The Alternating Knot 9₄

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Acknowledgement

KnotPlot

PD Presentation: X₁₆₂₇ X_3,12,4,13 X_7,18,8,1 X_9,16,10,17 X_15,10,16,11 X_17,8,18,9 X_5,14,6,15 X_11,2,12,3 X_13,4,14,5

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

DT (Dowker-Thistlethwaite) Code: 6 12 14 18 16 2 4 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 2 2 / 4--7 1

Alexander Polynomial: 3t^-2 - 5t^-1 + 5 - 5t + 3t²

Conway Polynomial: 1 + 7z² + 3z⁴

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

Determinant and Signature: {21, -4}

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

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

A2 (sl(3)) Invariant: - q^-34 - q^-32 - q^-30 - q^-28 + q^-26 + q^-24 + q^-22 + q^-20 + q^-16 + q^-10 + q^-6

HOMFLY-PT Polynomial: a⁴ + 3a⁴z² + a⁴z⁴ + 2a⁶z² + a⁶z⁴ + 2a⁸ + 3a⁸z² + a⁸z⁴ - 2a¹⁰ - a¹⁰z²

Kauffman Polynomial: a⁴ - 3a⁴z² + a⁴z⁴ - 2a⁵z³ + a⁵z⁵ + a⁶z² - 2a⁶z⁴ + a⁶z⁶ + 4a⁷z³ - 3a⁷z⁵ + a⁷z⁷ + 2a⁸ - 7a⁸z² + 11a⁸z⁴ - 5a⁸z⁶ + a⁸z⁸ - 4a⁹z + 12a⁹z³ - 8a⁹z⁵ + 2a⁹z⁷ + 2a¹⁰ - 10a¹⁰z² + 11a¹⁰z⁴ - 5a¹⁰z⁶ + a¹⁰z⁸ - a¹¹z + 2a¹¹z³ - 3a¹¹z⁵ + a¹¹z⁷ + a¹²z² - 3a¹²z⁴ + a¹²z⁶ + 3a¹³z - 4a¹³z³ + a¹³z⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {7, -19}

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 9₄. 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 1 1

j = -7 1

j = -9 2 1

j = -11 2 1

j = -13 1 2

j = -15 2 2

j = -17 1

j = -19 1 2

j = -21

j = -23 1

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

3 - q^-60 + q^-59 - q^-56 + 2q^-55 - q^-53 - 2q^-52 + 4q^-51 + q^-50 - 3q^-49 - 5q^-48 + 5q^-47 + 5q^-46 - 3q^-45 - 7q^-44 + 2q^-43 + 8q^-42 - 8q^-40 - 3q^-39 + 9q^-38 + 4q^-37 - 7q^-36 - 8q^-35 + 9q^-34 + 6q^-33 - 7q^-32 - 10q^-31 + 10q^-30 + 6q^-29 - 6q^-28 - 8q^-27 + 6q^-26 + 6q^-25 - 3q^-24 - 5q^-23 + 2q^-22 + 2q^-21 + q^-20 - 2q^-19 + q^-18 - q^-17 + q^-16 - q^-15 + 2q^-14 - q^-13 - q^-11 + 2q^-10 - q^-7 + q^-6

4 q^-98 - q^-97 - q^-94 + 2q^-93 - 2q^-92 + q^-91 + q^-90 - 3q^-89 + 3q^-88 - 4q^-87 + 2q^-86 + 5q^-85 - 4q^-84 + 4q^-83 - 9q^-82 + 8q^-80 - 2q^-79 + 9q^-78 - 14q^-77 - 4q^-76 + 7q^-75 - 3q^-74 + 17q^-73 - 13q^-72 - 4q^-71 + 3q^-70 - 11q^-69 + 22q^-68 - 8q^-67 + q^-66 + 2q^-65 - 22q^-64 + 21q^-63 - 4q^-62 + 10q^-61 + 3q^-60 - 32q^-59 + 18q^-58 - 2q^-57 + 16q^-56 + 4q^-55 - 37q^-54 + 17q^-53 - 2q^-52 + 18q^-51 + 4q^-50 - 39q^-49 + 18q^-48 - 2q^-47 + 18q^-46 + 6q^-45 - 36q^-44 + 13q^-43 - 4q^-42 + 17q^-41 + 9q^-40 - 27q^-39 + 7q^-38 - 9q^-37 + 11q^-36 + 13q^-35 - 13q^-34 + 4q^-33 - 13q^-32 + 3q^-31 + 11q^-30 - 4q^-29 + 7q^-28 - 11q^-27 - 2q^-26 + 5q^-25 - q^-24 + 8q^-23 - 6q^-22 - 2q^-21 + q^-20 - 2q^-19 + 6q^-18 - 2q^-17 - q^-16 - 2q^-14 + 3q^-13 - q^-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, 4]]

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

In[3]:=
GaussCode[Knot[9, 4]]

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

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

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

In[5]:=
br = BR[Knot[9, 4]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[9, 4]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[9, 4]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[9, 4]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[9, 4]][t]

Out[10]=
3 5 2 5 + -- - - - 5 t + 3 t 2 t t

In[11]:=
Conway[Knot[9, 4]][z]

Out[11]=
2 4 1 + 7 z + 3 z

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

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

In[13]:=
{KnotDet[Knot[9, 4]], KnotSignature[Knot[9, 4]]}

Out[13]=
{21, -4}

In[14]:=
Jones[Knot[9, 4]][q]

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

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

Out[15]=
{Knot[9, 4]}

In[16]:=
A2Invariant[Knot[9, 4]][q]

Out[16]=
-34 -32 -30 -28 -26 -24 -22 -20 -16 -10 -6 -q - q - q - q + q + q + q + q + q + q + q

In[17]:=
HOMFLYPT[Knot[9, 4]][a, z]

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

In[18]:=
Kauffman[Knot[9, 4]][a, z]

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

In[19]:=
{Vassiliev[2][Knot[9, 4]], Vassiliev[3][Knot[9, 4]]}

Out[19]=
{7, -19}

In[20]:=
Kh[Knot[9, 4]][q, t]

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

In[21]:=
ColouredJones[Knot[9, 4], 2][q]

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

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 9₄

9₃
9₅

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-31 - q^-30 + 2q^-28 - 3q^-27 - q^-26 + 5q^-25 - 4q^-24 - 3q^-23 + 8q^-22 - 4q^-21 - 6q^-20 + 10q^-19 - 3q^-18 - 8q^-17 + 11q^-16 - 3q^-15 - 7q^-14 + 8q^-13 - q^-12 - 4q^-11 + 4q^-10 - 2q^-8 + 2q^-7 - q^-5 + q^-4
3	- q^-60 + q^-59 - q^-56 + 2q^-55 - q^-53 - 2q^-52 + 4q^-51 + q^-50 - 3q^-49 - 5q^-48 + 5q^-47 + 5q^-46 - 3q^-45 - 7q^-44 + 2q^-43 + 8q^-42 - 8q^-40 - 3q^-39 + 9q^-38 + 4q^-37 - 7q^-36 - 8q^-35 + 9q^-34 + 6q^-33 - 7q^-32 - 10q^-31 + 10q^-30 + 6q^-29 - 6q^-28 - 8q^-27 + 6q^-26 + 6q^-25 - 3q^-24 - 5q^-23 + 2q^-22 + 2q^-21 + q^-20 - 2q^-19 + q^-18 - q^-17 + q^-16 - q^-15 + 2q^-14 - q^-13 - q^-11 + 2q^-10 - q^-7 + q^-6
4	q^-98 - q^-97 - q^-94 + 2q^-93 - 2q^-92 + q^-91 + q^-90 - 3q^-89 + 3q^-88 - 4q^-87 + 2q^-86 + 5q^-85 - 4q^-84 + 4q^-83 - 9q^-82 + 8q^-80 - 2q^-79 + 9q^-78 - 14q^-77 - 4q^-76 + 7q^-75 - 3q^-74 + 17q^-73 - 13q^-72 - 4q^-71 + 3q^-70 - 11q^-69 + 22q^-68 - 8q^-67 + q^-66 + 2q^-65 - 22q^-64 + 21q^-63 - 4q^-62 + 10q^-61 + 3q^-60 - 32q^-59 + 18q^-58 - 2q^-57 + 16q^-56 + 4q^-55 - 37q^-54 + 17q^-53 - 2q^-52 + 18q^-51 + 4q^-50 - 39q^-49 + 18q^-48 - 2q^-47 + 18q^-46 + 6q^-45 - 36q^-44 + 13q^-43 - 4q^-42 + 17q^-41 + 9q^-40 - 27q^-39 + 7q^-38 - 9q^-37 + 11q^-36 + 13q^-35 - 13q^-34 + 4q^-33 - 13q^-32 + 3q^-31 + 11q^-30 - 4q^-29 + 7q^-28 - 11q^-27 - 2q^-26 + 5q^-25 - q^-24 + 8q^-23 - 6q^-22 - 2q^-21 + q^-20 - 2q^-19 + 6q^-18 - 2q^-17 - q^-16 - 2q^-14 + 3q^-13 - q^-9 + q^-8

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

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

The Alternating Knot 94

The Alternating Knot 9₄