The Alternating Knot 10₂₀

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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_5,18,6,19 X_7,20,8,1 X_19,6,20,7 X_9,16,10,17 X_15,10,16,11 X_17,8,18,9

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

DT (Dowker-Thistlethwaite) Code: 4 12 18 20 16 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: - 3t^-2 + 9t^-1 - 11 + 9t - 3t²

Conway Polynomial: 1 - 3z² - 3z⁴

Other knots with the same Alexander/Conway Polynomial: {10₁₆₂, K11n117, ...}

Determinant and Signature: {35, -2}

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

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

A2 (sl(3)) Invariant: q^-28 + q^-22 - q^-20 - q^-14 - q^-10 - q^-4 + 2q^-2 + 1 + q² + q⁴

HOMFLY-PT Polynomial: 2 + z² - a² - 2a²z² - a²z⁴ - a⁴z² - a⁴z⁴ - a⁶ - 2a⁶z² - a⁶z⁴ + a⁸ + a⁸z²

Kauffman Polynomial: 2 - 3z² + z⁴ - az - az³ + az⁵ + a² - 2a²z² + a²z⁶ - a³z + 2a³z³ - a³z⁵ + a³z⁷ + 3a⁴z⁴ - 2a⁴z⁶ + a⁴z⁸ + 3a⁵z - 8a⁵z³ + 9a⁵z⁵ - 4a⁵z⁷ + a⁵z⁹ + a⁶ - 9a⁶z² + 17a⁶z⁴ - 12a⁶z⁶ + 3a⁶z⁸ + 2a⁷z - 4a⁷z³ + 3a⁷z⁵ - 3a⁷z⁷ + a⁷z⁹ + a⁸ - 5a⁸z² + 9a⁸z⁴ - 8a⁸z⁶ + 2a⁸z⁸ - a⁹z + 7a⁹z³ - 8a⁹z⁵ + 2a⁹z⁷ + 3a¹⁰z² - 4a¹⁰z⁴ + a¹⁰z⁶

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-3, 6}

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 10₂₀. 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

j = -1 3 1

j = -3 2 1

j = -5 3 2

j = -7 3 2

j = -9 2 3

j = -11 2 3

j = -13 1 2

j = -15 1 2

j = -17 1

j = -19 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-26 - 2q^-25 + 5q^-23 - 6q^-22 - 3q^-21 + 12q^-20 - 7q^-19 - 9q^-18 + 17q^-17 - 5q^-16 - 15q^-15 + 19q^-14 - q^-13 - 19q^-12 + 19q^-11 + 2q^-10 - 19q^-9 + 15q^-8 + 2q^-7 - 13q^-6 + 10q^-5 - 7q^-3 + 6q^-2 - q^-1 - 3 + 3q - q³ + q⁴

3 q^-51 - 2q^-50 + 2q^-48 + 2q^-47 - 5q^-46 - 4q^-45 + 7q^-44 + 9q^-43 - 9q^-42 - 13q^-41 + 6q^-40 + 20q^-39 - 4q^-38 - 23q^-37 - q^-36 + 24q^-35 + 6q^-34 - 22q^-33 - 10q^-32 + 18q^-31 + 12q^-30 - 12q^-29 - 12q^-28 + 5q^-27 + 13q^-26 - 11q^-24 - 8q^-23 + 12q^-22 + 9q^-21 - 7q^-20 - 16q^-19 + 8q^-18 + 15q^-17 - 2q^-16 - 15q^-15 - 2q^-14 + 11q^-13 + 6q^-12 - 5q^-11 - 9q^-10 + q^-9 + 6q^-8 + 8q^-7 - 9q^-6 - 4q^-5 + 11q^-3 - 4q^-2 - 4q^-1 - 3 + 7q - q² - q³ - 3q⁴ + 3q⁵ - q⁸ + q⁹

4 q^-84 - 2q^-83 + 2q^-81 - q^-80 + 3q^-79 - 7q^-78 + 7q^-76 - q^-75 + 10q^-74 - 19q^-73 - 8q^-72 + 13q^-71 + 3q^-70 + 29q^-69 - 28q^-68 - 23q^-67 + 6q^-66 - 2q^-65 + 57q^-64 - 22q^-63 - 29q^-62 - 4q^-61 - 22q^-60 + 69q^-59 - 16q^-58 - 16q^-57 + 6q^-56 - 41q^-55 + 59q^-54 - 33q^-53 - 5q^-52 + 40q^-51 - 34q^-50 + 45q^-49 - 74q^-48 - 13q^-47 + 80q^-46 - 5q^-45 + 42q^-44 - 119q^-43 - 38q^-42 + 110q^-41 + 29q^-40 + 47q^-39 - 153q^-38 - 63q^-37 + 129q^-36 + 56q^-35 + 53q^-34 - 174q^-33 - 83q^-32 + 137q^-31 + 77q^-30 + 57q^-29 - 180q^-28 - 99q^-27 + 125q^-26 + 89q^-25 + 73q^-24 - 159q^-23 - 114q^-22 + 85q^-21 + 81q^-20 + 94q^-19 - 109q^-18 - 106q^-17 + 33q^-16 + 45q^-15 + 96q^-14 - 50q^-13 - 73q^-12 - q^-11 + 8q^-10 + 73q^-9 - 13q^-8 - 37q^-7 - 9q^-6 - 11q^-5 + 44q^-4 - 14q^-2 - 6q^-1 - 13 + 22q + q² - 3q³ - 2q⁴ - 9q⁵ + 9q⁶ - 4q¹⁰ + 3q¹¹ - q¹⁵ + 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, 20]]

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, 18, 6, 19], X[7, 20, 8, 1], X[19, 6, 20, 7], X[9, 16, 10, 17], > X[15, 10, 16, 11], X[17, 8, 18, 9]]

In[3]:=
GaussCode[Knot[10, 20]]

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

In[4]:=
DTCode[Knot[10, 20]]

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

In[5]:=
br = BR[Knot[10, 20]]

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

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

Out[6]=
{5, 12}

In[7]:=
BraidIndex[Knot[10, 20]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[10, 20]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 20]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 20]][t]

Out[10]=
3 9 2 -11 - -- + - + 9 t - 3 t 2 t t

In[11]:=
Conway[Knot[10, 20]][z]

Out[11]=
2 4 1 - 3 z - 3 z

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

Out[12]=
{Knot[10, 20], Knot[10, 162], Knot[11, NonAlternating, 117]}

In[13]:=
{KnotDet[Knot[10, 20]], KnotSignature[Knot[10, 20]]}

Out[13]=
{35, -2}

In[14]:=
Jones[Knot[10, 20]][q]

Out[14]=
-9 2 3 4 5 6 5 4 3 -1 + 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, 20]}

In[16]:=
A2Invariant[Knot[10, 20]][q]

Out[16]=
-28 -22 -20 -14 -10 -4 2 2 4 1 + q + q - q - q - q - q + -- + q + q 2 q

In[17]:=
HOMFLYPT[Knot[10, 20]][a, z]

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

In[18]:=
Kauffman[Knot[10, 20]][a, z]

Out[18]=
2 6 8 3 5 7 9 2 2 2 2 + a + a + a - a z - a z + 3 a z + 2 a z - a z - 3 z - 2 a z - 6 2 8 2 10 2 3 3 3 5 3 7 3 > 9 a z - 5 a z + 3 a z - a z + 2 a z - 8 a z - 4 a z + 9 3 4 4 4 6 4 8 4 10 4 5 3 5 > 7 a z + z + 3 a z + 17 a z + 9 a z - 4 a z + a z - a z + 5 5 7 5 9 5 2 6 4 6 6 6 8 6 > 9 a z + 3 a z - 8 a z + a z - 2 a z - 12 a z - 8 a z + 10 6 3 7 5 7 7 7 9 7 4 8 6 8 8 8 > a z + a z - 4 a z - 3 a z + 2 a z + a z + 3 a z + 2 a z + 5 9 7 9 > a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 20]], Vassiliev[3][Knot[10, 20]]}

Out[19]=
{-3, 6}

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

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

In[21]:=
ColouredJones[Knot[10, 20], 2][q]

Out[21]=
-26 2 5 6 3 12 7 9 17 5 15 19 -3 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 25 23 22 21 20 19 18 17 16 15 14 q q q q q q q q q q q -13 19 19 2 19 15 2 13 10 7 6 1 3 > q - --- + --- + --- - -- + -- + -- - -- + -- - -- + -- - - + 3 q - q + 12 11 10 9 8 7 6 5 3 2 q q q q q q q q q q q 4 > q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₂₀

10₁₉
10₂₁

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-26 - 2q^-25 + 5q^-23 - 6q^-22 - 3q^-21 + 12q^-20 - 7q^-19 - 9q^-18 + 17q^-17 - 5q^-16 - 15q^-15 + 19q^-14 - q^-13 - 19q^-12 + 19q^-11 + 2q^-10 - 19q^-9 + 15q^-8 + 2q^-7 - 13q^-6 + 10q^-5 - 7q^-3 + 6q^-2 - q^-1 - 3 + 3q - q³ + q⁴
3	q^-51 - 2q^-50 + 2q^-48 + 2q^-47 - 5q^-46 - 4q^-45 + 7q^-44 + 9q^-43 - 9q^-42 - 13q^-41 + 6q^-40 + 20q^-39 - 4q^-38 - 23q^-37 - q^-36 + 24q^-35 + 6q^-34 - 22q^-33 - 10q^-32 + 18q^-31 + 12q^-30 - 12q^-29 - 12q^-28 + 5q^-27 + 13q^-26 - 11q^-24 - 8q^-23 + 12q^-22 + 9q^-21 - 7q^-20 - 16q^-19 + 8q^-18 + 15q^-17 - 2q^-16 - 15q^-15 - 2q^-14 + 11q^-13 + 6q^-12 - 5q^-11 - 9q^-10 + q^-9 + 6q^-8 + 8q^-7 - 9q^-6 - 4q^-5 + 11q^-3 - 4q^-2 - 4q^-1 - 3 + 7q - q² - q³ - 3q⁴ + 3q⁵ - q⁸ + q⁹
4	q^-84 - 2q^-83 + 2q^-81 - q^-80 + 3q^-79 - 7q^-78 + 7q^-76 - q^-75 + 10q^-74 - 19q^-73 - 8q^-72 + 13q^-71 + 3q^-70 + 29q^-69 - 28q^-68 - 23q^-67 + 6q^-66 - 2q^-65 + 57q^-64 - 22q^-63 - 29q^-62 - 4q^-61 - 22q^-60 + 69q^-59 - 16q^-58 - 16q^-57 + 6q^-56 - 41q^-55 + 59q^-54 - 33q^-53 - 5q^-52 + 40q^-51 - 34q^-50 + 45q^-49 - 74q^-48 - 13q^-47 + 80q^-46 - 5q^-45 + 42q^-44 - 119q^-43 - 38q^-42 + 110q^-41 + 29q^-40 + 47q^-39 - 153q^-38 - 63q^-37 + 129q^-36 + 56q^-35 + 53q^-34 - 174q^-33 - 83q^-32 + 137q^-31 + 77q^-30 + 57q^-29 - 180q^-28 - 99q^-27 + 125q^-26 + 89q^-25 + 73q^-24 - 159q^-23 - 114q^-22 + 85q^-21 + 81q^-20 + 94q^-19 - 109q^-18 - 106q^-17 + 33q^-16 + 45q^-15 + 96q^-14 - 50q^-13 - 73q^-12 - q^-11 + 8q^-10 + 73q^-9 - 13q^-8 - 37q^-7 - 9q^-6 - 11q^-5 + 44q^-4 - 14q^-2 - 6q^-1 - 13 + 22q + q² - 3q³ - 2q⁴ - 9q⁵ + 9q⁶ - 4q¹⁰ + 3q¹¹ - q¹⁵ + q¹⁶

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

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 20]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 20]][t]
Out[10]=	3 9 2 -11 - -- + - + 9 t - 3 t 2 t t
In[11]:=	Conway[Knot[10, 20]][z]
Out[11]=	2 4 1 - 3 z - 3 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 20], Knot[10, 162], Knot[11, NonAlternating, 117]}
In[13]:=	{KnotDet[Knot[10, 20]], KnotSignature[Knot[10, 20]]}
Out[13]=	{35, -2}
In[14]:=	Jones[Knot[10, 20]][q]
Out[14]=	-9 2 3 4 5 6 5 4 3 -1 + 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, 20]}
In[16]:=	A2Invariant[Knot[10, 20]][q]
Out[16]=	-28 -22 -20 -14 -10 -4 2 2 4 1 + q + q - q - q - q - q + -- + q + q 2 q
In[17]:=	HOMFLYPT[Knot[10, 20]][a, z]
Out[17]=	2 6 8 2 2 2 4 2 6 2 8 2 2 4 4 4 2 - a - a + a + z - 2 a z - a z - 2 a z + a z - a z - a z - 6 4 > a z
In[18]:=	Kauffman[Knot[10, 20]][a, z]
Out[18]=	2 6 8 3 5 7 9 2 2 2 2 + a + a + a - a z - a z + 3 a z + 2 a z - a z - 3 z - 2 a z - 6 2 8 2 10 2 3 3 3 5 3 7 3 > 9 a z - 5 a z + 3 a z - a z + 2 a z - 8 a z - 4 a z + 9 3 4 4 4 6 4 8 4 10 4 5 3 5 > 7 a z + z + 3 a z + 17 a z + 9 a z - 4 a z + a z - a z + 5 5 7 5 9 5 2 6 4 6 6 6 8 6 > 9 a z + 3 a z - 8 a z + a z - 2 a z - 12 a z - 8 a z + 10 6 3 7 5 7 7 7 9 7 4 8 6 8 8 8 > a z + a z - 4 a z - 3 a z + 2 a z + a z + 3 a z + 2 a z + 5 9 7 9 > a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 20]], Vassiliev[3][Knot[10, 20]]}
Out[19]=	{-3, 6}
In[20]:=	Kh[Knot[10, 20]][q, t]
Out[20]=	-3 3 1 1 1 2 1 2 2 q + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q t q t q t q t q t q t q t 3 2 3 3 2 3 2 2 t 3 2 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + - + 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
In[21]:=	ColouredJones[Knot[10, 20], 2][q]
Out[21]=	-26 2 5 6 3 12 7 9 17 5 15 19 -3 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 25 23 22 21 20 19 18 17 16 15 14 q q q q q q q q q q q -13 19 19 2 19 15 2 13 10 7 6 1 3 > q - --- + --- + --- - -- + -- + -- - -- + -- - -- + -- - - + 3 q - q + 12 11 10 9 8 7 6 5 3 2 q q q q q q q q q q q 4 > q

The Alternating Knot 1020

The Alternating Knot 10₂₀