The Alternating Knot 9₁₈

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Acknowledgement

KnotPlot

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

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

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

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--6 1

Alexander Polynomial: 4t^-2 - 10t^-1 + 13 - 10t + 4t²

Conway Polynomial: 1 + 6z² + 4z⁴

Other knots with the same Alexander/Conway Polynomial: {K11a246, ...}

Determinant and Signature: {41, -4}

Jones Polynomial: - q^-11 + 2q^-10 - 4q^-9 + 6q^-8 - 7q^-7 + 7q^-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 - 2q^-28 + q^-26 + q^-20 - q^-18 + 2q^-16 + q^-12 + 2q^-10 - q^-8 + q^-6

HOMFLY-PT Polynomial: a⁴ + 2a⁴z² + a⁴z⁴ + a⁶ + 4a⁶z² + 2a⁶z⁴ + a⁸z² + a⁸z⁴ - a¹⁰ - a¹⁰z²

Kauffman Polynomial: a⁴ - 2a⁴z² + a⁴z⁴ - 2a⁵z³ + 2a⁵z⁵ - a⁶ + 3a⁶z² - 4a⁶z⁴ + 3a⁶z⁶ + 2a⁷z - 4a⁷z³ + a⁷z⁵ + 2a⁷z⁷ - 2a⁸z⁴ + 2a⁸z⁶ + a⁸z⁸ + a⁹z³ - 5a⁹z⁵ + 4a⁹z⁷ + a¹⁰ - 2a¹⁰z² - 2a¹⁰z⁴ + a¹⁰z⁶ + a¹⁰z⁸ - 3a¹¹z⁵ + 2a¹¹z⁷ + 3a¹²z² - 5a¹²z⁴ + 2a¹²z⁶ + 2a¹³z - 3a¹³z³ + a¹³z⁵

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=-4 is the signature of 9₁₈. 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 4 3

j = -13 3 3

j = -15 3 4

j = -17 1 3

j = -19 1 3

j = -21 1

j = -23 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-31 - 2q^-30 + 6q^-28 - 8q^-27 - 4q^-26 + 20q^-25 - 15q^-24 - 15q^-23 + 38q^-22 - 19q^-21 - 30q^-20 + 51q^-19 - 16q^-18 - 39q^-17 + 52q^-16 - 10q^-15 - 37q^-14 + 39q^-13 - 3q^-12 - 25q^-11 + 20q^-10 + q^-9 - 10q^-8 + 6q^-7 + q^-6 - 2q^-5 + q^-4

3 - q^-60 + 2q^-59 - 2q^-57 - 3q^-56 + 7q^-55 + 5q^-54 - 10q^-53 - 15q^-52 + 17q^-51 + 26q^-50 - 17q^-49 - 48q^-48 + 18q^-47 + 68q^-46 - 7q^-45 - 93q^-44 - 7q^-43 + 116q^-42 + 26q^-41 - 134q^-40 - 50q^-39 + 148q^-38 + 74q^-37 - 157q^-36 - 95q^-35 + 160q^-34 + 111q^-33 - 155q^-32 - 124q^-31 + 145q^-30 + 127q^-29 - 126q^-28 - 128q^-27 + 107q^-26 + 114q^-25 - 76q^-24 - 104q^-23 + 57q^-22 + 78q^-21 - 29q^-20 - 65q^-19 + 22q^-18 + 37q^-17 - 5q^-16 - 27q^-15 + 5q^-14 + 13q^-13 - 8q^-11 + 2q^-10 + 2q^-9 + q^-8 - 2q^-7 + q^-6

4 q^-98 - 2q^-97 + 2q^-95 - q^-94 + 4q^-93 - 9q^-92 - q^-91 + 10q^-90 + 15q^-88 - 31q^-87 - 17q^-86 + 23q^-85 + 17q^-84 + 58q^-83 - 67q^-82 - 70q^-81 + 7q^-80 + 42q^-79 + 171q^-78 - 73q^-77 - 151q^-76 - 83q^-75 + 19q^-74 + 343q^-73 + 4q^-72 - 195q^-71 - 239q^-70 - 105q^-69 + 506q^-68 + 155q^-67 - 153q^-66 - 397q^-65 - 308q^-64 + 599q^-63 + 323q^-62 - 36q^-61 - 512q^-60 - 520q^-59 + 621q^-58 + 456q^-57 + 97q^-56 - 567q^-55 - 679q^-54 + 582q^-53 + 529q^-52 + 218q^-51 - 557q^-50 - 762q^-49 + 486q^-48 + 527q^-47 + 310q^-46 - 465q^-45 - 748q^-44 + 333q^-43 + 433q^-42 + 355q^-41 - 300q^-40 - 626q^-39 + 168q^-38 + 266q^-37 + 321q^-36 - 123q^-35 - 423q^-34 + 53q^-33 + 103q^-32 + 218q^-31 - 15q^-30 - 221q^-29 + 14q^-28 + 10q^-27 + 106q^-26 + 15q^-25 - 89q^-24 + 10q^-23 - 12q^-22 + 39q^-21 + 9q^-20 - 31q^-19 + 9q^-18 - 7q^-17 + 11q^-16 + 3q^-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, 18]]

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

In[3]:=
GaussCode[Knot[9, 18]]

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

In[4]:=
DTCode[Knot[9, 18]]

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

In[5]:=
br = BR[Knot[9, 18]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[9, 18]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[9, 18]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 18]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[9, 18]][t]

Out[10]=
4 10 2 13 + -- - -- - 10 t + 4 t 2 t t

In[11]:=
Conway[Knot[9, 18]][z]

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

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

Out[12]=
{Knot[9, 18], Knot[11, Alternating, 246]}

In[13]:=
{KnotDet[Knot[9, 18]], KnotSignature[Knot[9, 18]]}

Out[13]=
{41, -4}

In[14]:=
Jones[Knot[9, 18]][q]

Out[14]=
-11 2 4 6 7 7 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, 18]}

In[16]:=
A2Invariant[Knot[9, 18]][q]

Out[16]=
-34 2 -26 -20 -18 2 -12 2 -8 -6 -q - --- + q + q - q + --- + q + --- - q + q 28 16 10 q q q

In[17]:=
HOMFLYPT[Knot[9, 18]][a, z]

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

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

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

In[19]:=
{Vassiliev[2][Knot[9, 18]], Vassiliev[3][Knot[9, 18]]}

Out[19]=
{6, -15}

In[20]:=
Kh[Knot[9, 18]][q, t]

Out[20]=
-5 -3 1 1 1 3 1 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 4 3 3 4 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, 18], 2][q]

Out[21]=
-31 2 6 8 4 20 15 15 38 19 30 51 16 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 39 52 10 37 39 3 25 20 -9 10 6 -6 2 > --- + --- - --- - --- + --- - --- - --- + --- + q - -- + -- + q - -- + 17 16 15 14 13 12 11 10 8 7 5 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 9₁₈

9₁₇
9₁₉

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-31 - 2q^-30 + 6q^-28 - 8q^-27 - 4q^-26 + 20q^-25 - 15q^-24 - 15q^-23 + 38q^-22 - 19q^-21 - 30q^-20 + 51q^-19 - 16q^-18 - 39q^-17 + 52q^-16 - 10q^-15 - 37q^-14 + 39q^-13 - 3q^-12 - 25q^-11 + 20q^-10 + q^-9 - 10q^-8 + 6q^-7 + q^-6 - 2q^-5 + q^-4
3	- q^-60 + 2q^-59 - 2q^-57 - 3q^-56 + 7q^-55 + 5q^-54 - 10q^-53 - 15q^-52 + 17q^-51 + 26q^-50 - 17q^-49 - 48q^-48 + 18q^-47 + 68q^-46 - 7q^-45 - 93q^-44 - 7q^-43 + 116q^-42 + 26q^-41 - 134q^-40 - 50q^-39 + 148q^-38 + 74q^-37 - 157q^-36 - 95q^-35 + 160q^-34 + 111q^-33 - 155q^-32 - 124q^-31 + 145q^-30 + 127q^-29 - 126q^-28 - 128q^-27 + 107q^-26 + 114q^-25 - 76q^-24 - 104q^-23 + 57q^-22 + 78q^-21 - 29q^-20 - 65q^-19 + 22q^-18 + 37q^-17 - 5q^-16 - 27q^-15 + 5q^-14 + 13q^-13 - 8q^-11 + 2q^-10 + 2q^-9 + q^-8 - 2q^-7 + q^-6
4	q^-98 - 2q^-97 + 2q^-95 - q^-94 + 4q^-93 - 9q^-92 - q^-91 + 10q^-90 + 15q^-88 - 31q^-87 - 17q^-86 + 23q^-85 + 17q^-84 + 58q^-83 - 67q^-82 - 70q^-81 + 7q^-80 + 42q^-79 + 171q^-78 - 73q^-77 - 151q^-76 - 83q^-75 + 19q^-74 + 343q^-73 + 4q^-72 - 195q^-71 - 239q^-70 - 105q^-69 + 506q^-68 + 155q^-67 - 153q^-66 - 397q^-65 - 308q^-64 + 599q^-63 + 323q^-62 - 36q^-61 - 512q^-60 - 520q^-59 + 621q^-58 + 456q^-57 + 97q^-56 - 567q^-55 - 679q^-54 + 582q^-53 + 529q^-52 + 218q^-51 - 557q^-50 - 762q^-49 + 486q^-48 + 527q^-47 + 310q^-46 - 465q^-45 - 748q^-44 + 333q^-43 + 433q^-42 + 355q^-41 - 300q^-40 - 626q^-39 + 168q^-38 + 266q^-37 + 321q^-36 - 123q^-35 - 423q^-34 + 53q^-33 + 103q^-32 + 218q^-31 - 15q^-30 - 221q^-29 + 14q^-28 + 10q^-27 + 106q^-26 + 15q^-25 - 89q^-24 + 10q^-23 - 12q^-22 + 39q^-21 + 9q^-20 - 31q^-19 + 9q^-18 - 7q^-17 + 11q^-16 + 3q^-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, 18]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[5, 14, 6, 15], X[9, 18, 10, 1], > X[17, 6, 18, 7], X[7, 16, 8, 17], X[15, 8, 16, 9], X[13, 10, 14, 11], > X[11, 2, 12, 3]]
In[3]:=	GaussCode[Knot[9, 18]]
Out[3]=	GaussCode[-1, 9, -2, 1, -3, 5, -6, 7, -4, 8, -9, 2, -8, 3, -7, 6, -5, 4]
In[4]:=	DTCode[Knot[9, 18]]
Out[4]=	DTCode[4, 12, 14, 16, 18, 2, 10, 8, 6]
In[5]:=	br = BR[Knot[9, 18]]
Out[5]=	BR[4, {-1, -1, -1, -2, 1, -2, -2, -2, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[9, 18]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 18]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 18]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 18]][t]
Out[10]=	4 10 2 13 + -- - -- - 10 t + 4 t 2 t t
In[11]:=	Conway[Knot[9, 18]][z]
Out[11]=	2 4 1 + 6 z + 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 18], Knot[11, Alternating, 246]}
In[13]:=	{KnotDet[Knot[9, 18]], KnotSignature[Knot[9, 18]]}
Out[13]=	{41, -4}
In[14]:=	Jones[Knot[9, 18]][q]
Out[14]=	-11 2 4 6 7 7 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, 18]}
In[16]:=	A2Invariant[Knot[9, 18]][q]
Out[16]=	-34 2 -26 -20 -18 2 -12 2 -8 -6 -q - --- + q + q - q + --- + q + --- - q + q 28 16 10 q q q
In[17]:=	HOMFLYPT[Knot[9, 18]][a, z]
Out[17]=	4 6 10 4 2 6 2 8 2 10 2 4 4 6 4 8 4 a + a - a + 2 a z + 4 a z + a z - a z + a z + 2 a z + a z
In[18]:=	Kauffman[Knot[9, 18]][a, z]
Out[18]=	4 6 10 7 13 4 2 6 2 10 2 12 2 a - a + a + 2 a z + 2 a z - 2 a z + 3 a z - 2 a z + 3 a z - 5 3 7 3 9 3 13 3 4 4 6 4 8 4 > 2 a z - 4 a z + a z - 3 a z + a z - 4 a z - 2 a z - 10 4 12 4 5 5 7 5 9 5 11 5 13 5 > 2 a z - 5 a z + 2 a z + a z - 5 a z - 3 a z + a z + 6 6 8 6 10 6 12 6 7 7 9 7 11 7 > 3 a z + 2 a z + a z + 2 a z + 2 a z + 4 a z + 2 a z + 8 8 10 8 > a z + a z
In[19]:=	{Vassiliev[2][Knot[9, 18]], Vassiliev[3][Knot[9, 18]]}
Out[19]=	{6, -15}
In[20]:=	Kh[Knot[9, 18]][q, t]
Out[20]=	-5 -3 1 1 1 3 1 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 4 3 3 4 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, 18], 2][q]
Out[21]=	-31 2 6 8 4 20 15 15 38 19 30 51 16 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 39 52 10 37 39 3 25 20 -9 10 6 -6 2 > --- + --- - --- - --- + --- - --- - --- + --- + q - -- + -- + q - -- + 17 16 15 14 13 12 11 10 8 7 5 q q q q q q q q q q q -4 > q

The Alternating Knot 918

The Alternating Knot 9₁₈