The Alternating Knot 10₅₀

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X₈₄₉₃ X₂₈₃₇ X_16,10,17,9 X_14,5,15,6 X_4,15,5,16 X_20,14,1,13 X_10,20,11,19 X_18,12,19,11 X_12,18,13,17

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

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

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 + 7t^-2 - 11t^-1 + 13 - 11t + 7t² - 2t³

Conway Polynomial: 1 - z² - 5z⁴ - 2z⁶

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

Determinant and Signature: {53, 4}

Jones Polynomial: 1 - 2q + 5q² - 6q³ + 8q⁴ - 9q⁵ + 8q⁶ - 7q⁷ + 4q⁸ - 2q⁹ + q¹⁰

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

A2 (sl(3)) Invariant: 1 + q⁴ + 2q⁶ + 3q¹⁰ - q¹² - q¹⁶ - 3q¹⁸ - 2q²² + q²⁴ + q²⁶ + q³⁰

HOMFLY-PT Polynomial: 2a^-8 + 3a^-8z² + a^-8z⁴ - 4a^-6 - 6a^-6z² - 4a^-6z⁴ - a^-6z⁶ + a^-4 - a^-4z² - 3a^-4z⁴ - a^-4z⁶ + 2a^-2 + 3a^-2z² + a^-2z⁴

Kauffman Polynomial: - 2a^-12z² + a^-12z⁴ - 3a^-11z³ + 2a^-11z⁵ + 3a^-10z² - 5a^-10z⁴ + 3a^-10z⁶ - 6a^-9z + 16a^-9z³ - 11a^-9z⁵ + 4a^-9z⁷ + 2a^-8 - 3a^-8z² + 9a^-8z⁴ - 7a^-8z⁶ + 3a^-8z⁸ - 10a^-7z + 22a^-7z³ - 15a^-7z⁵ + 3a^-7z⁷ + a^-7z⁹ + 4a^-6 - 13a^-6z² + 18a^-6z⁴ - 15a^-6z⁶ + 5a^-6z⁸ - 3a^-5z + 6a^-5z³ - 8a^-5z⁵ + a^-5z⁷ + a^-5z⁹ + a^-4 - a^-4z⁴ - 4a^-4z⁶ + 2a^-4z⁸ + a^-3z + 3a^-3z³ - 6a^-3z⁵ + 2a^-3z⁷ - 2a^-2 + 5a^-2z² - 4a^-2z⁴ + a^-2z⁶

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {-1, -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=4 is the signature of 10₅₀. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8

j = 21 1

j = 19 1

j = 17 3 1

j = 15 4 1

j = 13 4 3

j = 11 5 4

j = 9 3 4

j = 7 3 5

j = 5 2 3

j = 3 1 4

j = 1 1

j = -1 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-2 - 2q^-1 + 7q - 8q² - 5q³ + 22q⁴ - 14q⁵ - 19q⁶ + 39q⁷ - 12q⁸ - 39q⁹ + 50q¹⁰ - 2q¹¹ - 54q¹² + 51q¹³ + 9q¹⁴ - 57q¹⁵ + 41q¹⁶ + 14q¹⁷ - 43q¹⁸ + 24q¹⁹ + 11q²⁰ - 22q²¹ + 9q²² + 5q²³ - 8q²⁴ + 3q²⁵ + q²⁶ - 2q²⁷ + q²⁸

3 q^-6 - 2q^-5 + 2q^-3 + 4q^-2 - 7q^-1 - 6 + 9q + 17q² - 15q³ - 26q⁴ + 9q⁵ + 51q⁶ - 9q⁷ - 61q⁸ - 17q⁹ + 85q¹⁰ + 34q¹¹ - 84q¹² - 74q¹³ + 90q¹⁴ + 96q¹⁵ - 68q¹⁶ - 134q¹⁷ + 55q¹⁸ + 153q¹⁹ - 25q²⁰ - 174q²¹ - q²² + 186q²³ + 26q²⁴ - 188q²⁵ - 53q²⁶ + 188q²⁷ + 67q²⁸ - 167q²⁹ - 85q³⁰ + 147q³¹ + 84q³² - 109q³³ - 87q³⁴ + 82q³⁵ + 67q³⁶ - 44q³⁷ - 56q³⁸ + 24q³⁹ + 37q⁴⁰ - 10q⁴¹ - 20q⁴² + 2q⁴³ + 12q⁴⁴ - 3q⁴⁵ - 3q⁴⁶ + 3q⁴⁸ - 3q⁴⁹ + q⁵⁰ + q⁵² - 2q⁵³ + q⁵⁴

4 q^-12 - 2q^-11 + 2q^-9 - q^-8 + 5q^-7 - 9q^-6 - 2q^-5 + 9q^-4 - q^-3 + 18q^-2 - 29q^-1 - 16 + 19q + 9q² + 62q³ - 56q⁴ - 60q⁵ - 3q⁶ + 10q⁷ + 169q⁸ - 35q⁹ - 106q¹⁰ - 93q¹¹ - 77q¹² + 286q¹³ + 80q¹⁴ - 44q¹⁵ - 184q¹⁶ - 298q¹⁷ + 279q¹⁸ + 204q¹⁹ + 174q²⁰ - 133q²¹ - 543q²² + 92q²³ + 189q²⁴ + 448q²⁵ + 94q²⁶ - 671q²⁷ - 172q²⁸ + 14q²⁹ + 646q³⁰ + 393q³¹ - 659q³² - 402q³³ - 219q³⁴ + 743q³⁵ + 655q³⁶ - 574q³⁷ - 562q³⁸ - 425q³⁹ + 759q⁴⁰ + 840q⁴¹ - 442q⁴² - 645q⁴³ - 591q⁴⁴ + 679q⁴⁵ + 925q⁴⁶ - 246q⁴⁷ - 600q⁴⁸ - 697q⁴⁹ + 460q⁵⁰ + 857q⁵¹ - 13q⁵² - 396q⁵³ - 674q⁵⁴ + 167q⁵⁵ + 610q⁵⁶ + 138q⁵⁷ - 117q⁵⁸ - 487q⁵⁹ - 46q⁶⁰ + 295q⁶¹ + 137q⁶² + 65q⁶³ - 243q⁶⁴ - 90q⁶⁵ + 81q⁶⁶ + 52q⁶⁷ + 95q⁶⁸ - 82q⁶⁹ - 46q⁷⁰ + 10q⁷¹ - 2q⁷² + 54q⁷³ - 24q⁷⁴ - 11q⁷⁵ + 3q⁷⁶ - 11q⁷⁷ + 21q⁷⁸ - 8q⁷⁹ - 2q⁸⁰ + 3q⁸¹ - 6q⁸² + 6q⁸³ - 2q⁸⁴ + q⁸⁶ - 2q⁸⁷ + q⁸⁸

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, 50]]

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

In[3]:=
GaussCode[Knot[10, 50]]

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

In[4]:=
DTCode[Knot[10, 50]]

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

In[5]:=
br = BR[Knot[10, 50]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 50]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 50]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 50]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 50]][t]

Out[10]=
2 7 11 2 3 13 - -- + -- - -- - 11 t + 7 t - 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 50]][z]

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

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

Out[12]=
{Knot[10, 50]}

In[13]:=
{KnotDet[Knot[10, 50]], KnotSignature[Knot[10, 50]]}

Out[13]=
{53, 4}

In[14]:=
Jones[Knot[10, 50]][q]

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

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

Out[15]=
{Knot[10, 50]}

In[16]:=
A2Invariant[Knot[10, 50]][q]

Out[16]=
4 6 10 12 16 18 22 24 26 30 1 + q + 2 q + 3 q - q - q - 3 q - 2 q + q + q + q

In[17]:=
HOMFLYPT[Knot[10, 50]][a, z]

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

In[18]:=
Kauffman[Knot[10, 50]][a, z]

Out[18]=
2 2 2 2 2 4 -4 2 6 z 10 z 3 z z 2 z 3 z 3 z 13 z -- + -- + a - -- - --- - ---- - --- + -- - ---- + ---- - ---- - ----- + 8 6 2 9 7 5 3 12 10 8 6 a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 5 z 3 z 16 z 22 z 6 z 3 z z 5 z 9 z 18 z > ---- - ---- + ----- + ----- + ---- + ---- + --- - ---- + ---- + ----- - 2 11 9 7 5 3 12 10 8 6 a a a a a a a a a a 4 4 5 5 5 5 5 6 6 6 z 4 z 2 z 11 z 15 z 8 z 6 z 3 z 7 z 15 z > -- - ---- + ---- - ----- - ----- - ---- - ---- + ---- - ---- - ----- - 4 2 11 9 7 5 3 10 8 6 a a a a a a a a a a 6 6 7 7 7 7 8 8 8 9 9 4 z z 4 z 3 z z 2 z 3 z 5 z 2 z z z > ---- + -- + ---- + ---- + -- + ---- + ---- + ---- + ---- + -- + -- 4 2 9 7 5 3 8 6 4 7 5 a a a a a a a a a a a

In[19]:=
{Vassiliev[2][Knot[10, 50]], Vassiliev[3][Knot[10, 50]]}

Out[19]=
{-1, -5}

In[20]:=
Kh[Knot[10, 50]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 50], 2][q]

Out[21]=
-2 2 2 3 4 5 6 7 8 9 q - - + 7 q - 8 q - 5 q + 22 q - 14 q - 19 q + 39 q - 12 q - 39 q + q 10 11 12 13 14 15 16 17 > 50 q - 2 q - 54 q + 51 q + 9 q - 57 q + 41 q + 14 q - 18 19 20 21 22 23 24 25 26 > 43 q + 24 q + 11 q - 22 q + 9 q + 5 q - 8 q + 3 q + q - 27 28 > 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₅₀

10₄₉
10₅₁

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-2 - 2q^-1 + 7q - 8q² - 5q³ + 22q⁴ - 14q⁵ - 19q⁶ + 39q⁷ - 12q⁸ - 39q⁹ + 50q¹⁰ - 2q¹¹ - 54q¹² + 51q¹³ + 9q¹⁴ - 57q¹⁵ + 41q¹⁶ + 14q¹⁷ - 43q¹⁸ + 24q¹⁹ + 11q²⁰ - 22q²¹ + 9q²² + 5q²³ - 8q²⁴ + 3q²⁵ + q²⁶ - 2q²⁷ + q²⁸
3	q^-6 - 2q^-5 + 2q^-3 + 4q^-2 - 7q^-1 - 6 + 9q + 17q² - 15q³ - 26q⁴ + 9q⁵ + 51q⁶ - 9q⁷ - 61q⁸ - 17q⁹ + 85q¹⁰ + 34q¹¹ - 84q¹² - 74q¹³ + 90q¹⁴ + 96q¹⁵ - 68q¹⁶ - 134q¹⁷ + 55q¹⁸ + 153q¹⁹ - 25q²⁰ - 174q²¹ - q²² + 186q²³ + 26q²⁴ - 188q²⁵ - 53q²⁶ + 188q²⁷ + 67q²⁸ - 167q²⁹ - 85q³⁰ + 147q³¹ + 84q³² - 109q³³ - 87q³⁴ + 82q³⁵ + 67q³⁶ - 44q³⁷ - 56q³⁸ + 24q³⁹ + 37q⁴⁰ - 10q⁴¹ - 20q⁴² + 2q⁴³ + 12q⁴⁴ - 3q⁴⁵ - 3q⁴⁶ + 3q⁴⁸ - 3q⁴⁹ + q⁵⁰ + q⁵² - 2q⁵³ + q⁵⁴
4	q^-12 - 2q^-11 + 2q^-9 - q^-8 + 5q^-7 - 9q^-6 - 2q^-5 + 9q^-4 - q^-3 + 18q^-2 - 29q^-1 - 16 + 19q + 9q² + 62q³ - 56q⁴ - 60q⁵ - 3q⁶ + 10q⁷ + 169q⁸ - 35q⁹ - 106q¹⁰ - 93q¹¹ - 77q¹² + 286q¹³ + 80q¹⁴ - 44q¹⁵ - 184q¹⁶ - 298q¹⁷ + 279q¹⁸ + 204q¹⁹ + 174q²⁰ - 133q²¹ - 543q²² + 92q²³ + 189q²⁴ + 448q²⁵ + 94q²⁶ - 671q²⁷ - 172q²⁸ + 14q²⁹ + 646q³⁰ + 393q³¹ - 659q³² - 402q³³ - 219q³⁴ + 743q³⁵ + 655q³⁶ - 574q³⁷ - 562q³⁸ - 425q³⁹ + 759q⁴⁰ + 840q⁴¹ - 442q⁴² - 645q⁴³ - 591q⁴⁴ + 679q⁴⁵ + 925q⁴⁶ - 246q⁴⁷ - 600q⁴⁸ - 697q⁴⁹ + 460q⁵⁰ + 857q⁵¹ - 13q⁵² - 396q⁵³ - 674q⁵⁴ + 167q⁵⁵ + 610q⁵⁶ + 138q⁵⁷ - 117q⁵⁸ - 487q⁵⁹ - 46q⁶⁰ + 295q⁶¹ + 137q⁶² + 65q⁶³ - 243q⁶⁴ - 90q⁶⁵ + 81q⁶⁶ + 52q⁶⁷ + 95q⁶⁸ - 82q⁶⁹ - 46q⁷⁰ + 10q⁷¹ - 2q⁷² + 54q⁷³ - 24q⁷⁴ - 11q⁷⁵ + 3q⁷⁶ - 11q⁷⁷ + 21q⁷⁸ - 8q⁷⁹ - 2q⁸⁰ + 3q⁸¹ - 6q⁸² + 6q⁸³ - 2q⁸⁴ + q⁸⁶ - 2q⁸⁷ + q⁸⁸

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 50]]
Out[2]=	PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[2, 8, 3, 7], X[16, 10, 17, 9], > X[14, 5, 15, 6], X[4, 15, 5, 16], X[20, 14, 1, 13], X[10, 20, 11, 19], > X[18, 12, 19, 11], X[12, 18, 13, 17]]
In[3]:=	GaussCode[Knot[10, 50]]
Out[3]=	GaussCode[1, -3, 2, -6, 5, -1, 3, -2, 4, -8, 9, -10, 7, -5, 6, -4, 10, -9, 8, > -7]
In[4]:=	DTCode[Knot[10, 50]]
Out[4]=	DTCode[6, 8, 14, 2, 16, 18, 20, 4, 12, 10]
In[5]:=	br = BR[Knot[10, 50]]
Out[5]=	BR[4, {1, 1, 2, -1, 2, 2, -3, 2, 2, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 50]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 50]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 50]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 50]][t]
Out[10]=	2 7 11 2 3 13 - -- + -- - -- - 11 t + 7 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 50]][z]
Out[11]=	2 4 6 1 - z - 5 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 50]}
In[13]:=	{KnotDet[Knot[10, 50]], KnotSignature[Knot[10, 50]]}
Out[13]=	{53, 4}
In[14]:=	Jones[Knot[10, 50]][q]
Out[14]=	2 3 4 5 6 7 8 9 10 1 - 2 q + 5 q - 6 q + 8 q - 9 q + 8 q - 7 q + 4 q - 2 q + q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 50]}
In[16]:=	A2Invariant[Knot[10, 50]][q]
Out[16]=	4 6 10 12 16 18 22 24 26 30 1 + q + 2 q + 3 q - q - q - 3 q - 2 q + q + q + q
In[17]:=	HOMFLYPT[Knot[10, 50]][a, z]
Out[17]=	2 2 2 2 4 4 4 4 6 6 2 4 -4 2 3 z 6 z z 3 z z 4 z 3 z z z z -- - -- + a + -- + ---- - ---- - -- + ---- + -- - ---- - ---- + -- - -- - -- 8 6 2 8 6 4 2 8 6 4 2 6 4 a a a a a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 50]][a, z]
Out[18]=	2 2 2 2 2 4 -4 2 6 z 10 z 3 z z 2 z 3 z 3 z 13 z -- + -- + a - -- - --- - ---- - --- + -- - ---- + ---- - ---- - ----- + 8 6 2 9 7 5 3 12 10 8 6 a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 5 z 3 z 16 z 22 z 6 z 3 z z 5 z 9 z 18 z > ---- - ---- + ----- + ----- + ---- + ---- + --- - ---- + ---- + ----- - 2 11 9 7 5 3 12 10 8 6 a a a a a a a a a a 4 4 5 5 5 5 5 6 6 6 z 4 z 2 z 11 z 15 z 8 z 6 z 3 z 7 z 15 z > -- - ---- + ---- - ----- - ----- - ---- - ---- + ---- - ---- - ----- - 4 2 11 9 7 5 3 10 8 6 a a a a a a a a a a 6 6 7 7 7 7 8 8 8 9 9 4 z z 4 z 3 z z 2 z 3 z 5 z 2 z z z > ---- + -- + ---- + ---- + -- + ---- + ---- + ---- + ---- + -- + -- 4 2 9 7 5 3 8 6 4 7 5 a a a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 50]], Vassiliev[3][Knot[10, 50]]}
Out[19]=	{-1, -5}
In[20]:=	Kh[Knot[10, 50]][q, t]
Out[20]=	3 3 5 1 q q 5 7 7 2 9 2 9 3 4 q + 2 q + ---- + - + -- + 3 q t + 3 q t + 5 q t + 3 q t + 4 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 17 6 > 5 q t + 4 q t + 4 q t + 3 q t + 4 q t + q t + 3 q t + 17 7 19 7 21 8 > q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 50], 2][q]
Out[21]=	-2 2 2 3 4 5 6 7 8 9 q - - + 7 q - 8 q - 5 q + 22 q - 14 q - 19 q + 39 q - 12 q - 39 q + q 10 11 12 13 14 15 16 17 > 50 q - 2 q - 54 q + 51 q + 9 q - 57 q + 41 q + 14 q - 18 19 20 21 22 23 24 25 26 > 43 q + 24 q + 11 q - 22 q + 9 q + 5 q - 8 q + 3 q + q - 27 28 > 2 q + q

The Alternating Knot 1050

The Alternating Knot 10₅₀