The Alternating Knot 9₂₃

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Acknowledgement

KnotPlot

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

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

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

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 2 2 / 4--7 1

Alexander Polynomial: 4t^-2 - 11t^-1 + 15 - 11t + 4t²

Conway Polynomial: 1 + 5z² + 4z⁴

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

Determinant and Signature: {45, -4}

Jones Polynomial: - q^-11 + 3q^-10 - 5q^-9 + 6q^-8 - 8q^-7 + 8q^-6 - 6q^-5 + 5q^-4 - 2q^-3 + q^-2

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

A2 (sl(3)) Invariant: - q^-34 + q^-32 + q^-30 - 2q^-28 - 2q^-24 - q^-22 + q^-20 + 3q^-16 + q^-12 + 2q^-10 - q^-8 + q^-6

HOMFLY-PT Polynomial: a⁴ + 2a⁴z² + a⁴z⁴ + 2a⁶ + 4a⁶z² + 2a⁶z⁴ - 2a⁸ + a⁸z⁴ - a¹⁰z²

Kauffman Polynomial: a⁴ - 2a⁴z² + a⁴z⁴ - 2a⁵z³ + 2a⁵z⁵ - 2a⁶ + 4a⁶z² - 4a⁶z⁴ + 3a⁶z⁶ + 4a⁷z - 6a⁷z³ + 2a⁷z⁵ + 2a⁷z⁷ - 2a⁸ + 6a⁸z² - 8a⁸z⁴ + 4a⁸z⁶ + a⁸z⁸ + 4a⁹z - 2a⁹z³ - 6a⁹z⁵ + 5a⁹z⁷ + 3a¹⁰z² - 10a¹⁰z⁴ + 4a¹⁰z⁶ + a¹⁰z⁸ + a¹¹z - 5a¹¹z⁵ + 3a¹¹z⁷ + 3a¹²z² - 7a¹²z⁴ + 3a¹²z⁶ + a¹³z - 2a¹³z³ + a¹³z⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {5, -11}

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 9₂₃. 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 = -3 1

j = -5 2 1

j = -7 3

j = -9 3 2

j = -11 5 3

j = -13 3 3

j = -15 3 5

j = -17 2 3

j = -19 1 3

j = -21 2

j = -23 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-31 - 3q^-30 + 10q^-28 - 11q^-27 - 8q^-26 + 28q^-25 - 17q^-24 - 24q^-23 + 47q^-22 - 16q^-21 - 41q^-20 + 59q^-19 - 11q^-18 - 51q^-17 + 56q^-16 - 4q^-15 - 45q^-14 + 39q^-13 + 2q^-12 - 27q^-11 + 19q^-10 + 2q^-9 - 10q^-8 + 6q^-7 + q^-6 - 2q^-5 + q^-4

3 - q^-60 + 3q^-59 - 5q^-57 - 5q^-56 + 11q^-55 + 15q^-54 - 17q^-53 - 30q^-52 + 17q^-51 + 52q^-50 - 10q^-49 - 79q^-48 - 4q^-47 + 103q^-46 + 29q^-45 - 125q^-44 - 57q^-43 + 137q^-42 + 95q^-41 - 151q^-40 - 122q^-39 + 148q^-38 + 157q^-37 - 151q^-36 - 177q^-35 + 139q^-34 + 197q^-33 - 126q^-32 - 205q^-31 + 110q^-30 + 196q^-29 - 80q^-28 - 188q^-27 + 62q^-26 + 155q^-25 - 32q^-24 - 132q^-23 + 23q^-22 + 89q^-21 - 3q^-20 - 69q^-19 + 7q^-18 + 37q^-17 + 2q^-16 - 26q^-15 + 3q^-14 + 12q^-13 + q^-12 - 8q^-11 + 2q^-10 + 2q^-9 + q^-8 - 2q^-7 + q^-6

4 q^-98 - 3q^-97 + 5q^-95 + 5q^-93 - 18q^-92 - 8q^-91 + 17q^-90 + 12q^-89 + 39q^-88 - 52q^-87 - 54q^-86 + 8q^-85 + 33q^-84 + 145q^-83 - 58q^-82 - 130q^-81 - 81q^-80 - 5q^-79 + 322q^-78 + 35q^-77 - 160q^-76 - 245q^-75 - 174q^-74 + 472q^-73 + 222q^-72 - 53q^-71 - 399q^-70 - 465q^-69 + 508q^-68 + 421q^-67 + 176q^-66 - 469q^-65 - 779q^-64 + 433q^-63 + 559q^-62 + 442q^-61 - 455q^-60 - 1032q^-59 + 307q^-58 + 622q^-57 + 666q^-56 - 388q^-55 - 1184q^-54 + 160q^-53 + 613q^-52 + 815q^-51 - 276q^-50 - 1198q^-49 + 6q^-48 + 508q^-47 + 851q^-46 - 112q^-45 - 1041q^-44 - 126q^-43 + 308q^-42 + 745q^-41 + 50q^-40 - 743q^-39 - 167q^-38 + 94q^-37 + 512q^-36 + 129q^-35 - 415q^-34 - 112q^-33 - 33q^-32 + 264q^-31 + 108q^-30 - 184q^-29 - 33q^-28 - 52q^-27 + 103q^-26 + 51q^-25 - 72q^-24 + 6q^-23 - 27q^-22 + 34q^-21 + 15q^-20 - 28q^-19 + 10q^-18 - 9q^-17 + 10q^-16 + 4q^-15 - 9q^-14 + 4q^-13 - 2q^-12 + 2q^-11 + q^-10 - 2q^-9 + q^-8

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, 23]]

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

In[3]:=
GaussCode[Knot[9, 23]]

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

In[4]:=
DTCode[Knot[9, 23]]

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

In[5]:=
br = BR[Knot[9, 23]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[9, 23]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[9, 23]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 23]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[9, 23]][t]

Out[10]=
4 11 2 15 + -- - -- - 11 t + 4 t 2 t t

In[11]:=
Conway[Knot[9, 23]][z]

Out[11]=
2 4 1 + 5 z + 4 z

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

Out[12]=
{Knot[9, 23]}

In[13]:=
{KnotDet[Knot[9, 23]], KnotSignature[Knot[9, 23]]}

Out[13]=
{45, -4}

In[14]:=
Jones[Knot[9, 23]][q]

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

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

Out[15]=
{Knot[9, 23]}

In[16]:=
A2Invariant[Knot[9, 23]][q]

Out[16]=
-34 -32 -30 2 2 -22 -20 3 -12 2 -8 -6 -q + q + q - --- - --- - q + q + --- + q + --- - q + q 28 24 16 10 q q q q

In[17]:=
HOMFLYPT[Knot[9, 23]][a, z]

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

In[18]:=
Kauffman[Knot[9, 23]][a, z]

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

In[19]:=
{Vassiliev[2][Knot[9, 23]], Vassiliev[3][Knot[9, 23]]}

Out[19]=
{5, -11}

In[20]:=
Kh[Knot[9, 23]][q, t]

Out[20]=
-5 -3 1 2 1 3 2 3 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q t q t q t q t q t q t q t 5 3 3 5 3 3 2 3 2 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 5 q t q t q t q t q t q t q t q t q t

In[21]:=
ColouredJones[Knot[9, 23], 2][q]

Out[21]=
-31 3 10 11 8 28 17 24 47 16 41 59 11 q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - --- - 30 28 27 26 25 24 23 22 21 20 19 18 q q q q q q q q q q q q 51 56 4 45 39 2 27 19 2 10 6 -6 2 > --- + --- - --- - --- + --- + --- - --- + --- + -- - -- + -- + q - -- + 17 16 15 14 13 12 11 10 9 8 7 5 q q q q q q q q q q q q -4 > q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₂₃

9₂₂
9₂₄

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-31 - 3q^-30 + 10q^-28 - 11q^-27 - 8q^-26 + 28q^-25 - 17q^-24 - 24q^-23 + 47q^-22 - 16q^-21 - 41q^-20 + 59q^-19 - 11q^-18 - 51q^-17 + 56q^-16 - 4q^-15 - 45q^-14 + 39q^-13 + 2q^-12 - 27q^-11 + 19q^-10 + 2q^-9 - 10q^-8 + 6q^-7 + q^-6 - 2q^-5 + q^-4
3	- q^-60 + 3q^-59 - 5q^-57 - 5q^-56 + 11q^-55 + 15q^-54 - 17q^-53 - 30q^-52 + 17q^-51 + 52q^-50 - 10q^-49 - 79q^-48 - 4q^-47 + 103q^-46 + 29q^-45 - 125q^-44 - 57q^-43 + 137q^-42 + 95q^-41 - 151q^-40 - 122q^-39 + 148q^-38 + 157q^-37 - 151q^-36 - 177q^-35 + 139q^-34 + 197q^-33 - 126q^-32 - 205q^-31 + 110q^-30 + 196q^-29 - 80q^-28 - 188q^-27 + 62q^-26 + 155q^-25 - 32q^-24 - 132q^-23 + 23q^-22 + 89q^-21 - 3q^-20 - 69q^-19 + 7q^-18 + 37q^-17 + 2q^-16 - 26q^-15 + 3q^-14 + 12q^-13 + q^-12 - 8q^-11 + 2q^-10 + 2q^-9 + q^-8 - 2q^-7 + q^-6
4	q^-98 - 3q^-97 + 5q^-95 + 5q^-93 - 18q^-92 - 8q^-91 + 17q^-90 + 12q^-89 + 39q^-88 - 52q^-87 - 54q^-86 + 8q^-85 + 33q^-84 + 145q^-83 - 58q^-82 - 130q^-81 - 81q^-80 - 5q^-79 + 322q^-78 + 35q^-77 - 160q^-76 - 245q^-75 - 174q^-74 + 472q^-73 + 222q^-72 - 53q^-71 - 399q^-70 - 465q^-69 + 508q^-68 + 421q^-67 + 176q^-66 - 469q^-65 - 779q^-64 + 433q^-63 + 559q^-62 + 442q^-61 - 455q^-60 - 1032q^-59 + 307q^-58 + 622q^-57 + 666q^-56 - 388q^-55 - 1184q^-54 + 160q^-53 + 613q^-52 + 815q^-51 - 276q^-50 - 1198q^-49 + 6q^-48 + 508q^-47 + 851q^-46 - 112q^-45 - 1041q^-44 - 126q^-43 + 308q^-42 + 745q^-41 + 50q^-40 - 743q^-39 - 167q^-38 + 94q^-37 + 512q^-36 + 129q^-35 - 415q^-34 - 112q^-33 - 33q^-32 + 264q^-31 + 108q^-30 - 184q^-29 - 33q^-28 - 52q^-27 + 103q^-26 + 51q^-25 - 72q^-24 + 6q^-23 - 27q^-22 + 34q^-21 + 15q^-20 - 28q^-19 + 10q^-18 - 9q^-17 + 10q^-16 + 4q^-15 - 9q^-14 + 4q^-13 - 2q^-12 + 2q^-11 + q^-10 - 2q^-9 + q^-8

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[9, 23]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 12, 6, 13], X[7, 16, 8, 17], > X[13, 18, 14, 1], X[17, 14, 18, 15], X[15, 6, 16, 7], X[11, 8, 12, 9], > X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[9, 23]]
Out[3]=	GaussCode[-1, 9, -2, 1, -3, 7, -4, 8, -9, 2, -8, 3, -5, 6, -7, 4, -6, 5]
In[4]:=	DTCode[Knot[9, 23]]
Out[4]=	DTCode[4, 10, 12, 16, 2, 8, 18, 6, 14]
In[5]:=	br = BR[Knot[9, 23]]
Out[5]=	BR[4, {-1, -1, -1, -2, 1, -2, -2, -3, 2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[9, 23]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 23]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 23]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, {4, 7}, 1}
In[10]:=	alex = Alexander[Knot[9, 23]][t]
Out[10]=	4 11 2 15 + -- - -- - 11 t + 4 t 2 t t
In[11]:=	Conway[Knot[9, 23]][z]
Out[11]=	2 4 1 + 5 z + 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 23]}
In[13]:=	{KnotDet[Knot[9, 23]], KnotSignature[Knot[9, 23]]}
Out[13]=	{45, -4}
In[14]:=	Jones[Knot[9, 23]][q]
Out[14]=	-11 3 5 6 8 8 6 5 2 -2 -q + --- - -- + -- - -- + -- - -- + -- - -- + q 10 9 8 7 6 5 4 3 q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 23]}
In[16]:=	A2Invariant[Knot[9, 23]][q]
Out[16]=	-34 -32 -30 2 2 -22 -20 3 -12 2 -8 -6 -q + q + q - --- - --- - q + q + --- + q + --- - q + q 28 24 16 10 q q q q
In[17]:=	HOMFLYPT[Knot[9, 23]][a, z]
Out[17]=	4 6 8 4 2 6 2 10 2 4 4 6 4 8 4 a + 2 a - 2 a + 2 a z + 4 a z - a z + a z + 2 a z + a z
In[18]:=	Kauffman[Knot[9, 23]][a, z]
Out[18]=	4 6 8 7 9 11 13 4 2 6 2 a - 2 a - 2 a + 4 a z + 4 a z + a z + a z - 2 a z + 4 a z + 8 2 10 2 12 2 5 3 7 3 9 3 13 3 > 6 a z + 3 a z + 3 a z - 2 a z - 6 a z - 2 a z - 2 a z + 4 4 6 4 8 4 10 4 12 4 5 5 7 5 > a z - 4 a z - 8 a z - 10 a z - 7 a z + 2 a z + 2 a z - 9 5 11 5 13 5 6 6 8 6 10 6 12 6 > 6 a z - 5 a z + a z + 3 a z + 4 a z + 4 a z + 3 a z + 7 7 9 7 11 7 8 8 10 8 > 2 a z + 5 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[9, 23]], Vassiliev[3][Knot[9, 23]]}
Out[19]=	{5, -11}
In[20]:=	Kh[Knot[9, 23]][q, t]
Out[20]=	-5 -3 1 2 1 3 2 3 3 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q t q t q t q t q t q t q t 5 3 3 5 3 3 2 3 2 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 5 q t q t q t q t q t q t q t q t q t
In[21]:=	ColouredJones[Knot[9, 23], 2][q]
Out[21]=	-31 3 10 11 8 28 17 24 47 16 41 59 11 q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - --- - 30 28 27 26 25 24 23 22 21 20 19 18 q q q q q q q q q q q q 51 56 4 45 39 2 27 19 2 10 6 -6 2 > --- + --- - --- - --- + --- + --- - --- + --- + -- - -- + -- + q - -- + 17 16 15 14 13 12 11 10 9 8 7 5 q q q q q q q q q q q q -4 > q

The Alternating Knot 923

The Alternating Knot 9₂₃