The Alternating Knot 10₂₁

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_5,14,6,15 X_3,13,4,12 X_13,3,14,2 X_7,16,8,17 X_11,20,12,1 X_15,6,16,7 X_19,8,20,9 X_9,18,10,19 X_17,10,18,11

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

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

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 2 / NotAvailable 1

Alexander Polynomial: - 2t^-3 + 7t^-2 - 9t^-1 + 9 - 9t + 7t² - 2t³

Conway Polynomial: 1 + z² - 5z⁴ - 2z⁶

Other knots with the same Alexander/Conway Polynomial: {K11n69, ...}

Determinant and Signature: {45, -4}

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

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

A2 (sl(3)) Invariant: q^-30 - 2q^-22 - 2q^-18 + q^-14 + 3q^-10 + q^-6 + 1

HOMFLY-PT Polynomial: a² + 3a²z² + a²z⁴ + 2a⁴ - 3a⁴z⁴ - a⁴z⁶ - 3a⁶ - 5a⁶z² - 4a⁶z⁴ - a⁶z⁶ + a⁸ + 3a⁸z² + a⁸z⁴

Kauffman Polynomial: - a² + 4a²z² - 4a²z⁴ + a²z⁶ + 5a³z³ - 7a³z⁵ + 2a³z⁷ + 2a⁴ - 3a⁴z² + 4a⁴z⁴ - 6a⁴z⁶ + 2a⁴z⁸ - 2a⁵z + 3a⁵z³ - 3a⁵z⁵ - a⁵z⁷ + a⁵z⁹ + 3a⁶ - 14a⁶z² + 20a⁶z⁴ - 14a⁶z⁶ + 4a⁶z⁸ - a⁷z + 2a⁷z³ - a⁷z⁷ + a⁷z⁹ + a⁸ - 5a⁸z² + 9a⁸z⁴ - 5a⁸z⁶ + 2a⁸z⁸ + 3a⁹z - 2a⁹z⁵ + 2a⁹z⁷ - 2a¹⁰z⁴ + 2a¹⁰z⁶ + 2a¹¹z - 4a¹¹z³ + 2a¹¹z⁵ - 2a¹²z² + a¹²z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {1, 0}

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 10₂₁. 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 r = 1 r = 2

j = 1 1

j = -1 1

j = -3 3 1

j = -5 3 2

j = -7 4 2

j = -9 3 3

j = -11 4 4

j = -13 2 3

j = -15 1 4

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 + q^-23 + 9q^-22 - 14q^-21 + 4q^-20 + 19q^-19 - 28q^-18 + 6q^-17 + 29q^-16 - 38q^-15 + 4q^-14 + 35q^-13 - 36q^-12 - 2q^-11 + 35q^-10 - 27q^-9 - 8q^-8 + 28q^-7 - 14q^-6 - 10q^-5 + 17q^-4 - 4q^-3 - 7q^-2 + 6q^-1 - 2q + q²

3 q^-54 - 2q^-53 + q^-51 + 3q^-50 - 4q^-49 - 2q^-48 + 2q^-47 + 5q^-46 - 3q^-45 - 4q^-44 + 3q^-43 + q^-42 - 2q^-41 + 2q^-40 + 8q^-39 - 14q^-38 - 10q^-37 + 15q^-36 + 30q^-35 - 33q^-34 - 36q^-33 + 33q^-32 + 56q^-31 - 37q^-30 - 67q^-29 + 34q^-28 + 75q^-27 - 25q^-26 - 82q^-25 + 19q^-24 + 76q^-23 - q^-22 - 79q^-21 - 7q^-20 + 66q^-19 + 27q^-18 - 64q^-17 - 32q^-16 + 46q^-15 + 47q^-14 - 38q^-13 - 46q^-12 + 19q^-11 + 49q^-10 - 9q^-9 - 39q^-8 - 5q^-7 + 34q^-6 + 7q^-5 - 19q^-4 - 12q^-3 + 13q^-2 + 8q^-1 - 5 - 6q + 3q² + 2q³ - 2q⁵ + q⁶

4 q^-88 - 2q^-87 + q^-85 + 5q^-83 - 8q^-82 + q^-80 - q^-79 + 18q^-78 - 16q^-77 - 3q^-76 - 5q^-75 - 6q^-74 + 43q^-73 - 19q^-72 - 4q^-71 - 24q^-70 - 27q^-69 + 76q^-68 - 4q^-67 + 15q^-66 - 55q^-65 - 87q^-64 + 97q^-63 + 36q^-62 + 84q^-61 - 76q^-60 - 197q^-59 + 74q^-58 + 71q^-57 + 211q^-56 - 46q^-55 - 322q^-54 - 8q^-53 + 64q^-52 + 346q^-51 + 35q^-50 - 392q^-49 - 94q^-48 + 7q^-47 + 416q^-46 + 121q^-45 - 389q^-44 - 133q^-43 - 63q^-42 + 410q^-41 + 167q^-40 - 336q^-39 - 121q^-38 - 125q^-37 + 352q^-36 + 183q^-35 - 253q^-34 - 82q^-33 - 177q^-32 + 256q^-31 + 178q^-30 - 145q^-29 - 23q^-28 - 215q^-27 + 136q^-26 + 145q^-25 - 40q^-24 + 53q^-23 - 208q^-22 + 23q^-21 + 72q^-20 + 18q^-19 + 124q^-18 - 144q^-17 - 38q^-16 - 10q^-15 + 10q^-14 + 144q^-13 - 56q^-12 - 31q^-11 - 51q^-10 - 27q^-9 + 103q^-8 - 3q^-7 + 3q^-6 - 38q^-5 - 40q^-4 + 45q^-3 + 5q^-2 + 16q^-1 - 12 - 24q + 14q² - q³ + 8q⁴ - q⁵ - 8q⁶ + 4q⁷ - q⁸ + 2q⁹ - 2q¹¹ + q¹²

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, 21]]

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

In[3]:=
GaussCode[Knot[10, 21]]

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

In[4]:=
DTCode[Knot[10, 21]]

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

In[5]:=
br = BR[Knot[10, 21]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 21]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 21]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 21]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 21]][t]

Out[10]=
2 7 9 2 3 9 - -- + -- - - - 9 t + 7 t - 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 21]][z]

Out[11]=
2 4 6 1 + z - 5 z - 2 z

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

Out[12]=
{Knot[10, 21], Knot[11, NonAlternating, 69]}

In[13]:=
{KnotDet[Knot[10, 21]], KnotSignature[Knot[10, 21]]}

Out[13]=
{45, -4}

In[14]:=
Jones[Knot[10, 21]][q]

Out[14]=
-10 2 3 6 7 7 7 5 4 2 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 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, 21]}

In[16]:=
A2Invariant[Knot[10, 21]][q]

Out[16]=
-30 2 2 -14 3 -6 1 + q - --- - --- + q + --- + q 22 18 10 q q q

In[17]:=
HOMFLYPT[Knot[10, 21]][a, z]

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

In[18]:=
Kauffman[Knot[10, 21]][a, z]

Out[18]=
2 4 6 8 5 7 9 11 2 2 4 2 -a + 2 a + 3 a + a - 2 a z - a z + 3 a z + 2 a z + 4 a z - 3 a z - 6 2 8 2 12 2 3 3 5 3 7 3 11 3 > 14 a z - 5 a z - 2 a z + 5 a z + 3 a z + 2 a z - 4 a z - 2 4 4 4 6 4 8 4 10 4 12 4 3 5 > 4 a z + 4 a z + 20 a z + 9 a z - 2 a z + a z - 7 a z - 5 5 9 5 11 5 2 6 4 6 6 6 8 6 > 3 a z - 2 a z + 2 a z + a z - 6 a z - 14 a z - 5 a z + 10 6 3 7 5 7 7 7 9 7 4 8 6 8 > 2 a z + 2 a z - a z - a z + 2 a z + 2 a z + 4 a z + 8 8 5 9 7 9 > 2 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 21]], Vassiliev[3][Knot[10, 21]]}

Out[19]=
{1, 0}

In[20]:=
Kh[Knot[10, 21]][q, t]

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

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

Out[21]=
-28 2 4 6 -23 9 14 4 19 28 6 29 38 q - --- + --- - --- + q + --- - --- + --- + --- - --- + --- + --- - --- + 27 25 24 22 21 20 19 18 17 16 15 q q q q q q q q q q q 4 35 36 2 35 27 8 28 14 10 17 4 7 6 > --- + --- - --- - --- + --- - -- - -- + -- - -- - -- + -- - -- - -- + - - 14 13 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q q q 2 > 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₂₁

10₂₀
10₂₂

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-28 - 2q^-27 + 4q^-25 - 6q^-24 + q^-23 + 9q^-22 - 14q^-21 + 4q^-20 + 19q^-19 - 28q^-18 + 6q^-17 + 29q^-16 - 38q^-15 + 4q^-14 + 35q^-13 - 36q^-12 - 2q^-11 + 35q^-10 - 27q^-9 - 8q^-8 + 28q^-7 - 14q^-6 - 10q^-5 + 17q^-4 - 4q^-3 - 7q^-2 + 6q^-1 - 2q + q²
3	q^-54 - 2q^-53 + q^-51 + 3q^-50 - 4q^-49 - 2q^-48 + 2q^-47 + 5q^-46 - 3q^-45 - 4q^-44 + 3q^-43 + q^-42 - 2q^-41 + 2q^-40 + 8q^-39 - 14q^-38 - 10q^-37 + 15q^-36 + 30q^-35 - 33q^-34 - 36q^-33 + 33q^-32 + 56q^-31 - 37q^-30 - 67q^-29 + 34q^-28 + 75q^-27 - 25q^-26 - 82q^-25 + 19q^-24 + 76q^-23 - q^-22 - 79q^-21 - 7q^-20 + 66q^-19 + 27q^-18 - 64q^-17 - 32q^-16 + 46q^-15 + 47q^-14 - 38q^-13 - 46q^-12 + 19q^-11 + 49q^-10 - 9q^-9 - 39q^-8 - 5q^-7 + 34q^-6 + 7q^-5 - 19q^-4 - 12q^-3 + 13q^-2 + 8q^-1 - 5 - 6q + 3q² + 2q³ - 2q⁵ + q⁶
4	q^-88 - 2q^-87 + q^-85 + 5q^-83 - 8q^-82 + q^-80 - q^-79 + 18q^-78 - 16q^-77 - 3q^-76 - 5q^-75 - 6q^-74 + 43q^-73 - 19q^-72 - 4q^-71 - 24q^-70 - 27q^-69 + 76q^-68 - 4q^-67 + 15q^-66 - 55q^-65 - 87q^-64 + 97q^-63 + 36q^-62 + 84q^-61 - 76q^-60 - 197q^-59 + 74q^-58 + 71q^-57 + 211q^-56 - 46q^-55 - 322q^-54 - 8q^-53 + 64q^-52 + 346q^-51 + 35q^-50 - 392q^-49 - 94q^-48 + 7q^-47 + 416q^-46 + 121q^-45 - 389q^-44 - 133q^-43 - 63q^-42 + 410q^-41 + 167q^-40 - 336q^-39 - 121q^-38 - 125q^-37 + 352q^-36 + 183q^-35 - 253q^-34 - 82q^-33 - 177q^-32 + 256q^-31 + 178q^-30 - 145q^-29 - 23q^-28 - 215q^-27 + 136q^-26 + 145q^-25 - 40q^-24 + 53q^-23 - 208q^-22 + 23q^-21 + 72q^-20 + 18q^-19 + 124q^-18 - 144q^-17 - 38q^-16 - 10q^-15 + 10q^-14 + 144q^-13 - 56q^-12 - 31q^-11 - 51q^-10 - 27q^-9 + 103q^-8 - 3q^-7 + 3q^-6 - 38q^-5 - 40q^-4 + 45q^-3 + 5q^-2 + 16q^-1 - 12 - 24q + 14q² - q³ + 8q⁴ - q⁵ - 8q⁶ + 4q⁷ - q⁸ + 2q⁹ - 2q¹¹ + q¹²

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 21]]
Out[2]=	PD[X[1, 4, 2, 5], X[5, 14, 6, 15], X[3, 13, 4, 12], X[13, 3, 14, 2], > X[7, 16, 8, 17], X[11, 20, 12, 1], X[15, 6, 16, 7], X[19, 8, 20, 9], > X[9, 18, 10, 19], X[17, 10, 18, 11]]
In[3]:=	GaussCode[Knot[10, 21]]
Out[3]=	GaussCode[-1, 4, -3, 1, -2, 7, -5, 8, -9, 10, -6, 3, -4, 2, -7, 5, -10, 9, -8, > 6]
In[4]:=	DTCode[Knot[10, 21]]
Out[4]=	DTCode[4, 12, 14, 16, 18, 20, 2, 6, 10, 8]
In[5]:=	br = BR[Knot[10, 21]]
Out[5]=	BR[4, {-1, -1, -2, 1, -2, -2, -2, -2, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 21]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 21]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 21]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 21]][t]
Out[10]=	2 7 9 2 3 9 - -- + -- - - - 9 t + 7 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 21]][z]
Out[11]=	2 4 6 1 + z - 5 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 21], Knot[11, NonAlternating, 69]}
In[13]:=	{KnotDet[Knot[10, 21]], KnotSignature[Knot[10, 21]]}
Out[13]=	{45, -4}
In[14]:=	Jones[Knot[10, 21]][q]
Out[14]=	-10 2 3 6 7 7 7 5 4 2 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 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, 21]}
In[16]:=	A2Invariant[Knot[10, 21]][q]
Out[16]=	-30 2 2 -14 3 -6 1 + q - --- - --- + q + --- + q 22 18 10 q q q
In[17]:=	HOMFLYPT[Knot[10, 21]][a, z]
Out[17]=	2 4 6 8 2 2 6 2 8 2 2 4 4 4 a + 2 a - 3 a + a + 3 a z - 5 a z + 3 a z + a z - 3 a z - 6 4 8 4 4 6 6 6 > 4 a z + a z - a z - a z
In[18]:=	Kauffman[Knot[10, 21]][a, z]
Out[18]=	2 4 6 8 5 7 9 11 2 2 4 2 -a + 2 a + 3 a + a - 2 a z - a z + 3 a z + 2 a z + 4 a z - 3 a z - 6 2 8 2 12 2 3 3 5 3 7 3 11 3 > 14 a z - 5 a z - 2 a z + 5 a z + 3 a z + 2 a z - 4 a z - 2 4 4 4 6 4 8 4 10 4 12 4 3 5 > 4 a z + 4 a z + 20 a z + 9 a z - 2 a z + a z - 7 a z - 5 5 9 5 11 5 2 6 4 6 6 6 8 6 > 3 a z - 2 a z + 2 a z + a z - 6 a z - 14 a z - 5 a z + 10 6 3 7 5 7 7 7 9 7 4 8 6 8 > 2 a z + 2 a z - a z - a z + 2 a z + 2 a z + 4 a z + 8 8 5 9 7 9 > 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 21]], Vassiliev[3][Knot[10, 21]]}
Out[19]=	{1, 0}
In[20]:=	Kh[Knot[10, 21]][q, t]
Out[20]=	2 3 1 1 1 2 1 4 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 3 4 4 3 3 4 2 3 t t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + -- + - + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q t q t q 2 > q t
In[21]:=	ColouredJones[Knot[10, 21], 2][q]
Out[21]=	-28 2 4 6 -23 9 14 4 19 28 6 29 38 q - --- + --- - --- + q + --- - --- + --- + --- - --- + --- + --- - --- + 27 25 24 22 21 20 19 18 17 16 15 q q q q q q q q q q q 4 35 36 2 35 27 8 28 14 10 17 4 7 6 > --- + --- - --- - --- + --- - -- - -- + -- - -- - -- + -- - -- - -- + - - 14 13 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q q q 2 > 2 q + q

The Alternating Knot 1021

The Alternating Knot 10₂₁