The Alternating Knot 9₁₃

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Acknowledgement

KnotPlot

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

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

DT (Dowker-Thistlethwaite) Code: 6 12 14 16 18 4 2 10 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 2--3 2 2 / 4--6 1

Alexander Polynomial: 4t^-2 - 9t^-1 + 11 - 9t + 4t²

Conway Polynomial: 1 + 7z² + 4z⁴

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

Determinant and Signature: {37, 4}

Jones Polynomial: q² - 2q³ + 4q⁴ - 5q⁵ + 7q⁶ - 6q⁷ + 5q⁸ - 4q⁹ + 2q¹⁰ - q¹¹

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

A2 (sl(3)) Invariant: q⁶ - q⁸ + q¹⁰ + 3q¹⁶ + q¹⁸ + 2q²⁰ - q²⁴ - 2q²⁸ - q³⁴

HOMFLY-PT Polynomial: - a^-10 - a^-10z² - a^-8 + a^-8z² + a^-8z⁴ + 3a^-6 + 5a^-6z² + 2a^-6z⁴ + 2a^-4z² + a^-4z⁴

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

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {7, 18}

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 9₁₃. 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 = 23 1

j = 21 1

j = 19 3 1

j = 17 2 1

j = 15 4 3

j = 13 3 2

j = 11 2 4

j = 9 2 3

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 + 5q⁷ - 8q⁸ + q⁹ + 14q¹⁰ - 18q¹¹ - q¹² + 28q¹³ - 28q¹⁴ - 6q¹⁵ + 39q¹⁶ - 31q¹⁷ - 10q¹⁸ + 38q¹⁹ - 24q²⁰ - 13q²¹ + 29q²² - 13q²³ - 12q²⁴ + 17q²⁵ - 4q²⁶ - 7q²⁷ + 6q²⁸ - 2q³⁰ + q³¹

3 q⁶ - 2q⁷ + q⁸ + 2q⁹ + q¹⁰ - 6q¹¹ + q¹² + 8q¹³ + 2q¹⁴ - 16q¹⁵ + 21q¹⁷ + 7q¹⁸ - 36q¹⁹ - 9q²⁰ + 45q²¹ + 24q²² - 63q²³ - 33q²⁴ + 71q²⁵ + 52q²⁶ - 85q²⁷ - 60q²⁸ + 82q²⁹ + 79q³⁰ - 89q³¹ - 79q³² + 77q³³ + 87q³⁴ - 71q³⁵ - 85q³⁶ + 55q³⁷ + 83q³⁸ - 40q³⁹ - 79q⁴⁰ + 26q⁴¹ + 69q⁴² - 9q⁴³ - 61q⁴⁴ + 46q⁴⁶ + 11q⁴⁷ - 36q⁴⁸ - 10q⁴⁹ + 20q⁵⁰ + 13q⁵¹ - 13q⁵² - 8q⁵³ + 5q⁵⁴ + 6q⁵⁵ - 3q⁵⁶ - 2q⁵⁷ + 2q⁵⁹ - q⁶⁰

4 q⁸ - 2q⁹ + q¹⁰ + 2q¹¹ - 2q¹² + 3q¹³ - 7q¹⁴ + 4q¹⁵ + 7q¹⁶ - 9q¹⁷ + 9q¹⁸ - 17q¹⁹ + 10q²⁰ + 18q²¹ - 22q²² + 13q²³ - 38q²⁴ + 25q²⁵ + 47q²⁶ - 33q²⁷ + 8q²⁸ - 92q²⁹ + 36q³⁰ + 109q³¹ - 11q³² + 5q³³ - 196q³⁴ + 11q³⁵ + 186q³⁶ + 60q³⁷ + 36q³⁸ - 324q³⁹ - 58q⁴⁰ + 238q⁴¹ + 147q⁴² + 103q⁴³ - 418q⁴⁴ - 136q⁴⁵ + 241q⁴⁶ + 205q⁴⁷ + 178q⁴⁸ - 450q⁴⁹ - 189q⁵⁰ + 209q⁵¹ + 219q⁵² + 227q⁵³ - 419q⁵⁴ - 207q⁵⁵ + 148q⁵⁶ + 197q⁵⁷ + 259q⁵⁸ - 340q⁵⁹ - 202q⁶⁰ + 65q⁶¹ + 149q⁶² + 275q⁶³ - 227q⁶⁴ - 173q⁶⁵ - 24q⁶⁶ + 78q⁶⁷ + 262q⁶⁸ - 105q⁶⁹ - 116q⁷⁰ - 80q⁷¹ - q⁷² + 202q⁷³ - 15q⁷⁴ - 44q⁷⁵ - 81q⁷⁶ - 48q⁷⁷ + 115q⁷⁸ + 15q⁷⁹ + 6q⁸⁰ - 43q⁸¹ - 48q⁸² + 45q⁸³ + 8q⁸⁴ + 17q⁸⁵ - 12q⁸⁶ - 25q⁸⁷ + 14q⁸⁸ - q⁸⁹ + 8q⁹⁰ - q⁹¹ - 8q⁹² + 4q⁹³ - q⁹⁴ + 2q⁹⁵ - 2q⁹⁷ + q⁹⁸

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, 13]]

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

In[3]:=
GaussCode[Knot[9, 13]]

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

In[4]:=
DTCode[Knot[9, 13]]

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

In[5]:=
br = BR[Knot[9, 13]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[9, 13]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[9, 13]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 13]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, {2, 3}, 2, 2, {4, 6}, 1}

In[10]:=
alex = Alexander[Knot[9, 13]][t]

Out[10]=
4 9 2 11 + -- - - - 9 t + 4 t 2 t t

In[11]:=
Conway[Knot[9, 13]][z]

Out[11]=
2 4 1 + 7 z + 4 z

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

Out[12]=
{Knot[9, 13]}

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

Out[13]=
{37, 4}

In[14]:=
Jones[Knot[9, 13]][q]

Out[14]=
2 3 4 5 6 7 8 9 10 11 q - 2 q + 4 q - 5 q + 7 q - 6 q + 5 q - 4 q + 2 q - q

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

Out[15]=
{Knot[9, 13]}

In[16]:=
A2Invariant[Knot[9, 13]][q]

Out[16]=
6 8 10 16 18 20 24 28 34 q - q + q + 3 q + q + 2 q - q - 2 q - q

In[17]:=
HOMFLYPT[Knot[9, 13]][a, z]

Out[17]=
2 2 2 2 4 4 4 -10 -8 3 z z 5 z 2 z z 2 z z -a - a + -- - --- + -- + ---- + ---- + -- + ---- + -- 6 10 8 6 4 8 6 4 a a a a a a a a

In[18]:=
Kauffman[Knot[9, 13]][a, z]

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

In[19]:=
{Vassiliev[2][Knot[9, 13]], Vassiliev[3][Knot[9, 13]]}

Out[19]=
{7, 18}

In[20]:=
Kh[Knot[9, 13]][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 + 3 q t + 2 q t + 4 q t + 13 4 13 5 15 5 15 6 17 6 17 7 19 7 > 3 q t + 2 q t + 4 q t + 3 q t + 2 q t + q t + 3 q t + 19 8 21 8 23 9 > q t + q t + q t

In[21]:=
ColouredJones[Knot[9, 13], 2][q]

Out[21]=
4 5 6 7 8 9 10 11 12 13 14 q - 2 q + q + 5 q - 8 q + q + 14 q - 18 q - q + 28 q - 28 q - 15 16 17 18 19 20 21 22 > 6 q + 39 q - 31 q - 10 q + 38 q - 24 q - 13 q + 29 q - 23 24 25 26 27 28 30 31 > 13 q - 12 q + 17 q - 4 q - 7 q + 6 q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₁₃

9₁₂
9₁₄

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q⁴ - 2q⁵ + q⁶ + 5q⁷ - 8q⁸ + q⁹ + 14q¹⁰ - 18q¹¹ - q¹² + 28q¹³ - 28q¹⁴ - 6q¹⁵ + 39q¹⁶ - 31q¹⁷ - 10q¹⁸ + 38q¹⁹ - 24q²⁰ - 13q²¹ + 29q²² - 13q²³ - 12q²⁴ + 17q²⁵ - 4q²⁶ - 7q²⁷ + 6q²⁸ - 2q³⁰ + q³¹
3	q⁶ - 2q⁷ + q⁸ + 2q⁹ + q¹⁰ - 6q¹¹ + q¹² + 8q¹³ + 2q¹⁴ - 16q¹⁵ + 21q¹⁷ + 7q¹⁸ - 36q¹⁹ - 9q²⁰ + 45q²¹ + 24q²² - 63q²³ - 33q²⁴ + 71q²⁵ + 52q²⁶ - 85q²⁷ - 60q²⁸ + 82q²⁹ + 79q³⁰ - 89q³¹ - 79q³² + 77q³³ + 87q³⁴ - 71q³⁵ - 85q³⁶ + 55q³⁷ + 83q³⁸ - 40q³⁹ - 79q⁴⁰ + 26q⁴¹ + 69q⁴² - 9q⁴³ - 61q⁴⁴ + 46q⁴⁶ + 11q⁴⁷ - 36q⁴⁸ - 10q⁴⁹ + 20q⁵⁰ + 13q⁵¹ - 13q⁵² - 8q⁵³ + 5q⁵⁴ + 6q⁵⁵ - 3q⁵⁶ - 2q⁵⁷ + 2q⁵⁹ - q⁶⁰
4	q⁸ - 2q⁹ + q¹⁰ + 2q¹¹ - 2q¹² + 3q¹³ - 7q¹⁴ + 4q¹⁵ + 7q¹⁶ - 9q¹⁷ + 9q¹⁸ - 17q¹⁹ + 10q²⁰ + 18q²¹ - 22q²² + 13q²³ - 38q²⁴ + 25q²⁵ + 47q²⁶ - 33q²⁷ + 8q²⁸ - 92q²⁹ + 36q³⁰ + 109q³¹ - 11q³² + 5q³³ - 196q³⁴ + 11q³⁵ + 186q³⁶ + 60q³⁷ + 36q³⁸ - 324q³⁹ - 58q⁴⁰ + 238q⁴¹ + 147q⁴² + 103q⁴³ - 418q⁴⁴ - 136q⁴⁵ + 241q⁴⁶ + 205q⁴⁷ + 178q⁴⁸ - 450q⁴⁹ - 189q⁵⁰ + 209q⁵¹ + 219q⁵² + 227q⁵³ - 419q⁵⁴ - 207q⁵⁵ + 148q⁵⁶ + 197q⁵⁷ + 259q⁵⁸ - 340q⁵⁹ - 202q⁶⁰ + 65q⁶¹ + 149q⁶² + 275q⁶³ - 227q⁶⁴ - 173q⁶⁵ - 24q⁶⁶ + 78q⁶⁷ + 262q⁶⁸ - 105q⁶⁹ - 116q⁷⁰ - 80q⁷¹ - q⁷² + 202q⁷³ - 15q⁷⁴ - 44q⁷⁵ - 81q⁷⁶ - 48q⁷⁷ + 115q⁷⁸ + 15q⁷⁹ + 6q⁸⁰ - 43q⁸¹ - 48q⁸² + 45q⁸³ + 8q⁸⁴ + 17q⁸⁵ - 12q⁸⁶ - 25q⁸⁷ + 14q⁸⁸ - q⁸⁹ + 8q⁹⁰ - q⁹¹ - 8q⁹² + 4q⁹³ - q⁹⁴ + 2q⁹⁵ - 2q⁹⁷ + q⁹⁸

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

Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 13]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, {2, 3}, 2, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 13]][t]
Out[10]=	4 9 2 11 + -- - - - 9 t + 4 t 2 t t
In[11]:=	Conway[Knot[9, 13]][z]
Out[11]=	2 4 1 + 7 z + 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 13]}
In[13]:=	{KnotDet[Knot[9, 13]], KnotSignature[Knot[9, 13]]}
Out[13]=	{37, 4}
In[14]:=	Jones[Knot[9, 13]][q]
Out[14]=	2 3 4 5 6 7 8 9 10 11 q - 2 q + 4 q - 5 q + 7 q - 6 q + 5 q - 4 q + 2 q - q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 13]}
In[16]:=	A2Invariant[Knot[9, 13]][q]
Out[16]=	6 8 10 16 18 20 24 28 34 q - q + q + 3 q + q + 2 q - q - 2 q - q
In[17]:=	HOMFLYPT[Knot[9, 13]][a, z]
Out[17]=	2 2 2 2 4 4 4 -10 -8 3 z z 5 z 2 z z 2 z z -a - a + -- - --- + -- + ---- + ---- + -- + ---- + -- 6 10 8 6 4 8 6 4 a a a a a a a a
In[18]:=	Kauffman[Knot[9, 13]][a, z]
Out[18]=	2 2 2 2 2 -10 -8 3 2 z 2 z 3 z z 2 z 2 z 6 z 8 z 2 z a - a - -- + --- - --- - --- + -- + ---- - ---- + ---- + ---- - ---- - 6 13 11 9 7 12 10 8 6 4 a a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 5 3 z 2 z 9 z z 3 z 5 z z 4 z 7 z z z > ---- + ---- + ---- + -- - ---- - ---- - --- - ---- - ---- + -- + --- - 13 11 9 7 5 12 10 8 6 4 13 a a a a a a a a a a a 5 5 5 5 6 6 6 7 7 7 8 8 4 z 9 z 2 z 2 z 2 z z 3 z 2 z 4 z 2 z z z > ---- - ---- - ---- + ---- + ---- + -- + ---- + ---- + ---- + ---- + --- + -- 11 9 7 5 12 8 6 11 9 7 10 8 a a a a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[9, 13]], Vassiliev[3][Knot[9, 13]]}
Out[19]=	{7, 18}
In[20]:=	Kh[Knot[9, 13]][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 + 3 q t + 2 q t + 4 q t + 13 4 13 5 15 5 15 6 17 6 17 7 19 7 > 3 q t + 2 q t + 4 q t + 3 q t + 2 q t + q t + 3 q t + 19 8 21 8 23 9 > q t + q t + q t
In[21]:=	ColouredJones[Knot[9, 13], 2][q]
Out[21]=	4 5 6 7 8 9 10 11 12 13 14 q - 2 q + q + 5 q - 8 q + q + 14 q - 18 q - q + 28 q - 28 q - 15 16 17 18 19 20 21 22 > 6 q + 39 q - 31 q - 10 q + 38 q - 24 q - 13 q + 29 q - 23 24 25 26 27 28 30 31 > 13 q - 12 q + 17 q - 4 q - 7 q + 6 q - 2 q + q

The Alternating Knot 913

The Alternating Knot 9₁₃