The Alternating Knot 10₁₁

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Acknowledgement

KnotPlot

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

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

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

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--3 2 2 / NotAvailable 1

Alexander Polynomial: - 4t^-2 + 11t^-1 - 13 + 11t - 4t²

Conway Polynomial: 1 - 5z² - 4z⁴

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

Determinant and Signature: {43, -2}

Jones Polynomial: q^-7 - 2q^-6 + 4q^-5 - 6q^-4 + 7q^-3 - 7q^-2 + 6q^-1 - 5 + 3q - q² + q³

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

A2 (sl(3)) Invariant: q^-22 + 2q^-16 - q^-14 - q^-8 + q^-6 - 2q^-4 - 1 - q² + 2q⁴ + q⁶ + q⁸ + q¹⁰

HOMFLY-PT Polynomial: 2a^-2 + a^-2z² - 1 - 2z² - z⁴ - a² - 4a²z² - 2a²z⁴ - a⁴z² - a⁴z⁴ + a⁶ + a⁶z²

Kauffman Polynomial: - 2a^-2 + 7a^-2z² - 5a^-2z⁴ + a^-2z⁶ + a^-1z + a^-1z³ - 3a^-1z⁵ + a^-1z⁷ - 1 + 2z² - z⁴ - 2z⁶ + z⁸ + 5az - 16az³ + 11az⁵ - 4az⁷ + az⁹ + a² - 12a²z² + 16a²z⁴ - 10a²z⁶ + 3a²z⁸ + 2a³z - 5a³z³ + 5a³z⁵ - 2a³z⁷ + a³z⁹ + 5a⁴z⁴ - 4a⁴z⁶ + 2a⁴z⁸ - 2a⁵z + 9a⁵z³ - 7a⁵z⁵ + 3a⁵z⁷ - a⁶ + 5a⁶z² - 6a⁶z⁴ + 3a⁶z⁶ - 3a⁷z³ + 2a⁷z⁵ - 2a⁸z² + a⁸z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-5, 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 10₁₁. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4

j = 7 1

j = 5

j = 3 3 1

j = 1 2

j = -1 4 3

j = -3 4 3

j = -5 3 3

j = -7 3 4

j = -9 1 3

j = -11 1 3

j = -13 1

j = -15 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-20 - 2q^-19 + q^-18 + 4q^-17 - 8q^-16 + 2q^-15 + 12q^-14 - 19q^-13 + 2q^-12 + 27q^-11 - 32q^-10 - q^-9 + 39q^-8 - 37q^-7 - 5q^-6 + 41q^-5 - 31q^-4 - 10q^-3 + 35q^-2 - 19q^-1 - 12 + 23q - 8q² - 9q³ + 10q⁴ - 2q⁵ - 4q⁶ + 3q⁷ - q⁹ + q¹⁰

3 q^-39 - 2q^-38 + q^-37 + q^-36 + q^-35 - 5q^-34 + q^-33 + 5q^-32 + q^-31 - 11q^-30 + 2q^-29 + 14q^-28 + q^-27 - 27q^-26 + q^-25 + 39q^-24 + 4q^-23 - 56q^-22 - 12q^-21 + 76q^-20 + 20q^-19 - 90q^-18 - 32q^-17 + 101q^-16 + 43q^-15 - 105q^-14 - 53q^-13 + 103q^-12 + 60q^-11 - 95q^-10 - 67q^-9 + 84q^-8 + 69q^-7 - 66q^-6 - 74q^-5 + 52q^-4 + 72q^-3 - 34q^-2 - 68q^-1 + 17 + 61q - 4q² - 48q³ - 7q⁴ + 38q⁵ + 8q⁶ - 20q⁷ - 14q⁸ + 16q⁹ + 6q¹⁰ - 4q¹¹ - 8q¹² + 5q¹³ + q¹⁴ - 3q¹⁶ + 2q¹⁷ - q²⁰ + q²¹

4 q^-64 - 2q^-63 + q^-62 + q^-61 - 2q^-60 + 4q^-59 - 7q^-58 + 3q^-57 + 4q^-56 - 7q^-55 + 13q^-54 - 16q^-53 + 5q^-52 + 8q^-51 - 19q^-50 + 28q^-49 - 27q^-48 + 16q^-47 + 17q^-46 - 43q^-45 + 33q^-44 - 52q^-43 + 49q^-42 + 59q^-41 - 70q^-40 + 8q^-39 - 122q^-38 + 92q^-37 + 154q^-36 - 61q^-35 - 30q^-34 - 255q^-33 + 103q^-32 + 274q^-31 - 43q^-29 - 398q^-28 + 67q^-27 + 350q^-26 + 77q^-25 - 8q^-24 - 484q^-23 + 8q^-22 + 355q^-21 + 126q^-20 + 48q^-19 - 492q^-18 - 41q^-17 + 303q^-16 + 141q^-15 + 109q^-14 - 443q^-13 - 81q^-12 + 216q^-11 + 138q^-10 + 171q^-9 - 356q^-8 - 118q^-7 + 104q^-6 + 119q^-5 + 226q^-4 - 238q^-3 - 132q^-2 - 6q^-1 + 66 + 238q - 106q² - 98q³ - 75q⁴ - 7q⁵ + 188q⁶ - 13q⁷ - 33q⁸ - 74q⁹ - 52q¹⁰ + 100q¹¹ + 15q¹² + 16q¹³ - 35q¹⁴ - 49q¹⁵ + 35q¹⁶ + 3q¹⁷ + 22q¹⁸ - 5q¹⁹ - 24q²⁰ + 11q²¹ - 7q²² + 10q²³ + 2q²⁴ - 8q²⁵ + 6q²⁶ - 5q²⁷ + 2q²⁸ + q²⁹ - 3q³⁰ + 3q³¹ - q³² - q³⁵ + q³⁶

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, 11]]

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

In[3]:=
GaussCode[Knot[10, 11]]

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

In[4]:=
DTCode[Knot[10, 11]]

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

In[5]:=
br = BR[Knot[10, 11]]

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

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

Out[6]=
{5, 12}

In[7]:=
BraidIndex[Knot[10, 11]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[10, 11]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 11]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 11]][t]

Out[10]=
4 11 2 -13 - -- + -- + 11 t - 4 t 2 t t

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

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

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

Out[12]=
{Knot[10, 11]}

In[13]:=
{KnotDet[Knot[10, 11]], KnotSignature[Knot[10, 11]]}

Out[13]=
{43, -2}

In[14]:=
Jones[Knot[10, 11]][q]

Out[14]=
-7 2 4 6 7 7 6 2 3 -5 + q - -- + -- - -- + -- - -- + - + 3 q - q + q 6 5 4 3 2 q q q q q q

In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]

Out[15]=
{Knot[10, 11]}

In[16]:=
A2Invariant[Knot[10, 11]][q]

Out[16]=
-22 2 -14 -8 -6 2 2 4 6 8 10 -1 + q + --- - q - q + q - -- - q + 2 q + q + q + q 16 4 q q

In[17]:=
HOMFLYPT[Knot[10, 11]][a, z]

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

In[18]:=
Kauffman[Knot[10, 11]][a, z]

Out[18]=
2 2 2 6 z 3 5 2 7 z 2 2 -1 - -- + a - a + - + 5 a z + 2 a z - 2 a z + 2 z + ---- - 12 a z + 2 a 2 a a 3 6 2 8 2 z 3 3 3 5 3 7 3 4 > 5 a z - 2 a z + -- - 16 a z - 5 a z + 9 a z - 3 a z - z - a 4 5 5 z 2 4 4 4 6 4 8 4 3 z 5 3 5 > ---- + 16 a z + 5 a z - 6 a z + a z - ---- + 11 a z + 5 a z - 2 a a 6 7 5 5 7 5 6 z 2 6 4 6 6 6 z > 7 a z + 2 a z - 2 z + -- - 10 a z - 4 a z + 3 a z + -- - 2 a a 7 3 7 5 7 8 2 8 4 8 9 3 9 > 4 a z - 2 a z + 3 a z + z + 3 a z + 2 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 11]], Vassiliev[3][Knot[10, 11]]}

Out[19]=
{-5, 4}

In[20]:=
Kh[Knot[10, 11]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 11], 2][q]

Out[21]=
-20 2 -18 4 8 2 12 19 2 27 32 -9 -12 + q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- - q + 19 17 16 15 14 13 12 11 10 q q q q q q q q q 39 37 5 41 31 10 35 19 2 3 4 5 > -- - -- - -- + -- - -- - -- + -- - -- + 23 q - 8 q - 9 q + 10 q - 2 q - 8 7 6 5 4 3 2 q q q q q q q q 6 7 9 10 > 4 q + 3 q - q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₁

10₁₀
10₁₂

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-20 - 2q^-19 + q^-18 + 4q^-17 - 8q^-16 + 2q^-15 + 12q^-14 - 19q^-13 + 2q^-12 + 27q^-11 - 32q^-10 - q^-9 + 39q^-8 - 37q^-7 - 5q^-6 + 41q^-5 - 31q^-4 - 10q^-3 + 35q^-2 - 19q^-1 - 12 + 23q - 8q² - 9q³ + 10q⁴ - 2q⁵ - 4q⁶ + 3q⁷ - q⁹ + q¹⁰
3	q^-39 - 2q^-38 + q^-37 + q^-36 + q^-35 - 5q^-34 + q^-33 + 5q^-32 + q^-31 - 11q^-30 + 2q^-29 + 14q^-28 + q^-27 - 27q^-26 + q^-25 + 39q^-24 + 4q^-23 - 56q^-22 - 12q^-21 + 76q^-20 + 20q^-19 - 90q^-18 - 32q^-17 + 101q^-16 + 43q^-15 - 105q^-14 - 53q^-13 + 103q^-12 + 60q^-11 - 95q^-10 - 67q^-9 + 84q^-8 + 69q^-7 - 66q^-6 - 74q^-5 + 52q^-4 + 72q^-3 - 34q^-2 - 68q^-1 + 17 + 61q - 4q² - 48q³ - 7q⁴ + 38q⁵ + 8q⁶ - 20q⁷ - 14q⁸ + 16q⁹ + 6q¹⁰ - 4q¹¹ - 8q¹² + 5q¹³ + q¹⁴ - 3q¹⁶ + 2q¹⁷ - q²⁰ + q²¹
4	q^-64 - 2q^-63 + q^-62 + q^-61 - 2q^-60 + 4q^-59 - 7q^-58 + 3q^-57 + 4q^-56 - 7q^-55 + 13q^-54 - 16q^-53 + 5q^-52 + 8q^-51 - 19q^-50 + 28q^-49 - 27q^-48 + 16q^-47 + 17q^-46 - 43q^-45 + 33q^-44 - 52q^-43 + 49q^-42 + 59q^-41 - 70q^-40 + 8q^-39 - 122q^-38 + 92q^-37 + 154q^-36 - 61q^-35 - 30q^-34 - 255q^-33 + 103q^-32 + 274q^-31 - 43q^-29 - 398q^-28 + 67q^-27 + 350q^-26 + 77q^-25 - 8q^-24 - 484q^-23 + 8q^-22 + 355q^-21 + 126q^-20 + 48q^-19 - 492q^-18 - 41q^-17 + 303q^-16 + 141q^-15 + 109q^-14 - 443q^-13 - 81q^-12 + 216q^-11 + 138q^-10 + 171q^-9 - 356q^-8 - 118q^-7 + 104q^-6 + 119q^-5 + 226q^-4 - 238q^-3 - 132q^-2 - 6q^-1 + 66 + 238q - 106q² - 98q³ - 75q⁴ - 7q⁵ + 188q⁶ - 13q⁷ - 33q⁸ - 74q⁹ - 52q¹⁰ + 100q¹¹ + 15q¹² + 16q¹³ - 35q¹⁴ - 49q¹⁵ + 35q¹⁶ + 3q¹⁷ + 22q¹⁸ - 5q¹⁹ - 24q²⁰ + 11q²¹ - 7q²² + 10q²³ + 2q²⁴ - 8q²⁵ + 6q²⁶ - 5q²⁷ + 2q²⁸ + q²⁹ - 3q³⁰ + 3q³¹ - q³² - q³⁵ + q³⁶

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

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 11]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, {2, 3}, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 11]][t]
Out[10]=	4 11 2 -13 - -- + -- + 11 t - 4 t 2 t t
In[11]:=	Conway[Knot[10, 11]][z]
Out[11]=	2 4 1 - 5 z - 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 11]}
In[13]:=	{KnotDet[Knot[10, 11]], KnotSignature[Knot[10, 11]]}
Out[13]=	{43, -2}
In[14]:=	Jones[Knot[10, 11]][q]
Out[14]=	-7 2 4 6 7 7 6 2 3 -5 + q - -- + -- - -- + -- - -- + - + 3 q - q + q 6 5 4 3 2 q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 11]}
In[16]:=	A2Invariant[Knot[10, 11]][q]
Out[16]=	-22 2 -14 -8 -6 2 2 4 6 8 10 -1 + q + --- - q - q + q - -- - q + 2 q + q + q + q 16 4 q q
In[17]:=	HOMFLYPT[Knot[10, 11]][a, z]
Out[17]=	2 2 2 6 2 z 2 2 4 2 6 2 4 2 4 4 4 -1 + -- - a + a - 2 z + -- - 4 a z - a z + a z - z - 2 a z - a z 2 2 a a
In[18]:=	Kauffman[Knot[10, 11]][a, z]
Out[18]=	2 2 2 6 z 3 5 2 7 z 2 2 -1 - -- + a - a + - + 5 a z + 2 a z - 2 a z + 2 z + ---- - 12 a z + 2 a 2 a a 3 6 2 8 2 z 3 3 3 5 3 7 3 4 > 5 a z - 2 a z + -- - 16 a z - 5 a z + 9 a z - 3 a z - z - a 4 5 5 z 2 4 4 4 6 4 8 4 3 z 5 3 5 > ---- + 16 a z + 5 a z - 6 a z + a z - ---- + 11 a z + 5 a z - 2 a a 6 7 5 5 7 5 6 z 2 6 4 6 6 6 z > 7 a z + 2 a z - 2 z + -- - 10 a z - 4 a z + 3 a z + -- - 2 a a 7 3 7 5 7 8 2 8 4 8 9 3 9 > 4 a z - 2 a z + 3 a z + z + 3 a z + 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 11]], Vassiliev[3][Knot[10, 11]]}
Out[19]=	{-5, 4}
In[20]:=	Kh[Knot[10, 11]][q, t]
Out[20]=	3 4 1 1 1 3 1 3 3 4 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 3 3 4 3 t 3 2 3 3 7 4 > ----- + ---- + ---- + --- + 2 q t + 3 q t + q t + q t 5 2 5 3 q q t q t q t
In[21]:=	ColouredJones[Knot[10, 11], 2][q]
Out[21]=	-20 2 -18 4 8 2 12 19 2 27 32 -9 -12 + q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- - q + 19 17 16 15 14 13 12 11 10 q q q q q q q q q 39 37 5 41 31 10 35 19 2 3 4 5 > -- - -- - -- + -- - -- - -- + -- - -- + 23 q - 8 q - 9 q + 10 q - 2 q - 8 7 6 5 4 3 2 q q q q q q q q 6 7 9 10 > 4 q + 3 q - q + q

The Alternating Knot 1011

The Alternating Knot 10₁₁