The Alternating Knot 10₂₄

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Acknowledgement

KnotPlot

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

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

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

Minimum Braid Representative:

Length is 12, width is 5
Braid index is 5

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index

Reversible 2 2 2 / NotAvailable 1

Alexander Polynomial: - 4t^-2 + 14t^-1 - 19 + 14t - 4t²

Conway Polynomial: 1 - 2z² - 4z⁴

Other knots with the same Alexander/Conway Polynomial: {10₁₈, ...}

Determinant and Signature: {55, -2}

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

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

A2 (sl(3)) Invariant: q^-28 + 2q^-22 - 2q^-20 - q^-18 - 2q^-14 + q^-12 - q^-10 + q^-8 + q^-6 - q^-4 + 3q^-2 + q⁴

HOMFLY-PT Polynomial: 1 + z² + a² - a²z⁴ - a⁴ - 3a⁴z² - 2a⁴z⁴ - a⁶ - a⁶z² - a⁶z⁴ + a⁸ + a⁸z²

Kauffman Polynomial: 1 - 2z² + z⁴ - 2az³ + 2az⁵ - a² + 2a²z² - 3a²z⁴ + 3a²z⁶ + 2a³z - 2a³z⁵ + 3a³z⁷ - a⁴ + 5a⁴z² - 5a⁴z⁴ + a⁴z⁶ + 2a⁴z⁸ + 4a⁵z - 7a⁵z³ + a⁵z⁵ + a⁵z⁷ + a⁵z⁹ + a⁶ - 5a⁶z² + 6a⁶z⁴ - 8a⁶z⁶ + 4a⁶z⁸ - 2a⁷z³ - 2a⁷z⁵ + a⁷z⁹ + a⁸ - 2a⁸z² + 3a⁸z⁴ - 5a⁸z⁶ + 2a⁸z⁸ - 2a⁹z + 7a⁹z³ - 7a⁹z⁵ + 2a⁹z⁷ + 4a¹⁰z² - 4a¹⁰z⁴ + a¹⁰z⁶

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-2, 5}

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 10₂₄. 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 = 3 1

j = 1 1

j = -1 4 1

j = -3 4 2

j = -5 5 3

j = -7 4 4

j = -9 4 5

j = -11 3 4

j = -13 1 4

j = -15 1 3

j = -17 1

j = -19 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-26 - 2q^-25 + 6q^-23 - 8q^-22 - 5q^-21 + 21q^-20 - 14q^-19 - 20q^-18 + 43q^-17 - 15q^-16 - 42q^-15 + 60q^-14 - 7q^-13 - 61q^-12 + 65q^-11 + 3q^-10 - 67q^-9 + 56q^-8 + 9q^-7 - 53q^-6 + 36q^-5 + 9q^-4 - 29q^-3 + 16q^-2 + 4q^-1 - 10 + 5q + q² - 2q³ + q⁴

3 q^-51 - 2q^-50 + 2q^-48 + 3q^-47 - 7q^-46 - 5q^-45 + 9q^-44 + 16q^-43 - 15q^-42 - 28q^-41 + 11q^-40 + 52q^-39 - 5q^-38 - 73q^-37 - 15q^-36 + 96q^-35 + 44q^-34 - 116q^-33 - 75q^-32 + 120q^-31 + 119q^-30 - 124q^-29 - 154q^-28 + 109q^-27 + 196q^-26 - 99q^-25 - 221q^-24 + 72q^-23 + 250q^-22 - 52q^-21 - 262q^-20 + 28q^-19 + 263q^-18 - 3q^-17 - 254q^-16 - 13q^-15 + 224q^-14 + 33q^-13 - 194q^-12 - 34q^-11 + 147q^-10 + 39q^-9 - 112q^-8 - 27q^-7 + 70q^-6 + 26q^-5 - 52q^-4 - 8q^-3 + 25q^-2 + 9q^-1 - 19 + 8q² + 2q³ - 7q⁴ + 2q⁵ + q⁶ + q⁷ - 2q⁸ + q⁹

4 q^-84 - 2q^-83 + 2q^-81 - q^-80 + 4q^-79 - 9q^-78 - q^-77 + 10q^-76 - q^-75 + 16q^-74 - 29q^-73 - 18q^-72 + 20q^-71 + 11q^-70 + 66q^-69 - 53q^-68 - 71q^-67 - 12q^-66 + 11q^-65 + 190q^-64 - 19q^-63 - 125q^-62 - 127q^-61 - 89q^-60 + 342q^-59 + 118q^-58 - 70q^-57 - 264q^-56 - 343q^-55 + 398q^-54 + 289q^-53 + 151q^-52 - 290q^-51 - 673q^-50 + 282q^-49 + 373q^-48 + 475q^-47 - 154q^-46 - 952q^-45 + 49q^-44 + 323q^-43 + 786q^-42 + 85q^-41 - 1112q^-40 - 214q^-39 + 183q^-38 + 1028q^-37 + 339q^-36 - 1165q^-35 - 444q^-34 + 12q^-33 + 1165q^-32 + 562q^-31 - 1102q^-30 - 604q^-29 - 176q^-28 + 1151q^-27 + 718q^-26 - 898q^-25 - 633q^-24 - 352q^-23 + 942q^-22 + 741q^-21 - 586q^-20 - 493q^-19 - 435q^-18 + 602q^-17 + 589q^-16 - 295q^-15 - 249q^-14 - 371q^-13 + 286q^-12 + 346q^-11 - 127q^-10 - 54q^-9 - 222q^-8 + 106q^-7 + 146q^-6 - 68q^-5 + 27q^-4 - 95q^-3 + 37q^-2 + 46q^-1 - 43 + 30q - 32q² + 16q³ + 13q⁴ - 24q⁵ + 15q⁶ - 9q⁷ + 6q⁸ + 4q⁹ - 9q¹⁰ + 5q¹¹ - 2q¹² + q¹³ + q¹⁴ - 2q¹⁵ + q¹⁶

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, 24]]

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

In[3]:=
GaussCode[Knot[10, 24]]

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

In[4]:=
DTCode[Knot[10, 24]]

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

In[5]:=
br = BR[Knot[10, 24]]

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

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

Out[6]=
{5, 12}

In[7]:=
BraidIndex[Knot[10, 24]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[10, 24]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 24]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 24]][t]

Out[10]=
4 14 2 -19 - -- + -- + 14 t - 4 t 2 t t

In[11]:=
Conway[Knot[10, 24]][z]

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

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

Out[12]=
{Knot[10, 18], Knot[10, 24]}

In[13]:=
{KnotDet[Knot[10, 24]], KnotSignature[Knot[10, 24]]}

Out[13]=
{55, -2}

In[14]:=
Jones[Knot[10, 24]][q]

Out[14]=
-9 2 4 7 8 9 9 7 5 -2 + q - -- + -- - -- + -- - -- + -- - -- + - + q 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, 24]}

In[16]:=
A2Invariant[Knot[10, 24]][q]

Out[16]=
-28 2 2 -18 2 -12 -10 -8 -6 -4 3 4 q + --- - --- - q - --- + q - q + q + q - q + -- + q 22 20 14 2 q q q q

In[17]:=
HOMFLYPT[Knot[10, 24]][a, z]

Out[17]=
2 4 6 8 2 4 2 6 2 8 2 2 4 4 4 6 4 1 + a - a - a + a + z - 3 a z - a z + a z - a z - 2 a z - a z

In[18]:=
Kauffman[Knot[10, 24]][a, z]

Out[18]=
2 4 6 8 3 5 9 2 2 2 4 2 1 - a - a + a + a + 2 a z + 4 a z - 2 a z - 2 z + 2 a z + 5 a z - 6 2 8 2 10 2 3 5 3 7 3 9 3 4 > 5 a z - 2 a z + 4 a z - 2 a z - 7 a z - 2 a z + 7 a z + z - 2 4 4 4 6 4 8 4 10 4 5 3 5 > 3 a z - 5 a z + 6 a z + 3 a z - 4 a z + 2 a z - 2 a z + 5 5 7 5 9 5 2 6 4 6 6 6 8 6 10 6 > a z - 2 a z - 7 a z + 3 a z + a z - 8 a z - 5 a z + a z + 3 7 5 7 9 7 4 8 6 8 8 8 5 9 7 9 > 3 a z + a z + 2 a z + 2 a z + 4 a z + 2 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 24]], Vassiliev[3][Knot[10, 24]]}

Out[19]=
{-2, 5}

In[20]:=
Kh[Knot[10, 24]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 24], 2][q]

Out[21]=
-26 2 6 8 5 21 14 20 43 15 42 60 -10 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 25 23 22 21 20 19 18 17 16 15 14 q q q q q q q q q q q 7 61 65 3 67 56 9 53 36 9 29 16 4 > --- - --- + --- + --- - -- + -- + -- - -- + -- + -- - -- + -- + - + 5 q + 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 2 3 4 > q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₂₄

10₂₃
10₂₅

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-26 - 2q^-25 + 6q^-23 - 8q^-22 - 5q^-21 + 21q^-20 - 14q^-19 - 20q^-18 + 43q^-17 - 15q^-16 - 42q^-15 + 60q^-14 - 7q^-13 - 61q^-12 + 65q^-11 + 3q^-10 - 67q^-9 + 56q^-8 + 9q^-7 - 53q^-6 + 36q^-5 + 9q^-4 - 29q^-3 + 16q^-2 + 4q^-1 - 10 + 5q + q² - 2q³ + q⁴
3	q^-51 - 2q^-50 + 2q^-48 + 3q^-47 - 7q^-46 - 5q^-45 + 9q^-44 + 16q^-43 - 15q^-42 - 28q^-41 + 11q^-40 + 52q^-39 - 5q^-38 - 73q^-37 - 15q^-36 + 96q^-35 + 44q^-34 - 116q^-33 - 75q^-32 + 120q^-31 + 119q^-30 - 124q^-29 - 154q^-28 + 109q^-27 + 196q^-26 - 99q^-25 - 221q^-24 + 72q^-23 + 250q^-22 - 52q^-21 - 262q^-20 + 28q^-19 + 263q^-18 - 3q^-17 - 254q^-16 - 13q^-15 + 224q^-14 + 33q^-13 - 194q^-12 - 34q^-11 + 147q^-10 + 39q^-9 - 112q^-8 - 27q^-7 + 70q^-6 + 26q^-5 - 52q^-4 - 8q^-3 + 25q^-2 + 9q^-1 - 19 + 8q² + 2q³ - 7q⁴ + 2q⁵ + q⁶ + q⁷ - 2q⁸ + q⁹
4	q^-84 - 2q^-83 + 2q^-81 - q^-80 + 4q^-79 - 9q^-78 - q^-77 + 10q^-76 - q^-75 + 16q^-74 - 29q^-73 - 18q^-72 + 20q^-71 + 11q^-70 + 66q^-69 - 53q^-68 - 71q^-67 - 12q^-66 + 11q^-65 + 190q^-64 - 19q^-63 - 125q^-62 - 127q^-61 - 89q^-60 + 342q^-59 + 118q^-58 - 70q^-57 - 264q^-56 - 343q^-55 + 398q^-54 + 289q^-53 + 151q^-52 - 290q^-51 - 673q^-50 + 282q^-49 + 373q^-48 + 475q^-47 - 154q^-46 - 952q^-45 + 49q^-44 + 323q^-43 + 786q^-42 + 85q^-41 - 1112q^-40 - 214q^-39 + 183q^-38 + 1028q^-37 + 339q^-36 - 1165q^-35 - 444q^-34 + 12q^-33 + 1165q^-32 + 562q^-31 - 1102q^-30 - 604q^-29 - 176q^-28 + 1151q^-27 + 718q^-26 - 898q^-25 - 633q^-24 - 352q^-23 + 942q^-22 + 741q^-21 - 586q^-20 - 493q^-19 - 435q^-18 + 602q^-17 + 589q^-16 - 295q^-15 - 249q^-14 - 371q^-13 + 286q^-12 + 346q^-11 - 127q^-10 - 54q^-9 - 222q^-8 + 106q^-7 + 146q^-6 - 68q^-5 + 27q^-4 - 95q^-3 + 37q^-2 + 46q^-1 - 43 + 30q - 32q² + 16q³ + 13q⁴ - 24q⁵ + 15q⁶ - 9q⁷ + 6q⁸ + 4q⁹ - 9q¹⁰ + 5q¹¹ - 2q¹² + q¹³ + q¹⁴ - 2q¹⁵ + q¹⁶

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 24]]
Out[2]=	PD[X[1, 4, 2, 5], X[11, 14, 12, 15], X[3, 13, 4, 12], X[13, 3, 14, 2], > X[5, 16, 6, 17], X[9, 20, 10, 1], X[19, 6, 20, 7], X[7, 18, 8, 19], > X[17, 8, 18, 9], X[15, 10, 16, 11]]
In[3]:=	GaussCode[Knot[10, 24]]
Out[3]=	GaussCode[-1, 4, -3, 1, -5, 7, -8, 9, -6, 10, -2, 3, -4, 2, -10, 5, -9, 8, -7, > 6]
In[4]:=	DTCode[Knot[10, 24]]
Out[4]=	DTCode[4, 12, 16, 18, 20, 14, 2, 10, 8, 6]
In[5]:=	br = BR[Knot[10, 24]]
Out[5]=	BR[5, {-1, -1, -2, 1, -2, -2, -2, -3, 2, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 24]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 24]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 24]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 24]][t]
Out[10]=	4 14 2 -19 - -- + -- + 14 t - 4 t 2 t t
In[11]:=	Conway[Knot[10, 24]][z]
Out[11]=	2 4 1 - 2 z - 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 18], Knot[10, 24]}
In[13]:=	{KnotDet[Knot[10, 24]], KnotSignature[Knot[10, 24]]}
Out[13]=	{55, -2}
In[14]:=	Jones[Knot[10, 24]][q]
Out[14]=	-9 2 4 7 8 9 9 7 5 -2 + q - -- + -- - -- + -- - -- + -- - -- + - + q 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, 24]}
In[16]:=	A2Invariant[Knot[10, 24]][q]
Out[16]=	-28 2 2 -18 2 -12 -10 -8 -6 -4 3 4 q + --- - --- - q - --- + q - q + q + q - q + -- + q 22 20 14 2 q q q q
In[17]:=	HOMFLYPT[Knot[10, 24]][a, z]
Out[17]=	2 4 6 8 2 4 2 6 2 8 2 2 4 4 4 6 4 1 + a - a - a + a + z - 3 a z - a z + a z - a z - 2 a z - a z
In[18]:=	Kauffman[Knot[10, 24]][a, z]
Out[18]=	2 4 6 8 3 5 9 2 2 2 4 2 1 - a - a + a + a + 2 a z + 4 a z - 2 a z - 2 z + 2 a z + 5 a z - 6 2 8 2 10 2 3 5 3 7 3 9 3 4 > 5 a z - 2 a z + 4 a z - 2 a z - 7 a z - 2 a z + 7 a z + z - 2 4 4 4 6 4 8 4 10 4 5 3 5 > 3 a z - 5 a z + 6 a z + 3 a z - 4 a z + 2 a z - 2 a z + 5 5 7 5 9 5 2 6 4 6 6 6 8 6 10 6 > a z - 2 a z - 7 a z + 3 a z + a z - 8 a z - 5 a z + a z + 3 7 5 7 9 7 4 8 6 8 8 8 5 9 7 9 > 3 a z + a z + 2 a z + 2 a z + 4 a z + 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 24]], Vassiliev[3][Knot[10, 24]]}
Out[19]=	{-2, 5}
In[20]:=	Kh[Knot[10, 24]][q, t]
Out[20]=	2 4 1 1 1 3 1 4 3 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 4 4 5 4 4 5 3 4 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + - + q t + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 3 2 > q t
In[21]:=	ColouredJones[Knot[10, 24], 2][q]
Out[21]=	-26 2 6 8 5 21 14 20 43 15 42 60 -10 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 25 23 22 21 20 19 18 17 16 15 14 q q q q q q q q q q q 7 61 65 3 67 56 9 53 36 9 29 16 4 > --- - --- + --- + --- - -- + -- + -- - -- + -- + -- - -- + -- + - + 5 q + 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 2 3 4 > q - 2 q + q

The Alternating Knot 1024

The Alternating Knot 10₂₄