The Non Alternating Knot 9₄₉

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X_12,8,13,7 X_5,15,6,14 X_3,11,4,10 X_11,3,12,2 X_15,5,16,4 X_17,9,18,8 X_9,17,10,16 X_18,14,1,13

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

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

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 3 2 3 / 4--5 2

Alexander Polynomial: 3t^-2 - 6t^-1 + 7 - 6t + 3t²

Conway Polynomial: 1 + 6z² + 3z⁴

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

Determinant and Signature: {25, 4}

Jones Polynomial: q² - 2q³ + 4q⁴ - 4q⁵ + 5q⁶ - 4q⁷ + 3q⁸ - 2q⁹

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

A2 (sl(3)) Invariant: q⁶ - q⁸ + q¹⁰ + q¹⁴ + 3q¹⁶ + q¹⁸ + 2q²⁰ - q²² - q²⁴ - q²⁶ - 2q²⁸

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

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

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {6, 14}

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 9₄₉. 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

j = 19 2

j = 17 1

j = 15 3 2

j = 13 2 1

j = 11 2 3

j = 9 2 2

j = 7 2

j = 5 1 2

j = 3 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q⁴ - 2q⁵ + q⁶ + 6q⁷ - 8q⁸ - 3q⁹ + 16q¹⁰ - 11q¹¹ - 10q¹² + 25q¹³ - 11q¹⁴ - 17q¹⁵ + 27q¹⁶ - 9q¹⁷ - 17q¹⁸ + 21q¹⁹ - 3q²⁰ - 13q²¹ + 10q²² + q²³ - 6q²⁴ + 2q²⁵ + q²⁶

3 q⁶ - 2q⁷ + q⁸ + 3q⁹ + q¹⁰ - 8q¹¹ - 3q¹² + 13q¹³ + 12q¹⁴ - 18q¹⁵ - 21q¹⁶ + 14q¹⁷ + 39q¹⁸ - 14q¹⁹ - 48q²⁰ + 2q²¹ + 64q²² + 5q²³ - 70q²⁴ - 18q²⁵ + 80q²⁶ + 22q²⁷ - 78q²⁸ - 33q²⁹ + 80q³⁰ + 33q³¹ - 72q³² - 40q³³ + 65q³⁴ + 40q³⁵ - 51q³⁶ - 40q³⁷ + 35q³⁸ + 39q³⁹ - 22q⁴⁰ - 31q⁴¹ + 8q⁴² + 24q⁴³ - 2q⁴⁴ - 12q⁴⁵ - 5q⁴⁶ + 9q⁴⁷ + q⁴⁸ - 2q⁵⁰

4 q⁸ - 2q⁹ + q¹⁰ + 3q¹¹ - 2q¹² + q¹³ - 9q¹⁴ + 3q¹⁵ + 15q¹⁶ + 5q¹⁸ - 35q¹⁹ - 13q²⁰ + 30q²¹ + 25q²² + 44q²³ - 63q²⁴ - 65q²⁵ + q²⁶ + 46q²⁷ + 137q²⁸ - 43q²⁹ - 122q³⁰ - 84q³¹ + 19q³² + 240q³³ + 31q³⁴ - 137q³⁵ - 181q³⁶ - 50q³⁷ + 309q³⁸ + 110q³⁹ - 117q⁴⁰ - 248q⁴¹ - 116q⁴² + 333q⁴³ + 162q⁴⁴ - 86q⁴⁵ - 276q⁴⁶ - 158q⁴⁷ + 324q⁴⁸ + 187q⁴⁹ - 52q⁵⁰ - 269q⁵¹ - 183q⁵² + 274q⁵³ + 192q⁵⁴ - q⁵⁵ - 226q⁵⁶ - 196q⁵⁷ + 182q⁵⁸ + 168q⁵⁹ + 59q⁶⁰ - 142q⁶¹ - 180q⁶² + 70q⁶³ + 105q⁶⁴ + 90q⁶⁵ - 46q⁶⁶ - 120q⁶⁷ - 5q⁶⁸ + 31q⁶⁹ + 65q⁷⁰ + 12q⁷¹ - 45q⁷² - 18q⁷³ - 7q⁷⁴ + 21q⁷⁵ + 14q⁷⁶ - 5q⁷⁷ - 4q⁷⁸ - 6q⁷⁹ + 2q⁸¹ + q⁸²

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, 49]]

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

In[3]:=
GaussCode[Knot[9, 49]]

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

In[4]:=
DTCode[Knot[9, 49]]

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

In[5]:=
br = BR[Knot[9, 49]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[9, 49]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[9, 49]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 49]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 3, 2, 3, {4, 5}, 2}

In[10]:=
alex = Alexander[Knot[9, 49]][t]

Out[10]=
3 6 2 7 + -- - - - 6 t + 3 t 2 t t

In[11]:=
Conway[Knot[9, 49]][z]

Out[11]=
2 4 1 + 6 z + 3 z

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

Out[12]=
{Knot[9, 49]}

In[13]:=
{KnotDet[Knot[9, 49]], KnotSignature[Knot[9, 49]]}

Out[13]=
{25, 4}

In[14]:=
Jones[Knot[9, 49]][q]

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

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

Out[15]=
{Knot[9, 49]}

In[16]:=
A2Invariant[Knot[9, 49]][q]

Out[16]=
6 8 10 14 16 18 20 22 24 26 28 q - q + q + q + 3 q + q + 2 q - q - q - q - 2 q

In[17]:=
HOMFLYPT[Knot[9, 49]][a, z]

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

In[18]:=
Kauffman[Knot[9, 49]][a, z]

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

In[19]:=
{Vassiliev[2][Knot[9, 49]], Vassiliev[3][Knot[9, 49]]}

Out[19]=
{6, 14}

In[20]:=
Kh[Knot[9, 49]][q, t]

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

In[21]:=
ColouredJones[Knot[9, 49], 2][q]

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

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₄₉

9₄₈
10₁

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q⁴ - 2q⁵ + q⁶ + 6q⁷ - 8q⁸ - 3q⁹ + 16q¹⁰ - 11q¹¹ - 10q¹² + 25q¹³ - 11q¹⁴ - 17q¹⁵ + 27q¹⁶ - 9q¹⁷ - 17q¹⁸ + 21q¹⁹ - 3q²⁰ - 13q²¹ + 10q²² + q²³ - 6q²⁴ + 2q²⁵ + q²⁶
3	q⁶ - 2q⁷ + q⁸ + 3q⁹ + q¹⁰ - 8q¹¹ - 3q¹² + 13q¹³ + 12q¹⁴ - 18q¹⁵ - 21q¹⁶ + 14q¹⁷ + 39q¹⁸ - 14q¹⁹ - 48q²⁰ + 2q²¹ + 64q²² + 5q²³ - 70q²⁴ - 18q²⁵ + 80q²⁶ + 22q²⁷ - 78q²⁸ - 33q²⁹ + 80q³⁰ + 33q³¹ - 72q³² - 40q³³ + 65q³⁴ + 40q³⁵ - 51q³⁶ - 40q³⁷ + 35q³⁸ + 39q³⁹ - 22q⁴⁰ - 31q⁴¹ + 8q⁴² + 24q⁴³ - 2q⁴⁴ - 12q⁴⁵ - 5q⁴⁶ + 9q⁴⁷ + q⁴⁸ - 2q⁵⁰
4	q⁸ - 2q⁹ + q¹⁰ + 3q¹¹ - 2q¹² + q¹³ - 9q¹⁴ + 3q¹⁵ + 15q¹⁶ + 5q¹⁸ - 35q¹⁹ - 13q²⁰ + 30q²¹ + 25q²² + 44q²³ - 63q²⁴ - 65q²⁵ + q²⁶ + 46q²⁷ + 137q²⁸ - 43q²⁹ - 122q³⁰ - 84q³¹ + 19q³² + 240q³³ + 31q³⁴ - 137q³⁵ - 181q³⁶ - 50q³⁷ + 309q³⁸ + 110q³⁹ - 117q⁴⁰ - 248q⁴¹ - 116q⁴² + 333q⁴³ + 162q⁴⁴ - 86q⁴⁵ - 276q⁴⁶ - 158q⁴⁷ + 324q⁴⁸ + 187q⁴⁹ - 52q⁵⁰ - 269q⁵¹ - 183q⁵² + 274q⁵³ + 192q⁵⁴ - q⁵⁵ - 226q⁵⁶ - 196q⁵⁷ + 182q⁵⁸ + 168q⁵⁹ + 59q⁶⁰ - 142q⁶¹ - 180q⁶² + 70q⁶³ + 105q⁶⁴ + 90q⁶⁵ - 46q⁶⁶ - 120q⁶⁷ - 5q⁶⁸ + 31q⁶⁹ + 65q⁷⁰ + 12q⁷¹ - 45q⁷² - 18q⁷³ - 7q⁷⁴ + 21q⁷⁵ + 14q⁷⁶ - 5q⁷⁷ - 4q⁷⁸ - 6q⁷⁹ + 2q⁸¹ + q⁸²

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[9, 49]]
Out[2]=	PD[X[6, 2, 7, 1], X[12, 8, 13, 7], X[5, 15, 6, 14], X[3, 11, 4, 10], > X[11, 3, 12, 2], X[15, 5, 16, 4], X[17, 9, 18, 8], X[9, 17, 10, 16], > X[18, 14, 1, 13]]
In[3]:=	GaussCode[Knot[9, 49]]
Out[3]=	GaussCode[1, 5, -4, 6, -3, -1, 2, 7, -8, 4, -5, -2, 9, 3, -6, 8, -7, -9]
In[4]:=	DTCode[Knot[9, 49]]
Out[4]=	DTCode[6, -10, -14, 12, -16, -2, 18, -4, -8]
In[5]:=	br = BR[Knot[9, 49]]
Out[5]=	BR[4, {1, 1, 2, 1, 1, -3, 2, -1, 2, 3, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[9, 49]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 49]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 49]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 3, 2, 3, {4, 5}, 2}
In[10]:=	alex = Alexander[Knot[9, 49]][t]
Out[10]=	3 6 2 7 + -- - - - 6 t + 3 t 2 t t
In[11]:=	Conway[Knot[9, 49]][z]
Out[11]=	2 4 1 + 6 z + 3 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 49]}
In[13]:=	{KnotDet[Knot[9, 49]], KnotSignature[Knot[9, 49]]}
Out[13]=	{25, 4}
In[14]:=	Jones[Knot[9, 49]][q]
Out[14]=	2 3 4 5 6 7 8 9 q - 2 q + 4 q - 4 q + 5 q - 4 q + 3 q - 2 q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 49]}
In[16]:=	A2Invariant[Knot[9, 49]][q]
Out[16]=	6 8 10 14 16 18 20 22 24 26 28 q - q + q + q + 3 q + q + 2 q - q - q - q - 2 q
In[17]:=	HOMFLYPT[Knot[9, 49]][a, z]
Out[17]=	2 2 2 4 4 -3 4 2 z 6 z 2 z 2 z z -- + -- - ---- + ---- + ---- + ---- + -- 8 6 8 6 4 6 4 a a a a a a a
In[18]:=	Kauffman[Knot[9, 49]][a, z]
Out[18]=	2 2 2 2 3 3 3 -3 4 4 z 2 z 2 z z 10 z 9 z 2 z 3 z 3 z 3 z -- - -- - --- - --- + --- - --- + ----- + ---- - ---- + ---- + ---- - ---- - 8 6 11 9 7 10 8 6 4 11 9 7 a a a a a a a a a a a a 3 4 4 4 5 5 5 6 6 6 7 7 3 z 9 z 8 z z z z 2 z z 4 z 3 z z z > ---- - ---- - ---- + -- - -- + -- + ---- + --- + ---- + ---- + -- + -- 5 8 6 4 9 7 5 10 8 6 9 7 a a a a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[9, 49]], Vassiliev[3][Knot[9, 49]]}
Out[19]=	{6, 14}
In[20]:=	Kh[Knot[9, 49]][q, t]
Out[20]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 2 q t + 2 q t + 2 q t + 2 q t + 2 q t + 3 q t + 13 4 13 5 15 5 15 6 17 6 19 7 > 2 q t + q t + 3 q t + 2 q t + q t + 2 q t
In[21]:=	ColouredJones[Knot[9, 49], 2][q]
Out[21]=	4 5 6 7 8 9 10 11 12 13 q - 2 q + q + 6 q - 8 q - 3 q + 16 q - 11 q - 10 q + 25 q - 14 15 16 17 18 19 20 21 > 11 q - 17 q + 27 q - 9 q - 17 q + 21 q - 3 q - 13 q + 22 23 24 25 26 > 10 q + q - 6 q + 2 q + q

The Non Alternating Knot 949

The Non Alternating Knot 9₄₉