The Alternating Knot 9₅

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X_14,6,15,5 X_18,8,1,7 X_16,10,17,9 X_10,16,11,15 X_8,18,9,17 X_2,14,3,13 X_12,4,13,3 X_4,12,5,11

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

DT (Dowker-Thistlethwaite) Code: 6 12 14 18 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 1 2 / 4--6 1

Alexander Polynomial: 6t^-1 - 11 + 6t

Conway Polynomial: 1 + 6z²

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

Determinant and Signature: {23, 2}

Jones Polynomial: q - 2q² + 3q³ - 3q⁴ + 4q⁵ - 3q⁶ + 3q⁷ - 2q⁸ + q⁹ - q¹⁰

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

A2 (sl(3)) Invariant: q² - q⁴ + q⁸ + q¹² + q¹⁴ + q¹⁶ + q¹⁸ + q²² - q²⁶ - q³⁰ - q³²

HOMFLY-PT Polynomial: - a^-10 + a^-8z² + a^-6 + 2a^-6z² + a^-4 + 2a^-4z² + a^-2z²

Kauffman Polynomial: - 6a^-11z + 11a^-11z³ - 6a^-11z⁵ + a^-11z⁷ + a^-10 - 3a^-10z² + 7a^-10z⁴ - 5a^-10z⁶ + a^-10z⁸ - 6a^-9z + 18a^-9z³ - 14a^-9z⁵ + 3a^-9z⁷ + 4a^-8z² - 3a^-8z⁴ - 2a^-8z⁶ + a^-8z⁸ + a^-7z³ - 5a^-7z⁵ + 2a^-7z⁷ - a^-6 + 3a^-6z² - 7a^-6z⁴ + 3a^-6z⁶ - 4a^-5z³ + 3a^-5z⁵ + a^-4 - 3a^-4z² + 3a^-4z⁴ + 2a^-3z³ + a^-2z²

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {6, 15}

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 9₅. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9

j = 21 1

j = 19

j = 17 2 1

j = 15 1

j = 13 2 2

j = 11 2 1

j = 9 1 2

j = 7 2 2

j = 5 1

j = 3 1 2

j = 1 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q² - 2q³ + q⁴ + 3q⁵ - 5q⁶ + 3q⁷ + 3q⁸ - 7q⁹ + 5q¹⁰ + 3q¹¹ - 8q¹² + 5q¹³ + 5q¹⁴ - 10q¹⁵ + 4q¹⁶ + 6q¹⁷ - 9q¹⁸ + 2q¹⁹ + 6q²⁰ - 7q²¹ + 5q²³ - 4q²⁴ - q²⁵ + 3q²⁶ - q²⁷ - q²⁸ + q²⁹

3 q³ - 2q⁴ + q⁵ + q⁶ + q⁷ - 3q⁸ + 3q¹⁰ + q¹¹ - 3q¹² - q¹³ + 2q¹⁴ + 3q¹⁵ - 2q¹⁶ - 2q¹⁷ - 2q¹⁸ + 6q¹⁹ - 2q²¹ - 4q²² + 5q²³ - 2q²⁶ + 3q²⁷ - 4q²⁸ + q²⁹ + 2q³⁰ - 6q³² + 2q³³ + 5q³⁴ - q³⁵ - 6q³⁶ + q³⁷ + 7q³⁸ - q³⁹ - 7q⁴⁰ - q⁴¹ + 8q⁴² + q⁴³ - 6q⁴⁴ - 4q⁴⁵ + 7q⁴⁶ + 3q⁴⁷ - 3q⁴⁸ - 5q⁴⁹ + 3q⁵⁰ + 3q⁵¹ - 3q⁵³ + q⁵⁵ + q⁵⁶ - q⁵⁷

4 q⁴ - 2q⁵ + q⁶ + q⁷ - q⁸ + 3q⁹ - 6q¹⁰ + 3q¹¹ + 2q¹² - q¹³ + 6q¹⁴ - 14q¹⁵ + 5q¹⁶ + 5q¹⁷ + 2q¹⁸ + 7q¹⁹ - 24q²⁰ + 5q²¹ + 10q²² + 9q²³ + 8q²⁴ - 36q²⁵ + q²⁶ + 15q²⁷ + 18q²⁸ + 12q²⁹ - 46q³⁰ - 6q³¹ + 14q³² + 25q³³ + 20q³⁴ - 49q³⁵ - 8q³⁶ + 9q³⁷ + 22q³⁸ + 25q³⁹ - 44q⁴⁰ - 6q⁴¹ + 5q⁴² + 14q⁴³ + 25q⁴⁴ - 39q⁴⁵ - q⁴⁶ + 4q⁴⁷ + 7q⁴⁸ + 23q⁴⁹ - 33q⁵⁰ + 2q⁵¹ + 2q⁵² + 2q⁵³ + 22q⁵⁴ - 26q⁵⁵ + 5q⁵⁶ - 4q⁵⁸ + 20q⁵⁹ - 19q⁶⁰ + 8q⁶¹ - 8q⁶³ + 14q⁶⁴ - 16q⁶⁵ + 9q⁶⁶ + 4q⁶⁷ - 7q⁶⁸ + 8q⁶⁹ - 16q⁷⁰ + 7q⁷¹ + 7q⁷² - q⁷³ + 6q⁷⁴ - 16q⁷⁵ + q⁷⁶ + 5q⁷⁷ + 3q⁷⁸ + 8q⁷⁹ - 11q⁸⁰ - 3q⁸¹ + 2q⁸³ + 8q⁸⁴ - 4q⁸⁵ - 2q⁸⁶ - 2q⁸⁷ - q⁸⁸ + 4q⁸⁹ - q⁹² - q⁹³ + q⁹⁴

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[9, 5]]

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

In[3]:=
GaussCode[Knot[9, 5]]

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

In[4]:=
DTCode[Knot[9, 5]]

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

In[5]:=
br = BR[Knot[9, 5]]

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

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

Out[6]=
{5, 12}

In[7]:=
BraidIndex[Knot[9, 5]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[9, 5]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 5]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 2, 1, 2, {4, 6}, 1}

In[10]:=
alex = Alexander[Knot[9, 5]][t]

Out[10]=
6 -11 + - + 6 t t

In[11]:=
Conway[Knot[9, 5]][z]

Out[11]=
2 1 + 6 z

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

Out[12]=
{Knot[9, 5]}

In[13]:=
{KnotDet[Knot[9, 5]], KnotSignature[Knot[9, 5]]}

Out[13]=
{23, 2}

In[14]:=
Jones[Knot[9, 5]][q]

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

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

Out[15]=
{Knot[9, 5]}

In[16]:=
A2Invariant[Knot[9, 5]][q]

Out[16]=
2 4 8 12 14 16 18 22 26 30 32 q - q + q + q + q + q + q + q - q - q - q

In[17]:=
HOMFLYPT[Knot[9, 5]][a, z]

Out[17]=
2 2 2 2 -10 -6 -4 z 2 z 2 z z -a + a + a + -- + ---- + ---- + -- 8 6 4 2 a a a a

In[18]:=
Kauffman[Knot[9, 5]][a, z]

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

In[19]:=
{Vassiliev[2][Knot[9, 5]], Vassiliev[3][Knot[9, 5]]}

Out[19]=
{6, 15}

In[20]:=
Kh[Knot[9, 5]][q, t]

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

In[21]:=
ColouredJones[Knot[9, 5], 2][q]

Out[21]=
2 3 4 5 6 7 8 9 10 11 12 q - 2 q + q + 3 q - 5 q + 3 q + 3 q - 7 q + 5 q + 3 q - 8 q + 13 14 15 16 17 18 19 20 21 > 5 q + 5 q - 10 q + 4 q + 6 q - 9 q + 2 q + 6 q - 7 q + 23 24 25 26 27 28 29 > 5 q - 4 q - q + 3 q - q - q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₅

9₄
9₆

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q² - 2q³ + q⁴ + 3q⁵ - 5q⁶ + 3q⁷ + 3q⁸ - 7q⁹ + 5q¹⁰ + 3q¹¹ - 8q¹² + 5q¹³ + 5q¹⁴ - 10q¹⁵ + 4q¹⁶ + 6q¹⁷ - 9q¹⁸ + 2q¹⁹ + 6q²⁰ - 7q²¹ + 5q²³ - 4q²⁴ - q²⁵ + 3q²⁶ - q²⁷ - q²⁸ + q²⁹
3	q³ - 2q⁴ + q⁵ + q⁶ + q⁷ - 3q⁸ + 3q¹⁰ + q¹¹ - 3q¹² - q¹³ + 2q¹⁴ + 3q¹⁵ - 2q¹⁶ - 2q¹⁷ - 2q¹⁸ + 6q¹⁹ - 2q²¹ - 4q²² + 5q²³ - 2q²⁶ + 3q²⁷ - 4q²⁸ + q²⁹ + 2q³⁰ - 6q³² + 2q³³ + 5q³⁴ - q³⁵ - 6q³⁶ + q³⁷ + 7q³⁸ - q³⁹ - 7q⁴⁰ - q⁴¹ + 8q⁴² + q⁴³ - 6q⁴⁴ - 4q⁴⁵ + 7q⁴⁶ + 3q⁴⁷ - 3q⁴⁸ - 5q⁴⁹ + 3q⁵⁰ + 3q⁵¹ - 3q⁵³ + q⁵⁵ + q⁵⁶ - q⁵⁷
4	q⁴ - 2q⁵ + q⁶ + q⁷ - q⁸ + 3q⁹ - 6q¹⁰ + 3q¹¹ + 2q¹² - q¹³ + 6q¹⁴ - 14q¹⁵ + 5q¹⁶ + 5q¹⁷ + 2q¹⁸ + 7q¹⁹ - 24q²⁰ + 5q²¹ + 10q²² + 9q²³ + 8q²⁴ - 36q²⁵ + q²⁶ + 15q²⁷ + 18q²⁸ + 12q²⁹ - 46q³⁰ - 6q³¹ + 14q³² + 25q³³ + 20q³⁴ - 49q³⁵ - 8q³⁶ + 9q³⁷ + 22q³⁸ + 25q³⁹ - 44q⁴⁰ - 6q⁴¹ + 5q⁴² + 14q⁴³ + 25q⁴⁴ - 39q⁴⁵ - q⁴⁶ + 4q⁴⁷ + 7q⁴⁸ + 23q⁴⁹ - 33q⁵⁰ + 2q⁵¹ + 2q⁵² + 2q⁵³ + 22q⁵⁴ - 26q⁵⁵ + 5q⁵⁶ - 4q⁵⁸ + 20q⁵⁹ - 19q⁶⁰ + 8q⁶¹ - 8q⁶³ + 14q⁶⁴ - 16q⁶⁵ + 9q⁶⁶ + 4q⁶⁷ - 7q⁶⁸ + 8q⁶⁹ - 16q⁷⁰ + 7q⁷¹ + 7q⁷² - q⁷³ + 6q⁷⁴ - 16q⁷⁵ + q⁷⁶ + 5q⁷⁷ + 3q⁷⁸ + 8q⁷⁹ - 11q⁸⁰ - 3q⁸¹ + 2q⁸³ + 8q⁸⁴ - 4q⁸⁵ - 2q⁸⁶ - 2q⁸⁷ - q⁸⁸ + 4q⁸⁹ - q⁹² - q⁹³ + q⁹⁴

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

Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 5]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 1, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 5]][t]
Out[10]=	6 -11 + - + 6 t t
In[11]:=	Conway[Knot[9, 5]][z]
Out[11]=	2 1 + 6 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 5]}
In[13]:=	{KnotDet[Knot[9, 5]], KnotSignature[Knot[9, 5]]}
Out[13]=	{23, 2}
In[14]:=	Jones[Knot[9, 5]][q]
Out[14]=	2 3 4 5 6 7 8 9 10 q - 2 q + 3 q - 3 q + 4 q - 3 q + 3 q - 2 q + q - q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 5]}
In[16]:=	A2Invariant[Knot[9, 5]][q]
Out[16]=	2 4 8 12 14 16 18 22 26 30 32 q - q + q + q + q + q + q + q - q - q - q
In[17]:=	HOMFLYPT[Knot[9, 5]][a, z]
Out[17]=	2 2 2 2 -10 -6 -4 z 2 z 2 z z -a + a + a + -- + ---- + ---- + -- 8 6 4 2 a a a a
In[18]:=	Kauffman[Knot[9, 5]][a, z]
Out[18]=	2 2 2 2 2 3 3 -10 -6 -4 6 z 6 z 3 z 4 z 3 z 3 z z 11 z 18 z a - a + a - --- - --- - ---- + ---- + ---- - ---- + -- + ----- + ----- + 11 9 10 8 6 4 2 11 9 a a a a a a a a a 3 3 3 4 4 4 4 5 5 5 5 z 4 z 2 z 7 z 3 z 7 z 3 z 6 z 14 z 5 z 3 z > -- - ---- + ---- + ---- - ---- - ---- + ---- - ---- - ----- - ---- + ---- - 7 5 3 10 8 6 4 11 9 7 5 a a a a a a a a a a a 6 6 6 7 7 7 8 8 5 z 2 z 3 z z 3 z 2 z z z > ---- - ---- + ---- + --- + ---- + ---- + --- + -- 10 8 6 11 9 7 10 8 a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[9, 5]], Vassiliev[3][Knot[9, 5]]}
Out[19]=	{6, 15}
In[20]:=	Kh[Knot[9, 5]][q, t]
Out[20]=	3 3 5 2 7 2 7 3 9 3 9 4 11 4 q + q + 2 q t + q t + 2 q t + 2 q t + q t + 2 q t + 2 q t + 11 5 13 5 13 6 15 6 17 7 17 8 21 9 > q t + 2 q t + 2 q t + q t + 2 q t + q t + q t
In[21]:=	ColouredJones[Knot[9, 5], 2][q]
Out[21]=	2 3 4 5 6 7 8 9 10 11 12 q - 2 q + q + 3 q - 5 q + 3 q + 3 q - 7 q + 5 q + 3 q - 8 q + 13 14 15 16 17 18 19 20 21 > 5 q + 5 q - 10 q + 4 q + 6 q - 9 q + 2 q + 6 q - 7 q + 23 24 25 26 27 28 29 > 5 q - 4 q - q + 3 q - q - q + q

The Alternating Knot 95

The Alternating Knot 9₅