 90q^-12 + 705q^-11 + 147q^-10 - 603q^-9 - 224q^-8 + 505q^-7 + 249q^-6 - 362q^-5 - 268q^-4 + 247q^-3 + 232q^-2 - 129q^-1 - 190 + 58q + 130q² - 17q³ - 76q⁴ + 40q⁶ + 2q⁷ - 20q⁸ + 10q¹⁰ - 2q¹¹ - 3q¹² - q¹³ + 3q¹⁴ - q¹⁵

4 q^-74 - 3q^-73 + 2q^-72 + 3q^-71 - 6q^-70 + 5q^-69 - 9q^-68 + 14q^-67 + 12q^-66 - 39q^-65 + 5q^-64 - 20q^-63 + 73q^-62 + 74q^-61 - 126q^-60 - 77q^-59 - 121q^-58 + 216q^-57 + 354q^-56 - 124q^-55 - 278q^-54 - 592q^-53 + 181q^-52 + 908q^-51 + 378q^-50 - 198q^-49 - 1493q^-48 - 569q^-47 + 1195q^-46 + 1428q^-45 + 794q^-44 - 2182q^-43 - 2028q^-42 + 541q^-41 + 2318q^-40 + 2599q^-39 - 1987q^-38 - 3412q^-37 - 908q^-36 + 2426q^-35 + 4375q^-34 - 1071q^-33 - 4093q^-32 - 2375q^-31 + 1887q^-30 + 5474q^-29 - 19q^-28 - 4083q^-27 - 3391q^-26 + 1109q^-25 + 5841q^-24 + 902q^-23 - 3591q^-22 - 3945q^-21 + 193q^-20 + 5532q^-19 + 1732q^-18 - 2574q^-17 - 4017q^-16 - 909q^-15 + 4453q^-14 + 2315q^-13 - 1064q^-12 - 3361q^-11 - 1864q^-10 + 2680q^-9 + 2211q^-8 + 391q^-7 - 1988q^-6 - 2048q^-5 + 904q^-4 + 1331q^-3 + 1017q^-2 - 581q^-1 - 1367 - 49q + 349q² + 745q³ + 104q⁴ - 530q⁵ - 145q⁶ - 89q⁷ + 266q⁸ + 139q⁹ - 118q¹⁰ - 11q¹¹ - 87q¹² + 51q¹³ + 37q¹⁴ - 31q¹⁵ + 23q¹⁶ - 22q¹⁷ + 10q¹⁸ + 4q¹⁹ - 13q²⁰ + 8q²¹ - 3q²² + 3q²³ + q²⁴ - 3q²⁵ + q²⁶

5 - q^-110 + 3q^-109 - 2q^-108 - 3q^-107 + 6q^-106 - q^-105 - 6q^-104 + 3q^-103 - 3q^-102 - 2q^-101 + 25q^-100 + 12q^-99 - 37q^-98 - 41q^-97 - 21q^-96 + 36q^-95 + 125q^-94 + 109q^-93 - 77q^-92 - 266q^-91 - 261q^-90 + 5q^-89 + 446q^-88 + 640q^-87 + 241q^-86 - 581q^-85 - 1195q^-84 - 877q^-83 + 431q^-82 + 1817q^-81 + 1989q^-80 + 373q^-79 - 2172q^-78 - 3525q^-77 - 2036q^-76 + 1705q^-75 + 4988q^-74 + 4713q^-73 + 105q^-72 - 5816q^-71 - 7946q^-70 - 3476q^-69 + 5126q^-68 + 11063q^-67 + 8308q^-66 - 2526q^-65 - 13171q^-64 - 13857q^-63 - 2099q^-62 + 13533q^-61 + 19270q^-60 + 8232q^-59 - 11847q^-58 - 23661q^-57 - 15075q^-56 + 8385q^-55 + 26442q^-54 + 21653q^-53 - 3652q^-52 - 27508q^-51 - 27347q^-50 - 1485q^-49 + 27167q^-48 + 31610q^-47 + 6436q^-46 - 25775q^-45 - 34576q^-44 - 10754q^-43 + 23911q^-42 + 36397q^-41 + 14232q^-40 - 21810q^-39 - 37280q^-38 - 17184q^-37 + 19581q^-36 + 37727q^-35 + 19591q^-34 - 17209q^-33 - 37381q^-32 - 21994q^-31 + 14253q^-30 + 36701q^-29 + 24132q^-28 - 10731q^-27 - 34797q^-26 - 26227q^-25 + 6247q^-24 + 31996q^-23 + 27591q^-22 - 1273q^-21 - 27448q^-20 - 28061q^-19 - 4042q^-18 + 21869q^-17 + 26814q^-16 + 8685q^-15 - 15026q^-14 - 23999q^-13 - 12153q^-12 + 8337q^-11 + 19434q^-10 + 13593q^-9 - 2144q^-8 - 14015q^-7 - 13154q^-6 - 2221q^-5 + 8472q^-4 + 10896q^-3 + 4744q^-2 - 3775q^-1 - 7834 - 5289q + 529q² + 4662q³ + 4518q⁴ + 1184q⁵ - 2139q⁶ - 3127q⁷ - 1671q⁸ + 564q⁹ + 1756q¹⁰ + 1400q¹¹ + 192q¹² - 761q¹³ - 899q¹⁴ - 385q¹⁵ + 219q¹⁶ + 459q¹⁷ + 298q¹⁸ - q¹⁹ - 161q²⁰ - 174q²¹ - 65q²² + 56q²³ + 72q²⁴ + 27q²⁵ + 8q²⁶ - 12q²⁷ - 31q²⁸ - 5q²⁹ + 11q³⁰ - 6q³¹ + 6q³² + 9q³³ - 5q³⁴ - 3q³⁵ + 3q³⁶ - 3q³⁷ - q³⁸ + 3q³⁹ - q⁴⁰

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, 103]]

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

In[3]:=
GaussCode[Knot[10, 103]]

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

In[4]:=
DTCode[Knot[10, 103]]

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

In[5]:=
br = BR[Knot[10, 103]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 103]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 103]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 103]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 103]][t]

Out[10]=
2 8 17 2 3 -21 + -- - -- + -- + 17 t - 8 t + 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 103]][z]

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

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

Out[12]=
{Knot[10, 40], Knot[10, 103]}

In[13]:=
{KnotDet[Knot[10, 103]], KnotSignature[Knot[10, 103]]}

Out[13]=
{75, -2}

In[14]:=
Jones[Knot[10, 103]][q]

Out[14]=
-8 3 6 9 12 13 11 10 2 -6 - q + -- - -- + -- - -- + -- - -- + -- + 3 q - 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, 40], Knot[10, 103]}

In[16]:=
A2Invariant[Knot[10, 103]][q]

Out[16]=
-24 -22 -20 -18 2 3 -12 -8 4 -4 3 2 -1 - q + q - q - q + --- - --- + q + q + -- - q + -- - q + 16 14 6 2 q q q q 4 6 > q - q

In[17]:=
HOMFLYPT[Knot[10, 103]][a, z]

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

In[18]:=
Kauffman[Knot[10, 103]][a, z]

Out[18]=
2 6 z 3 5 7 2 2 2 -1 - 3 a + a + - + a z - 2 a z - 6 a z - 4 a z + 3 z + 2 a z - a 3 4 2 6 2 8 2 2 z 3 3 3 5 3 7 3 > 8 a z - 6 a z + a z - ---- - 2 a z + 9 a z + 21 a z + 10 a z - a 5 9 3 4 4 4 6 4 8 4 z 5 3 5 > 2 a z - 6 z + 25 a z + 13 a z - 6 a z + -- - 5 a z - 9 a z - a 5 5 7 5 9 5 6 2 6 4 6 6 6 > 16 a z - 12 a z + a z + 3 z - 5 a z - 23 a z - 12 a z + 8 6 7 3 7 5 7 7 7 2 8 4 8 > 3 a z + 4 a z + 2 a z + 3 a z + 5 a z + 4 a z + 9 a z + 6 8 3 9 5 9 > 5 a z + 2 a z + 2 a z

In[19]:=
{Vassiliev[2][Knot[10, 103]], Vassiliev[3][Knot[10, 103]]}

Out[19]=
{3, -4}

In[20]:=
Kh[Knot[10, 103]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 103], 2][q]

Out[21]=
-23 3 2 7 17 6 30 46 3 75 72 30 -44 + q - --- + --- + --- - --- + --- + --- - --- - --- + --- - --- - --- + 22 21 20 19 18 17 16 15 14 13 12 q q q q q q q q q q q 122 81 61 141 67 75 123 35 69 78 7 2 > --- - --- - -- + --- - -- - -- + --- - -- - -- + -- - - + 32 q + 3 q - 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 3 4 5 6 7 > 16 q + 8 q + q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₀₃

10₁₀₂
10₁₀₄

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-23 - 3q^-22 + 2q^-21 + 7q^-20 - 17q^-19 + 6q^-18 + 30q^-17 - 46q^-16 - 3q^-15 + 75q^-14 - 72q^-13 - 30q^-12 + 122q^-11 - 81q^-10 - 61q^-9 + 141q^-8 - 67q^-7 - 75q^-6 + 123q^-5 - 35q^-4 - 69q^-3 + 78q^-2 - 7q^-1 - 44 + 32q + 3q² - 16q³ + 8q⁴ + q⁵ - 3q⁶ + q⁷
3	- q^-45 + 3q^-44 - 2q^-43 - 3q^-42 + 2q^-41 + 10q^-40 - 8q^-39 - 22q^-38 + 15q^-37 + 49q^-36 - 20q^-35 - 91q^-34 + 4q^-33 + 155q^-32 + 35q^-31 - 218q^-30 - 115q^-29 + 273q^-28 + 225q^-27 - 301q^-26 - 355q^-25 + 299q^-24 + 482q^-23 - 262q^-22 - 602q^-21 + 212q^-20 + 688q^-19 - 145q^-18 - 739q^-17 + 65q^-16 + 769q^-15 - q^-14 - 742q^-13 - 90q^-12 + 705q^-11 + 147q^-10 - 603q^-9 - 224q^-8 + 505q^-7 + 249q^-6 - 362q^-5 - 268q^-4 + 247q^-3 + 232q^-2 - 129q^-1 - 190 + 58q + 130q² - 17q³ - 76q⁴ + 40q⁶ + 2q⁷ - 20q⁸ + 10q¹⁰ - 2q¹¹ - 3q¹² - q¹³ + 3q¹⁴ - q¹⁵
4	q^-74 - 3q^-73 + 2q^-72 + 3q^-71 - 6q^-70 + 5q^-69 - 9q^-68 + 14q^-67 + 12q^-66 - 39q^-65 + 5q^-64 - 20q^-63 + 73q^-62 + 74q^-61 - 126q^-60 - 77q^-59 - 121q^-58 + 216q^-57 + 354q^-56 - 124q^-55 - 278q^-54 - 592q^-53 + 181q^-52 + 908q^-51 + 378q^-50 - 198q^-49 - 1493q^-48 - 569q^-47 + 1195q^-46 + 1428q^-45 + 794q^-44 - 2182q^-43 - 2028q^-42 + 541q^-41 + 2318q^-40 + 2599q^-39 - 1987q^-38 - 3412q^-37 - 908q^-36 + 2426q^-35 + 4375q^-34 - 1071q^-33 - 4093q^-32 - 2375q^-31 + 1887q^-30 + 5474q^-29 - 19q^-28 - 4083q^-27 - 3391q^-26 + 1109q^-25 + 5841q^-24 + 902q^-23 - 3591q^-22 - 3945q^-21 + 193q^-20 + 5532q^-19 + 1732q^-18 - 2574q^-17 - 4017q^-16 - 909q^-15 + 4453q^-14 + 2315q^-13 - 1064q^-12 - 3361q^-11 - 1864q^-10 + 2680q^-9 + 2211q^-8 + 391q^-7 - 1988q^-6 - 2048q^-5 + 904q^-4 + 1331q^-3 + 1017q^-2 - 581q^-1 - 1367 - 49q + 349q² + 745q³ + 104q⁴ - 530q⁵ - 145q⁶ - 89q⁷ + 266q⁸ + 139q⁹ - 118q¹⁰ - 11q¹¹ - 87q¹² + 51q¹³ + 37q¹⁴ - 31q¹⁵ + 23q¹⁶ - 22q¹⁷ + 10q¹⁸ + 4q¹⁹ - 13q²⁰ + 8q²¹ - 3q²² + 3q²³ + q²⁴ - 3q²⁵ + q²⁶
5	- q^-110 + 3q^-109 - 2q^-108 - 3q^-107 + 6q^-106 - q^-105 - 6q^-104 + 3q^-103 - 3q^-102 - 2q^-101 + 25q^-100 + 12q^-99 - 37q^-98 - 41q^-97 - 21q^-96 + 36q^-95 + 125q^-94 + 109q^-93 - 77q^-92 - 266q^-91 - 261q^-90 + 5q^-89 + 446q^-88 + 640q^-87 + 241q^-86 - 581q^-85 - 1195q^-84 - 877q^-83 + 431q^-82 + 1817q^-81 + 1989q^-80 + 373q^-79 - 2172q^-78 - 3525q^-77 - 2036q^-76 + 1705q^-75 + 4988q^-74 + 4713q^-73 + 105q^-72 - 5816q^-71 - 7946q^-70 - 3476q^-69 + 5126q^-68 + 11063q^-67 + 8308q^-66 - 2526q^-65 - 13171q^-64 - 13857q^-63 - 2099q^-62 + 13533q^-61 + 19270q^-60 + 8232q^-59 - 11847q^-58 - 23661q^-57 - 15075q^-56 + 8385q^-55 + 26442q^-54 + 21653q^-53 - 3652q^-52 - 27508q^-51 - 27347q^-50 - 1485q^-49 + 27167q^-48 + 31610q^-47 + 6436q^-46 - 25775q^-45 - 34576q^-44 - 10754q^-43 + 23911q^-42 + 36397q^-41 + 14232q^-40 - 21810q^-39 - 37280q^-38 - 17184q^-37 + 19581q^-36 + 37727q^-35 + 19591q^-34 - 17209q^-33 - 37381q^-32 - 21994q^-31 + 14253q^-30 + 36701q^-29 + 24132q^-28 - 10731q^-27 - 34797q^-26 - 26227q^-25 + 6247q^-24 + 31996q^-23 + 27591q^-22 - 1273q^-21 - 27448q^-20 - 28061q^-19 - 4042q^-18 + 21869q^-17 + 26814q^-16 + 8685q^-15 - 15026q^-14 - 23999q^-13 - 12153q^-12 + 8337q^-11 + 19434q^-10 + 13593q^-9 - 2144q^-8 - 14015q^-7 - 13154q^-6 - 2221q^-5 + 8472q^-4 + 10896q^-3 + 4744q^-2 - 3775q^-1 - 7834 - 5289q + 529q² + 4662q³ + 4518q⁴ + 1184q⁵ - 2139q⁶ - 3127q⁷ - 1671q⁸ + 564q⁹ + 1756q¹⁰ + 1400q¹¹ + 192q¹² - 761q¹³ - 899q¹⁴ - 385q¹⁵ + 219q¹⁶ + 459q¹⁷ + 298q¹⁸ - q¹⁹ - 161q²⁰ - 174q²¹ - 65q²² + 56q²³ + 72q²⁴ + 27q²⁵ + 8q²⁶ - 12q²⁷ - 31q²⁸ - 5q²⁹ + 11q³⁰ - 6q³¹ + 6q³² + 9q³³ - 5q³⁴ - 3q³⁵ + 3q³⁶ - 3q³⁷ - q³⁸ + 3q³⁹ - q⁴⁰

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 103]]
Out[2]=	PD[X[6, 2, 7, 1], X[18, 6, 19, 5], X[20, 13, 1, 14], X[16, 7, 17, 8], > X[10, 3, 11, 4], X[4, 11, 5, 12], X[14, 9, 15, 10], X[8, 15, 9, 16], > X[12, 19, 13, 20], X[2, 18, 3, 17]]
In[3]:=	GaussCode[Knot[10, 103]]
Out[3]=	GaussCode[1, -10, 5, -6, 2, -1, 4, -8, 7, -5, 6, -9, 3, -7, 8, -4, 10, -2, 9, > -3]
In[4]:=	DTCode[Knot[10, 103]]
Out[4]=	DTCode[6, 10, 18, 16, 14, 4, 20, 8, 2, 12]
In[5]:=	br = BR[Knot[10, 103]]
Out[5]=	BR[4, {-1, -1, -2, 1, 3, -2, -2, 3, -2, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 103]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 103]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 103]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 3, 3, 3, NotAvailable, 2}
In[10]:=	alex = Alexander[Knot[10, 103]][t]
Out[10]=	2 8 17 2 3 -21 + -- - -- + -- + 17 t - 8 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 103]][z]
Out[11]=	2 4 6 1 + 3 z + 4 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 40], Knot[10, 103]}
In[13]:=	{KnotDet[Knot[10, 103]], KnotSignature[Knot[10, 103]]}
Out[13]=	{75, -2}
In[14]:=	Jones[Knot[10, 103]][q]
Out[14]=	-8 3 6 9 12 13 11 10 2 -6 - q + -- - -- + -- - -- + -- - -- + -- + 3 q - 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, 40], Knot[10, 103]}
In[16]:=	A2Invariant[Knot[10, 103]][q]
Out[16]=	-24 -22 -20 -18 2 3 -12 -8 4 -4 3 2 -1 - q + q - q - q + --- - --- + q + q + -- - q + -- - q + 16 14 6 2 q q q q 4 6 > q - q
In[17]:=	HOMFLYPT[Knot[10, 103]][a, z]
Out[17]=	2 6 2 2 2 4 2 6 2 4 2 4 4 4 -1 + 3 a - a - 2 z + 4 a z + 3 a z - 2 a z - z + 3 a z + 3 a z - 6 4 2 6 4 6 > a z + a z + a z
In[18]:=	Kauffman[Knot[10, 103]][a, z]
Out[18]=	2 6 z 3 5 7 2 2 2 -1 - 3 a + a + - + a z - 2 a z - 6 a z - 4 a z + 3 z + 2 a z - a 3 4 2 6 2 8 2 2 z 3 3 3 5 3 7 3 > 8 a z - 6 a z + a z - ---- - 2 a z + 9 a z + 21 a z + 10 a z - a 5 9 3 4 4 4 6 4 8 4 z 5 3 5 > 2 a z - 6 z + 25 a z + 13 a z - 6 a z + -- - 5 a z - 9 a z - a 5 5 7 5 9 5 6 2 6 4 6 6 6 > 16 a z - 12 a z + a z + 3 z - 5 a z - 23 a z - 12 a z + 8 6 7 3 7 5 7 7 7 2 8 4 8 > 3 a z + 4 a z + 2 a z + 3 a z + 5 a z + 4 a z + 9 a z + 6 8 3 9 5 9 > 5 a z + 2 a z + 2 a z
In[19]:=	{Vassiliev[2][Knot[10, 103]], Vassiliev[3][Knot[10, 103]]}
Out[19]=	{3, -4}
In[20]:=	Kh[Knot[10, 103]][q, t]
Out[20]=	5 6 1 2 1 4 2 5 4 7 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 5 6 7 5 6 2 t 2 3 2 5 3 > ----- + ----- + ----- + ---- + ---- + --- + 4 q t + q t + 2 q t + q t 7 3 7 2 5 2 5 3 q q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 103], 2][q]
Out[21]=	-23 3 2 7 17 6 30 46 3 75 72 30 -44 + q - --- + --- + --- - --- + --- + --- - --- - --- + --- - --- - --- + 22 21 20 19 18 17 16 15 14 13 12 q q q q q q q q q q q 122 81 61 141 67 75 123 35 69 78 7 2 > --- - --- - -- + --- - -- - -- + --- - -- - -- + -- - - + 32 q + 3 q - 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 3 4 5 6 7 > 16 q + 8 q + q - 3 q + q

The Alternating Knot 10103

The Alternating Knot 10₁₀₃