The Alternating Knot 10₃₄

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X₃₈₄₉ X_13,17,14,16 X_5,15,6,14 X_15,7,16,6 X_9,1,10,20 X_11,19,12,18 X_17,13,18,12 X_19,11,20,10 X₇₂₈₃

Gauss Code: {-1, 10, -2, 1, -4, 5, -10, 2, -6, 9, -7, 8, -3, 4, -5, 3, -8, 7, -9, 6}

DT (Dowker-Thistlethwaite) Code: 4 8 14 2 20 18 16 6 12 10

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 2 2 2 / NotAvailable 1

Alexander Polynomial: 3t^-2 - 9t^-1 + 13 - 9t + 3t²

Conway Polynomial: 1 + 3z² + 3z⁴

Other knots with the same Alexander/Conway Polynomial: {10₁₃₅, ...}

Determinant and Signature: {37, 0}

Jones Polynomial: - q^-3 + 2q^-2 - 3q^-1 + 5 - 5q + 6q² - 5q³ + 4q⁴ - 3q⁵ + 2q⁶ - q⁷

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

A2 (sl(3)) Invariant: - q^-10 - q^-4 + 2q^-2 + 1 + q² + q⁴ + q⁸ + q¹⁴ - q¹⁶ - q²²

HOMFLY-PT Polynomial: - a^-6 - a^-6z² + a^-4 + 2a^-4z² + a^-4z⁴ + a^-2z² + a^-2z⁴ + 2 + 2z² + z⁴ - a² - a²z²

Kauffman Polynomial: - 3a^-7z + 7a^-7z³ - 5a^-7z⁵ + a^-7z⁷ + a^-6 - 6a^-6z² + 14a^-6z⁴ - 10a^-6z⁶ + 2a^-6z⁸ - 4a^-5z + 12a^-5z³ - 6a^-5z⁵ - 2a^-5z⁷ + a^-5z⁹ + a^-4 - 8a^-4z² + 20a^-4z⁴ - 17a^-4z⁶ + 4a^-4z⁸ - a^-3z + 5a^-3z³ - 5a^-3z⁵ - a^-3z⁷ + a^-3z⁹ - 3a^-2z² + 4a^-2z⁴ - 5a^-2z⁶ + 2a^-2z⁸ - a^-1z³ - 2a^-1z⁵ + 2a^-1z⁷ + 2 - 3z² + 2z⁶ - az + 2az⁵ + a² - 2a²z² + 2a²z⁴ - a³z + a³z³

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

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=0 is the signature of 10₃₄. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

j = 15 1

j = 13 1

j = 11 2 1

j = 9 2 1

j = 7 3 2

j = 5 3 2

j = 3 2 3

j = 1 3 3

j = -1 1 3

j = -3 1 2

j = -5 1

j = -7 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-9 - 2q^-8 + 4q^-6 - 6q^-5 + q^-4 + 7q^-3 - 10q^-2 + 4q^-1 + 8 - 14q + 8q² + 9q³ - 18q⁴ + 8q⁵ + 13q⁶ - 20q⁷ + 5q⁸ + 15q⁹ - 18q¹⁰ + q¹¹ + 15q¹² - 13q¹³ - 3q¹⁴ + 12q¹⁵ - 6q¹⁶ - 4q¹⁷ + 6q¹⁸ - q¹⁹ - 2q²⁰ + q²¹

3 - q^-18 + 2q^-17 - q^-15 - 3q^-14 + 5q^-13 + 2q^-12 - 5q^-11 - 6q^-10 + 7q^-9 + 9q^-8 - 8q^-7 - 12q^-6 + 5q^-5 + 18q^-4 - 3q^-3 - 19q^-2 - 7q^-1 + 27 + 7q - 19q² - 20q³ + 25q⁴ + 16q⁵ - 14q⁶ - 21q⁷ + 17q⁸ + 12q⁹ - 9q¹⁰ - 11q¹¹ + 11q¹² + q¹³ - 7q¹⁴ + 2q¹⁵ + 7q¹⁶ - 9q¹⁷ - 5q¹⁸ + 13q¹⁹ + 6q²⁰ - 17q²¹ - 8q²² + 20q²³ + 10q²⁴ - 20q²⁵ - 14q²⁶ + 18q²⁷ + 17q²⁸ - 13q²⁹ - 19q³⁰ + 9q³¹ + 16q³² - 2q³³ - 14q³⁴ - q³⁵ + 9q³⁶ + 3q³⁷ - 5q³⁸ - 2q³⁹ + q⁴⁰ + 2q⁴¹ - q⁴²

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[10, 34]]

Out[2]=
PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[13, 17, 14, 16], X[5, 15, 6, 14], > X[15, 7, 16, 6], X[9, 1, 10, 20], X[11, 19, 12, 18], X[17, 13, 18, 12], > X[19, 11, 20, 10], X[7, 2, 8, 3]]

In[3]:=
GaussCode[Knot[10, 34]]

Out[3]=
GaussCode[-1, 10, -2, 1, -4, 5, -10, 2, -6, 9, -7, 8, -3, 4, -5, 3, -8, 7, -9, > 6]

In[4]:=
DTCode[Knot[10, 34]]

Out[4]=
DTCode[4, 8, 14, 2, 20, 18, 16, 6, 12, 10]

In[5]:=
br = BR[Knot[10, 34]]

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

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

Out[6]=
{5, 12}

In[7]:=
BraidIndex[Knot[10, 34]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[10, 34]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 34]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 2, 2, 2, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 34]][t]

Out[10]=
3 9 2 13 + -- - - - 9 t + 3 t 2 t t

In[11]:=
Conway[Knot[10, 34]][z]

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

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

Out[12]=
{Knot[10, 34], Knot[10, 135]}

In[13]:=
{KnotDet[Knot[10, 34]], KnotSignature[Knot[10, 34]]}

Out[13]=
{37, 0}

In[14]:=
Jones[Knot[10, 34]][q]

Out[14]=
-3 2 3 2 3 4 5 6 7 5 - q + -- - - - 5 q + 6 q - 5 q + 4 q - 3 q + 2 q - q 2 q q

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

Out[15]=
{Knot[10, 34]}

In[16]:=
A2Invariant[Knot[10, 34]][q]

Out[16]=
-10 -4 2 2 4 8 14 16 22 1 - q - q + -- + q + q + q + q - q - q 2 q

In[17]:=
HOMFLYPT[Knot[10, 34]][a, z]

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

In[18]:=
Kauffman[Knot[10, 34]][a, z]

Out[18]=
2 2 2 -6 -4 2 3 z 4 z z 3 2 6 z 8 z 3 z 2 + a + a + a - --- - --- - -- - a z - a z - 3 z - ---- - ---- - ---- - 7 5 3 6 4 2 a a a a a a 3 3 3 3 4 4 4 2 2 7 z 12 z 5 z z 3 3 14 z 20 z 4 z > 2 a z + ---- + ----- + ---- - -- + a z + ----- + ----- + ---- + 7 5 3 a 6 4 2 a a a a a a 5 5 5 5 6 6 2 4 5 z 6 z 5 z 2 z 5 6 10 z 17 z > 2 a z - ---- - ---- - ---- - ---- + 2 a z + 2 z - ----- - ----- - 7 5 3 a 6 4 a a a a a 6 7 7 7 7 8 8 8 9 9 5 z z 2 z z 2 z 2 z 4 z 2 z z z > ---- + -- - ---- - -- + ---- + ---- + ---- + ---- + -- + -- 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a

In[19]:=
{Vassiliev[2][Knot[10, 34]], Vassiliev[3][Knot[10, 34]]}

Out[19]=
{3, 3}

In[20]:=
Kh[Knot[10, 34]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 34], 2][q]

Out[21]=
-9 2 4 6 -4 7 10 4 2 3 4 8 + q - -- + -- - -- + q + -- - -- + - - 14 q + 8 q + 9 q - 18 q + 8 6 5 3 2 q q q q q q 5 6 7 8 9 10 11 12 13 > 8 q + 13 q - 20 q + 5 q + 15 q - 18 q + q + 15 q - 13 q - 14 15 16 17 18 19 20 21 > 3 q + 12 q - 6 q - 4 q + 6 q - q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₃₄

10₃₃
10₃₅

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-9 - 2q^-8 + 4q^-6 - 6q^-5 + q^-4 + 7q^-3 - 10q^-2 + 4q^-1 + 8 - 14q + 8q² + 9q³ - 18q⁴ + 8q⁵ + 13q⁶ - 20q⁷ + 5q⁸ + 15q⁹ - 18q¹⁰ + q¹¹ + 15q¹² - 13q¹³ - 3q¹⁴ + 12q¹⁵ - 6q¹⁶ - 4q¹⁷ + 6q¹⁸ - q¹⁹ - 2q²⁰ + q²¹
3	- q^-18 + 2q^-17 - q^-15 - 3q^-14 + 5q^-13 + 2q^-12 - 5q^-11 - 6q^-10 + 7q^-9 + 9q^-8 - 8q^-7 - 12q^-6 + 5q^-5 + 18q^-4 - 3q^-3 - 19q^-2 - 7q^-1 + 27 + 7q - 19q² - 20q³ + 25q⁴ + 16q⁵ - 14q⁶ - 21q⁷ + 17q⁸ + 12q⁹ - 9q¹⁰ - 11q¹¹ + 11q¹² + q¹³ - 7q¹⁴ + 2q¹⁵ + 7q¹⁶ - 9q¹⁷ - 5q¹⁸ + 13q¹⁹ + 6q²⁰ - 17q²¹ - 8q²² + 20q²³ + 10q²⁴ - 20q²⁵ - 14q²⁶ + 18q²⁷ + 17q²⁸ - 13q²⁹ - 19q³⁰ + 9q³¹ + 16q³² - 2q³³ - 14q³⁴ - q³⁵ + 9q³⁶ + 3q³⁷ - 5q³⁸ - 2q³⁹ + q⁴⁰ + 2q⁴¹ - q⁴²

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 34]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[13, 17, 14, 16], X[5, 15, 6, 14], > X[15, 7, 16, 6], X[9, 1, 10, 20], X[11, 19, 12, 18], X[17, 13, 18, 12], > X[19, 11, 20, 10], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 34]]
Out[3]=	GaussCode[-1, 10, -2, 1, -4, 5, -10, 2, -6, 9, -7, 8, -3, 4, -5, 3, -8, 7, -9, > 6]
In[4]:=	DTCode[Knot[10, 34]]
Out[4]=	DTCode[4, 8, 14, 2, 20, 18, 16, 6, 12, 10]
In[5]:=	br = BR[Knot[10, 34]]
Out[5]=	BR[5, {1, 1, 1, 2, -1, 2, 3, -2, -4, 3, -4, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 34]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 34]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 34]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 34]][t]
Out[10]=	3 9 2 13 + -- - - - 9 t + 3 t 2 t t
In[11]:=	Conway[Knot[10, 34]][z]
Out[11]=	2 4 1 + 3 z + 3 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 34], Knot[10, 135]}
In[13]:=	{KnotDet[Knot[10, 34]], KnotSignature[Knot[10, 34]]}
Out[13]=	{37, 0}
In[14]:=	Jones[Knot[10, 34]][q]
Out[14]=	-3 2 3 2 3 4 5 6 7 5 - q + -- - - - 5 q + 6 q - 5 q + 4 q - 3 q + 2 q - q 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 34]}
In[16]:=	A2Invariant[Knot[10, 34]][q]
Out[16]=	-10 -4 2 2 4 8 14 16 22 1 - q - q + -- + q + q + q + q - q - q 2 q
In[17]:=	HOMFLYPT[Knot[10, 34]][a, z]
Out[17]=	2 2 2 4 4 -6 -4 2 2 z 2 z z 2 2 4 z z 2 - a + a - a + 2 z - -- + ---- + -- - a z + z + -- + -- 6 4 2 4 2 a a a a a
In[18]:=	Kauffman[Knot[10, 34]][a, z]
Out[18]=	2 2 2 -6 -4 2 3 z 4 z z 3 2 6 z 8 z 3 z 2 + a + a + a - --- - --- - -- - a z - a z - 3 z - ---- - ---- - ---- - 7 5 3 6 4 2 a a a a a a 3 3 3 3 4 4 4 2 2 7 z 12 z 5 z z 3 3 14 z 20 z 4 z > 2 a z + ---- + ----- + ---- - -- + a z + ----- + ----- + ---- + 7 5 3 a 6 4 2 a a a a a a 5 5 5 5 6 6 2 4 5 z 6 z 5 z 2 z 5 6 10 z 17 z > 2 a z - ---- - ---- - ---- - ---- + 2 a z + 2 z - ----- - ----- - 7 5 3 a 6 4 a a a a a 6 7 7 7 7 8 8 8 9 9 5 z z 2 z z 2 z 2 z 4 z 2 z z z > ---- + -- - ---- - -- + ---- + ---- + ---- + ---- + -- + -- 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 34]], Vassiliev[3][Knot[10, 34]]}
Out[19]=	{3, 3}
In[20]:=	Kh[Knot[10, 34]][q, t]
Out[20]=	3 1 1 1 2 1 3 3 2 - + 3 q + ----- + ----- + ----- + ---- + --- + 3 q t + 2 q t + 3 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 7 4 9 4 9 5 11 5 > 3 q t + 2 q t + 3 q t + 2 q t + 2 q t + q t + 2 q t + 11 6 13 6 15 7 > q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 34], 2][q]
Out[21]=	-9 2 4 6 -4 7 10 4 2 3 4 8 + q - -- + -- - -- + q + -- - -- + - - 14 q + 8 q + 9 q - 18 q + 8 6 5 3 2 q q q q q q 5 6 7 8 9 10 11 12 13 > 8 q + 13 q - 20 q + 5 q + 15 q - 18 q + q + 15 q - 13 q - 14 15 16 17 18 19 20 21 > 3 q + 12 q - 6 q - 4 q + 6 q - q - 2 q + q

The Alternating Knot 1034

The Alternating Knot 10₃₄