The Alternating Knot 9₂

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_3,12,4,13 X_5,18,6,1 X_7,16,8,17 X_9,14,10,15 X_13,10,14,11 X_15,8,16,9 X_17,6,18,7 X_11,2,12,3

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

DT (Dowker-Thistlethwaite) Code: 4 12 18 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 1 1 2 / 4--7 1

Alexander Polynomial: 4t^-1 - 7 + 4t

Conway Polynomial: 1 + 4z²

Other knots with the same Alexander/Conway Polynomial: {7₄, ...}

Determinant and Signature: {15, -2}

Jones Polynomial: - q^-10 + q^-9 - q^-8 + 2q^-7 - 2q^-6 + 2q^-5 - 2q^-4 + 2q^-3 - q^-2 + q^-1

Other knots (up to mirrors) with the same Jones Polynomial: {K11n13, ...}

A2 (sl(3)) Invariant: - q^-32 - q^-30 + q^-24 + q^-22 + q^-8 + q^-6 + q^-2

HOMFLY-PT Polynomial: a² + a²z² + a⁴z² + a⁶z² + a⁸ + a⁸z² - a¹⁰

Kauffman Polynomial: - a² + a²z² + a³z³ + a⁴z⁴ - a⁵z³ + a⁵z⁵ - 2a⁶z⁴ + a⁶z⁶ + a⁷z³ - 3a⁷z⁵ + a⁷z⁷ + a⁸ - 6a⁸z² + 8a⁸z⁴ - 5a⁸z⁶ + a⁸z⁸ - 4a⁹z + 13a⁹z³ - 10a⁹z⁵ + 2a⁹z⁷ + a¹⁰ - 7a¹⁰z² + 11a¹⁰z⁴ - 6a¹⁰z⁶ + a¹⁰z⁸ - 4a¹¹z + 10a¹¹z³ - 6a¹¹z⁵ + a¹¹z⁷

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {4, -10}

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

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

j = -1 1

j = -3 1 1

j = -5 1

j = -7 1 1

j = -9 1 1

j = -11 1 1

j = -13 1 1

j = -15 1

j = -17 1 1

j = -19

j = -21 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-29 - q^-28 - q^-27 + 2q^-26 - q^-25 - 2q^-24 + 3q^-23 - 3q^-21 + 3q^-20 - 3q^-18 + 3q^-17 - 3q^-15 + 2q^-14 - 2q^-12 + 2q^-11 - 2q^-9 + 3q^-8 - 2q^-6 + 2q^-5 - q^-3 + q^-2

3 - q^-57 + q^-56 + q^-55 - 2q^-53 + 2q^-51 + q^-50 - 3q^-49 - q^-48 + 2q^-47 + 2q^-46 - 2q^-45 - 2q^-44 + 2q^-43 + 2q^-42 - 2q^-41 - 2q^-40 + 2q^-39 + 2q^-38 - 2q^-37 - 2q^-36 + 3q^-35 + 2q^-34 - 3q^-33 - 3q^-32 + 4q^-31 + 2q^-30 - 4q^-29 - 3q^-28 + 5q^-27 + 2q^-26 - 5q^-25 - 2q^-24 + 6q^-23 + 2q^-22 - 6q^-21 - 3q^-20 + 6q^-19 + 2q^-18 - 4q^-17 - 4q^-16 + 5q^-15 + 2q^-14 - 2q^-13 - 3q^-12 + 3q^-11 + q^-10 - 2q^-8 + 2q^-7 - q^-4 + q^-3

4 q^-94 - q^-93 - q^-92 + 3q^-89 - q^-88 - q^-87 - q^-86 - 2q^-85 + 5q^-84 - q^-82 - q^-81 - 4q^-80 + 5q^-79 - 5q^-75 + 5q^-74 - 5q^-70 + 5q^-69 - 6q^-65 + 5q^-64 + q^-63 + q^-62 + q^-61 - 8q^-60 + 4q^-59 + 2q^-58 + 2q^-57 + 2q^-56 - 10q^-55 + 2q^-54 + 3q^-53 + 3q^-52 + 3q^-51 - 11q^-50 + 4q^-48 + 4q^-47 + 4q^-46 - 11q^-45 - q^-44 + 5q^-43 + 4q^-42 + 4q^-41 - 12q^-40 - 2q^-39 + 5q^-38 + 5q^-37 + 5q^-36 - 13q^-35 - 3q^-34 + 5q^-33 + 5q^-32 + 5q^-31 - 12q^-30 - 2q^-29 + 3q^-28 + 4q^-27 + 5q^-26 - 10q^-25 + 2q^-23 + 3q^-22 + 3q^-21 - 7q^-20 + q^-19 + q^-18 + q^-17 + 2q^-16 - 4q^-15 + 2q^-14 + q^-11 - 2q^-10 + 2q^-9 - q^-5 + q^-4

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, 2]]

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

In[3]:=
GaussCode[Knot[9, 2]]

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

In[4]:=
DTCode[Knot[9, 2]]

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

In[5]:=
br = BR[Knot[9, 2]]

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

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

Out[6]=
{5, 12}

In[7]:=
BraidIndex[Knot[9, 2]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[9, 2]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 2]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 1, 1, 2, {4, 7}, 1}

In[10]:=
alex = Alexander[Knot[9, 2]][t]

Out[10]=
4 -7 + - + 4 t t

In[11]:=
Conway[Knot[9, 2]][z]

Out[11]=
2 1 + 4 z

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

Out[12]=
{Knot[7, 4], Knot[9, 2]}

In[13]:=
{KnotDet[Knot[9, 2]], KnotSignature[Knot[9, 2]]}

Out[13]=
{15, -2}

In[14]:=
Jones[Knot[9, 2]][q]

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

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

Out[15]=
{Knot[9, 2], Knot[11, NonAlternating, 13]}

In[16]:=
A2Invariant[Knot[9, 2]][q]

Out[16]=
-32 -30 -24 -22 -8 -6 -2 -q - q + q + q + q + q + q

In[17]:=
HOMFLYPT[Knot[9, 2]][a, z]

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

In[18]:=
Kauffman[Knot[9, 2]][a, z]

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

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

Out[19]=
{4, -10}

In[20]:=
Kh[Knot[9, 2]][q, t]

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

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

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

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₂

9₁
9₃

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-29 - q^-28 - q^-27 + 2q^-26 - q^-25 - 2q^-24 + 3q^-23 - 3q^-21 + 3q^-20 - 3q^-18 + 3q^-17 - 3q^-15 + 2q^-14 - 2q^-12 + 2q^-11 - 2q^-9 + 3q^-8 - 2q^-6 + 2q^-5 - q^-3 + q^-2
3	- q^-57 + q^-56 + q^-55 - 2q^-53 + 2q^-51 + q^-50 - 3q^-49 - q^-48 + 2q^-47 + 2q^-46 - 2q^-45 - 2q^-44 + 2q^-43 + 2q^-42 - 2q^-41 - 2q^-40 + 2q^-39 + 2q^-38 - 2q^-37 - 2q^-36 + 3q^-35 + 2q^-34 - 3q^-33 - 3q^-32 + 4q^-31 + 2q^-30 - 4q^-29 - 3q^-28 + 5q^-27 + 2q^-26 - 5q^-25 - 2q^-24 + 6q^-23 + 2q^-22 - 6q^-21 - 3q^-20 + 6q^-19 + 2q^-18 - 4q^-17 - 4q^-16 + 5q^-15 + 2q^-14 - 2q^-13 - 3q^-12 + 3q^-11 + q^-10 - 2q^-8 + 2q^-7 - q^-4 + q^-3
4	q^-94 - q^-93 - q^-92 + 3q^-89 - q^-88 - q^-87 - q^-86 - 2q^-85 + 5q^-84 - q^-82 - q^-81 - 4q^-80 + 5q^-79 - 5q^-75 + 5q^-74 - 5q^-70 + 5q^-69 - 6q^-65 + 5q^-64 + q^-63 + q^-62 + q^-61 - 8q^-60 + 4q^-59 + 2q^-58 + 2q^-57 + 2q^-56 - 10q^-55 + 2q^-54 + 3q^-53 + 3q^-52 + 3q^-51 - 11q^-50 + 4q^-48 + 4q^-47 + 4q^-46 - 11q^-45 - q^-44 + 5q^-43 + 4q^-42 + 4q^-41 - 12q^-40 - 2q^-39 + 5q^-38 + 5q^-37 + 5q^-36 - 13q^-35 - 3q^-34 + 5q^-33 + 5q^-32 + 5q^-31 - 12q^-30 - 2q^-29 + 3q^-28 + 4q^-27 + 5q^-26 - 10q^-25 + 2q^-23 + 3q^-22 + 3q^-21 - 7q^-20 + q^-19 + q^-18 + q^-17 + 2q^-16 - 4q^-15 + 2q^-14 + q^-11 - 2q^-10 + 2q^-9 - q^-5 + q^-4

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

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

The Alternating Knot 92

The Alternating Knot 9₂