The Alternating Knot 10₁₂₁

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Acknowledgement

KnotPlot

PD Presentation: X₁₆₂₇ X_7,20,8,1 X_9,19,10,18 X_3,11,4,10 X_17,5,18,4 X_5,12,6,13 X_11,16,12,17 X_19,14,20,15 X_13,8,14,9 X_15,2,16,3

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

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

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 2 3 3 / NotAvailable 1

Alexander Polynomial: 2t^-3 - 11t^-2 + 27t^-1 - 35 + 27t - 11t² + 2t³

Conway Polynomial: 1 + z² + z⁴ + 2z⁶

Other knots with the same Alexander/Conway Polynomial: {K11a41, K11a183, K11a198, K11a331, ...}

Determinant and Signature: {115, -2}

Jones Polynomial: - q^-8 + 4q^-7 - 9q^-6 + 14q^-5 - 18q^-4 + 20q^-3 - 18q^-2 + 15q^-1 - 10 + 5q - q²

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

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

HOMFLY-PT Polynomial: 1 - z⁴ - a² - a²z² + a²z⁴ + a²z⁶ + 2a⁴ + 3a⁴z² + 2a⁴z⁴ + a⁴z⁶ - a⁶ - a⁶z² - a⁶z⁴

Kauffman Polynomial: a^-1z⁵ + 1 - 5z⁴ + 5z⁶ + 4az³ - 15az⁵ + 10az⁷ + a² - 3a²z² + 3a²z⁴ - 13a²z⁶ + 10a²z⁸ - a³z + 14a³z³ - 30a³z⁵ + 11a³z⁷ + 4a³z⁹ + 2a⁴ - 7a⁴z² + 22a⁴z⁴ - 36a⁴z⁶ + 19a⁴z⁸ - 3a⁵z + 19a⁵z³ - 28a⁵z⁵ + 9a⁵z⁷ + 4a⁵z⁹ + a⁶ - 3a⁶z² + 9a⁶z⁴ - 14a⁶z⁶ + 9a⁶z⁸ - 2a⁷z + 8a⁷z³ - 13a⁷z⁵ + 8a⁷z⁷ + a⁸z² - 5a⁸z⁴ + 4a⁸z⁶ - a⁹z³ + a⁹z⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {1, -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=-2 is the signature of 10₁₂₁. 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 4

j = 1 6 1

j = -1 9 4

j = -3 10 7

j = -5 10 8

j = -7 8 10

j = -9 6 10

j = -11 3 8

j = -13 1 6

j = -15 3

j = -17 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-23 - 4q^-22 + 4q^-21 + 10q^-20 - 32q^-19 + 17q^-18 + 59q^-17 - 106q^-16 + 7q^-15 + 170q^-14 - 186q^-13 - 52q^-12 + 291q^-11 - 215q^-10 - 129q^-9 + 348q^-8 - 179q^-7 - 178q^-6 + 312q^-5 - 99q^-4 - 175q^-3 + 203q^-2 - 20q^-1 - 115 + 82q + 12q² - 41q³ + 15q⁴ + 5q⁵ - 5q⁶ + q⁷

3 - q^-45 + 4q^-44 - 4q^-43 - 5q^-42 + 8q^-41 + 18q^-40 - 25q^-39 - 54q^-38 + 60q^-37 + 134q^-36 - 89q^-35 - 296q^-34 + 72q^-33 + 556q^-32 + 34q^-31 - 866q^-30 - 297q^-29 + 1184q^-28 + 710q^-27 - 1429q^-26 - 1233q^-25 + 1543q^-24 + 1798q^-23 - 1511q^-22 - 2331q^-21 + 1356q^-20 + 2765q^-19 - 1109q^-18 - 3059q^-17 + 784q^-16 + 3233q^-15 - 449q^-14 - 3232q^-13 + 64q^-12 + 3110q^-11 + 294q^-10 - 2813q^-9 - 644q^-8 + 2402q^-7 + 896q^-6 - 1872q^-5 - 1038q^-4 + 1320q^-3 + 1013q^-2 - 788q^-1 - 874 + 387q + 641q² - 125q³ - 404q⁴ + 4q⁵ + 208q⁶ + 38q⁷ - 100q⁸ - 19q⁹ + 32q¹⁰ + 11q¹¹ - 10q¹² - 5q¹³ + 5q¹⁴ - q¹⁵

4 q^-74 - 4q^-73 + 4q^-72 + 5q^-71 - 13q^-70 + 6q^-69 - 10q^-68 + 36q^-67 + 25q^-66 - 111q^-65 - 26q^-64 - 6q^-63 + 278q^-62 + 247q^-61 - 477q^-60 - 528q^-59 - 354q^-58 + 1111q^-57 + 1685q^-56 - 533q^-55 - 2180q^-54 - 2867q^-53 + 1493q^-52 + 5538q^-51 + 2438q^-50 - 3362q^-49 - 9179q^-48 - 2283q^-47 + 9553q^-46 + 10327q^-45 + 500q^-44 - 16370q^-43 - 12097q^-42 + 8483q^-41 + 19624q^-40 + 11021q^-39 - 18806q^-38 - 23810q^-37 + 813q^-36 + 24646q^-35 + 23705q^-34 - 14982q^-33 - 31723q^-32 - 9405q^-31 + 23876q^-30 + 33219q^-29 - 7995q^-28 - 34142q^-27 - 18052q^-26 + 19437q^-25 + 37954q^-24 - 530q^-23 - 32180q^-22 - 24101q^-21 + 12749q^-20 + 38261q^-19 + 7018q^-18 - 26171q^-17 - 27397q^-16 + 3832q^-15 + 33559q^-14 + 13781q^-13 - 15946q^-12 - 26135q^-11 - 5597q^-10 + 23284q^-9 + 16524q^-8 - 4083q^-7 - 18889q^-6 - 11015q^-5 + 10561q^-4 + 12915q^-3 + 3822q^-2 - 8735q^-1 - 9653 + 1523q + 5955q² + 4894q³ - 1616q⁴ - 4720q⁵ - 1182q⁶ + 1127q⁷ + 2406q⁸ + 481q⁹ - 1236q¹⁰ - 644q¹¹ - 173q¹² + 590q¹³ + 289q¹⁴ - 176q¹⁵ - 97q¹⁶ - 106q¹⁷ + 82q¹⁸ + 52q¹⁹ - 25q²⁰ - 2q²¹ - 16q²² + 10q²³ + 5q²⁴ - 5q²⁵ + q²⁶

5 - q^-110 + 4q^-109 - 4q^-108 - 5q^-107 + 13q^-106 - q^-105 - 14q^-104 - q^-103 - 7q^-102 + 10q^-101 + 78q^-100 + 32q^-99 - 131q^-98 - 196q^-97 - 100q^-96 + 230q^-95 + 628q^-94 + 523q^-93 - 417q^-92 - 1582q^-91 - 1652q^-90 + 140q^-89 + 3074q^-88 + 4482q^-87 + 1710q^-86 - 4710q^-85 - 9673q^-84 - 6938q^-83 + 4466q^-82 + 16938q^-81 + 17892q^-80 + 1065q^-79 - 24141q^-78 - 35432q^-77 - 15987q^-76 + 26168q^-75 + 57614q^-74 + 43794q^-73 - 16583q^-72 - 79185q^-71 - 83915q^-70 - 10429q^-69 + 91411q^-68 + 131709q^-67 + 57493q^-66 - 86326q^-65 - 178297q^-64 - 121321q^-63 + 58443q^-62 + 213473q^-61 + 194245q^-60 - 7919q^-59 - 229518q^-58 - 265676q^-57 - 59569q^-56 + 223155q^-55 + 325835q^-54 + 135017q^-53 - 196351q^-52 - 368727q^-51 - 208778q^-50 + 155175q^-49 + 392638q^-48 + 273351q^-47 - 106875q^-46 - 399728q^-45 - 325057q^-44 + 57800q^-43 + 394335q^-42 + 363719q^-41 - 12099q^-40 - 380646q^-39 - 390960q^-38 - 29850q^-37 + 361491q^-36 + 410235q^-35 + 68646q^-34 - 337731q^-33 - 422549q^-32 - 107164q^-31 + 307561q^-30 + 429225q^-29 + 146528q^-28 - 269102q^-27 - 427285q^-26 - 187091q^-25 + 219600q^-24 + 414167q^-23 + 225845q^-22 - 159267q^-21 - 385285q^-20 - 257884q^-19 + 90509q^-18 + 338995q^-17 + 276333q^-16 - 19871q^-15 - 275665q^-14 - 275896q^-13 - 44019q^-12 + 201274q^-11 + 253523q^-10 + 91957q^-9 - 124190q^-8 - 211957q^-7 - 118030q^-6 + 55648q^-5 + 158188q^-4 + 120534q^-3 - 3736q^-2 - 102435q^-1 - 104297 - 26923q + 54320q² + 77330q³ + 37974q⁴ - 19965q⁵ - 48831q⁶ - 34927q⁷ + 298q⁸ + 25619q⁹ + 25362q¹⁰ + 7132q¹¹ - 10397q¹² - 14877q¹³ - 7674q¹⁴ + 2485q¹⁵ + 7390q¹⁶ + 5211q¹⁷ + 293q¹⁸ - 2813q¹⁹ - 2763q²⁰ - 872q²¹ + 903q²² + 1219q²³ + 497q²⁴ - 195q²⁵ - 408q²⁶ - 230q²⁷ + 4q²⁸ + 149q²⁹ + 90q³⁰ - 29q³¹ - 34q³² - 8q³³ - 5q³⁴ + 7q³⁵ + 16q³⁶ - 10q³⁷ - 5q³⁸ + 5q³⁹ - 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, 121]]

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

In[3]:=
GaussCode[Knot[10, 121]]

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

In[4]:=
DTCode[Knot[10, 121]]

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

In[5]:=
br = BR[Knot[10, 121]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 121]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 121]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 121]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 121]][t]

Out[10]=
2 11 27 2 3 -35 + -- - -- + -- + 27 t - 11 t + 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 121]][z]

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

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

Out[12]=
{Knot[10, 121], Knot[11, Alternating, 41], Knot[11, Alternating, 183], > Knot[11, Alternating, 198], Knot[11, Alternating, 331]}

In[13]:=
{KnotDet[Knot[10, 121]], KnotSignature[Knot[10, 121]]}

Out[13]=
{115, -2}

In[14]:=
Jones[Knot[10, 121]][q]

Out[14]=
-8 4 9 14 18 20 18 15 2 -10 - q + -- - -- + -- - -- + -- - -- + -- + 5 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, 121]}

In[16]:=
A2Invariant[Knot[10, 121]][q]

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

In[17]:=
HOMFLYPT[Knot[10, 121]][a, z]

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

In[18]:=
Kauffman[Knot[10, 121]][a, z]

Out[18]=
2 4 6 3 5 7 2 2 4 2 6 2 1 + a + 2 a + a - a z - 3 a z - 2 a z - 3 a z - 7 a z - 3 a z + 8 2 3 3 3 5 3 7 3 9 3 4 2 4 > a z + 4 a z + 14 a z + 19 a z + 8 a z - a z - 5 z + 3 a z + 5 4 4 6 4 8 4 z 5 3 5 5 5 > 22 a z + 9 a z - 5 a z + -- - 15 a z - 30 a z - 28 a z - a 7 5 9 5 6 2 6 4 6 6 6 8 6 > 13 a z + a z + 5 z - 13 a z - 36 a z - 14 a z + 4 a z + 7 3 7 5 7 7 7 2 8 4 8 6 8 > 10 a z + 11 a z + 9 a z + 8 a z + 10 a z + 19 a z + 9 a z + 3 9 5 9 > 4 a z + 4 a z

In[19]:=
{Vassiliev[2][Knot[10, 121]], Vassiliev[3][Knot[10, 121]]}

Out[19]=
{1, -2}

In[20]:=
Kh[Knot[10, 121]][q, t]

Out[20]=
7 9 1 3 1 6 3 8 6 10 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 8 10 10 8 10 4 t 2 3 2 5 3 > ----- + ----- + ----- + ---- + ---- + --- + 6 q t + q t + 4 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, 121], 2][q]

Out[21]=
-23 4 4 10 32 17 59 106 7 170 186 52 -115 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- - --- + 22 21 20 19 18 17 16 15 14 13 12 q q q q q q q q q q q 291 215 129 348 179 178 312 99 175 203 20 > --- - --- - --- + --- - --- - --- + --- - -- - --- + --- - -- + 82 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 > 12 q - 41 q + 15 q + 5 q - 5 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₂₁

10₁₂₀
10₁₂₂

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-23 - 4q^-22 + 4q^-21 + 10q^-20 - 32q^-19 + 17q^-18 + 59q^-17 - 106q^-16 + 7q^-15 + 170q^-14 - 186q^-13 - 52q^-12 + 291q^-11 - 215q^-10 - 129q^-9 + 348q^-8 - 179q^-7 - 178q^-6 + 312q^-5 - 99q^-4 - 175q^-3 + 203q^-2 - 20q^-1 - 115 + 82q + 12q² - 41q³ + 15q⁴ + 5q⁵ - 5q⁶ + q⁷
3	- q^-45 + 4q^-44 - 4q^-43 - 5q^-42 + 8q^-41 + 18q^-40 - 25q^-39 - 54q^-38 + 60q^-37 + 134q^-36 - 89q^-35 - 296q^-34 + 72q^-33 + 556q^-32 + 34q^-31 - 866q^-30 - 297q^-29 + 1184q^-28 + 710q^-27 - 1429q^-26 - 1233q^-25 + 1543q^-24 + 1798q^-23 - 1511q^-22 - 2331q^-21 + 1356q^-20 + 2765q^-19 - 1109q^-18 - 3059q^-17 + 784q^-16 + 3233q^-15 - 449q^-14 - 3232q^-13 + 64q^-12 + 3110q^-11 + 294q^-10 - 2813q^-9 - 644q^-8 + 2402q^-7 + 896q^-6 - 1872q^-5 - 1038q^-4 + 1320q^-3 + 1013q^-2 - 788q^-1 - 874 + 387q + 641q² - 125q³ - 404q⁴ + 4q⁵ + 208q⁶ + 38q⁷ - 100q⁸ - 19q⁹ + 32q¹⁰ + 11q¹¹ - 10q¹² - 5q¹³ + 5q¹⁴ - q¹⁵
4	q^-74 - 4q^-73 + 4q^-72 + 5q^-71 - 13q^-70 + 6q^-69 - 10q^-68 + 36q^-67 + 25q^-66 - 111q^-65 - 26q^-64 - 6q^-63 + 278q^-62 + 247q^-61 - 477q^-60 - 528q^-59 - 354q^-58 + 1111q^-57 + 1685q^-56 - 533q^-55 - 2180q^-54 - 2867q^-53 + 1493q^-52 + 5538q^-51 + 2438q^-50 - 3362q^-49 - 9179q^-48 - 2283q^-47 + 9553q^-46 + 10327q^-45 + 500q^-44 - 16370q^-43 - 12097q^-42 + 8483q^-41 + 19624q^-40 + 11021q^-39 - 18806q^-38 - 23810q^-37 + 813q^-36 + 24646q^-35 + 23705q^-34 - 14982q^-33 - 31723q^-32 - 9405q^-31 + 23876q^-30 + 33219q^-29 - 7995q^-28 - 34142q^-27 - 18052q^-26 + 19437q^-25 + 37954q^-24 - 530q^-23 - 32180q^-22 - 24101q^-21 + 12749q^-20 + 38261q^-19 + 7018q^-18 - 26171q^-17 - 27397q^-16 + 3832q^-15 + 33559q^-14 + 13781q^-13 - 15946q^-12 - 26135q^-11 - 5597q^-10 + 23284q^-9 + 16524q^-8 - 4083q^-7 - 18889q^-6 - 11015q^-5 + 10561q^-4 + 12915q^-3 + 3822q^-2 - 8735q^-1 - 9653 + 1523q + 5955q² + 4894q³ - 1616q⁴ - 4720q⁵ - 1182q⁶ + 1127q⁷ + 2406q⁸ + 481q⁹ - 1236q¹⁰ - 644q¹¹ - 173q¹² + 590q¹³ + 289q¹⁴ - 176q¹⁵ - 97q¹⁶ - 106q¹⁷ + 82q¹⁸ + 52q¹⁹ - 25q²⁰ - 2q²¹ - 16q²² + 10q²³ + 5q²⁴ - 5q²⁵ + q²⁶
5	- q^-110 + 4q^-109 - 4q^-108 - 5q^-107 + 13q^-106 - q^-105 - 14q^-104 - q^-103 - 7q^-102 + 10q^-101 + 78q^-100 + 32q^-99 - 131q^-98 - 196q^-97 - 100q^-96 + 230q^-95 + 628q^-94 + 523q^-93 - 417q^-92 - 1582q^-91 - 1652q^-90 + 140q^-89 + 3074q^-88 + 4482q^-87 + 1710q^-86 - 4710q^-85 - 9673q^-84 - 6938q^-83 + 4466q^-82 + 16938q^-81 + 17892q^-80 + 1065q^-79 - 24141q^-78 - 35432q^-77 - 15987q^-76 + 26168q^-75 + 57614q^-74 + 43794q^-73 - 16583q^-72 - 79185q^-71 - 83915q^-70 - 10429q^-69 + 91411q^-68 + 131709q^-67 + 57493q^-66 - 86326q^-65 - 178297q^-64 - 121321q^-63 + 58443q^-62 + 213473q^-61 + 194245q^-60 - 7919q^-59 - 229518q^-58 - 265676q^-57 - 59569q^-56 + 223155q^-55 + 325835q^-54 + 135017q^-53 - 196351q^-52 - 368727q^-51 - 208778q^-50 + 155175q^-49 + 392638q^-48 + 273351q^-47 - 106875q^-46 - 399728q^-45 - 325057q^-44 + 57800q^-43 + 394335q^-42 + 363719q^-41 - 12099q^-40 - 380646q^-39 - 390960q^-38 - 29850q^-37 + 361491q^-36 + 410235q^-35 + 68646q^-34 - 337731q^-33 - 422549q^-32 - 107164q^-31 + 307561q^-30 + 429225q^-29 + 146528q^-28 - 269102q^-27 - 427285q^-26 - 187091q^-25 + 219600q^-24 + 414167q^-23 + 225845q^-22 - 159267q^-21 - 385285q^-20 - 257884q^-19 + 90509q^-18 + 338995q^-17 + 276333q^-16 - 19871q^-15 - 275665q^-14 - 275896q^-13 - 44019q^-12 + 201274q^-11 + 253523q^-10 + 91957q^-9 - 124190q^-8 - 211957q^-7 - 118030q^-6 + 55648q^-5 + 158188q^-4 + 120534q^-3 - 3736q^-2 - 102435q^-1 - 104297 - 26923q + 54320q² + 77330q³ + 37974q⁴ - 19965q⁵ - 48831q⁶ - 34927q⁷ + 298q⁸ + 25619q⁹ + 25362q¹⁰ + 7132q¹¹ - 10397q¹² - 14877q¹³ - 7674q¹⁴ + 2485q¹⁵ + 7390q¹⁶ + 5211q¹⁷ + 293q¹⁸ - 2813q¹⁹ - 2763q²⁰ - 872q²¹ + 903q²² + 1219q²³ + 497q²⁴ - 195q²⁵ - 408q²⁶ - 230q²⁷ + 4q²⁸ + 149q²⁹ + 90q³⁰ - 29q³¹ - 34q³² - 8q³³ - 5q³⁴ + 7q³⁵ + 16q³⁶ - 10q³⁷ - 5q³⁸ + 5q³⁹ - q⁴⁰

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 121]]
Out[2]=	PD[X[1, 6, 2, 7], X[7, 20, 8, 1], X[9, 19, 10, 18], X[3, 11, 4, 10], > X[17, 5, 18, 4], X[5, 12, 6, 13], X[11, 16, 12, 17], X[19, 14, 20, 15], > X[13, 8, 14, 9], X[15, 2, 16, 3]]
In[3]:=	GaussCode[Knot[10, 121]]
Out[3]=	GaussCode[-1, 10, -4, 5, -6, 1, -2, 9, -3, 4, -7, 6, -9, 8, -10, 7, -5, 3, -8, > 2]
In[4]:=	DTCode[Knot[10, 121]]
Out[4]=	DTCode[6, 10, 12, 20, 18, 16, 8, 2, 4, 14]
In[5]:=	br = BR[Knot[10, 121]]
Out[5]=	BR[4, {-1, -1, -2, 3, -2, 1, -2, 3, -2, 3, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 121]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 121]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 121]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 121]][t]
Out[10]=	2 11 27 2 3 -35 + -- - -- + -- + 27 t - 11 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 121]][z]
Out[11]=	2 4 6 1 + z + z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 121], Knot[11, Alternating, 41], Knot[11, Alternating, 183], > Knot[11, Alternating, 198], Knot[11, Alternating, 331]}
In[13]:=	{KnotDet[Knot[10, 121]], KnotSignature[Knot[10, 121]]}
Out[13]=	{115, -2}
In[14]:=	Jones[Knot[10, 121]][q]
Out[14]=	-8 4 9 14 18 20 18 15 2 -10 - q + -- - -- + -- - -- + -- - -- + -- + 5 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, 121]}
In[16]:=	A2Invariant[Knot[10, 121]][q]
Out[16]=	-24 2 2 2 4 3 3 -8 3 4 4 2 -1 - q + --- - --- - --- + --- - --- + --- - q + -- - -- + -- - q + 22 20 18 16 14 12 6 4 2 q q q q q q q q q 4 6 > 3 q - q
In[17]:=	HOMFLYPT[Knot[10, 121]][a, z]
Out[17]=	2 4 6 2 2 4 2 6 2 4 2 4 4 4 6 4 1 - a + 2 a - a - a z + 3 a z - a z - z + a z + 2 a z - a z + 2 6 4 6 > a z + a z
In[18]:=	Kauffman[Knot[10, 121]][a, z]
Out[18]=	2 4 6 3 5 7 2 2 4 2 6 2 1 + a + 2 a + a - a z - 3 a z - 2 a z - 3 a z - 7 a z - 3 a z + 8 2 3 3 3 5 3 7 3 9 3 4 2 4 > a z + 4 a z + 14 a z + 19 a z + 8 a z - a z - 5 z + 3 a z + 5 4 4 6 4 8 4 z 5 3 5 5 5 > 22 a z + 9 a z - 5 a z + -- - 15 a z - 30 a z - 28 a z - a 7 5 9 5 6 2 6 4 6 6 6 8 6 > 13 a z + a z + 5 z - 13 a z - 36 a z - 14 a z + 4 a z + 7 3 7 5 7 7 7 2 8 4 8 6 8 > 10 a z + 11 a z + 9 a z + 8 a z + 10 a z + 19 a z + 9 a z + 3 9 5 9 > 4 a z + 4 a z
In[19]:=	{Vassiliev[2][Knot[10, 121]], Vassiliev[3][Knot[10, 121]]}
Out[19]=	{1, -2}
In[20]:=	Kh[Knot[10, 121]][q, t]
Out[20]=	7 9 1 3 1 6 3 8 6 10 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 8 10 10 8 10 4 t 2 3 2 5 3 > ----- + ----- + ----- + ---- + ---- + --- + 6 q t + q t + 4 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, 121], 2][q]
Out[21]=	-23 4 4 10 32 17 59 106 7 170 186 52 -115 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- - --- + 22 21 20 19 18 17 16 15 14 13 12 q q q q q q q q q q q 291 215 129 348 179 178 312 99 175 203 20 > --- - --- - --- + --- - --- - --- + --- - -- - --- + --- - -- + 82 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 > 12 q - 41 q + 15 q + 5 q - 5 q + q

The Alternating Knot 10121

The Alternating Knot 10₁₂₁