The Non Alternating Knot 10₁₃₅

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X₃₈₄₉ X_9,15,10,14 X_12,5,13,6 X_6,13,7,14 X_11,19,12,18 X_15,1,16,20 X_19,17,20,16 X_17,11,18,10 X₇₂₈₃

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

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

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 3 / NotAvailable 1

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

Conway Polynomial: 1 + 3z² + 3z⁴

Other knots with the same Alexander/Conway Polynomial: {10₃₄, ...}

Determinant and Signature: {37, 0}

Jones Polynomial: - q^-5 + 2q^-4 - 4q^-3 + 6q^-2 - 6q^-1 + 7 - 5q + 4q² - 2q³

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

A2 (sl(3)) Invariant: - q^-16 - 2q^-10 + q^-8 + q^-4 + 3q^-2 + 1 + 3q² - q⁴ - 2q¹⁰

HOMFLY-PT Polynomial: - 2a^-2 - 2a^-2z² + 4 + 5z² + 2z⁴ + a²z² + a²z⁴ - a⁴ - a⁴z²

Kauffman Polynomial: - 3a^-3z + 3a^-3z³ + 2a^-2 - 4a^-2z² + 2a^-2z⁴ + a^-2z⁶ - 6a^-1z + 9a^-1z³ - 4a^-1z⁵ + 2a^-1z⁷ + 4 - 6z² + 3z⁴ + z⁸ - 4az + 8az³ - 8az⁵ + 4az⁷ + a²z² - 4a²z⁴ + a²z⁶ + a²z⁸ + a³z - a³z³ - 3a³z⁵ + 2a³z⁷ - a⁴ + 3a⁴z² - 5a⁴z⁴ + 2a⁴z⁶ + 2a⁵z - 3a⁵z³ + a⁵z⁵

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {3, -1}

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

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

j = 7 2

j = 5 2

j = 3 3 2

j = 1 4 2

j = -1 3 4

j = -3 3 3

j = -5 1 3

j = -7 1 3

j = -9 1

j = -11 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-15 - 2q^-14 + 6q^-12 - 8q^-11 - 4q^-10 + 19q^-9 - 14q^-8 - 16q^-7 + 35q^-6 - 13q^-5 - 30q^-4 + 44q^-3 - 7q^-2 - 38q^-1 + 42 - q - 34q² + 28q³ + 4q⁴ - 21q⁵ + 11q⁶ + 4q⁷ - 7q⁸ + q⁹ + q¹⁰

3 - q^-30 + 2q^-29 - 2q^-27 - 3q^-26 + 7q^-25 + 5q^-24 - 10q^-23 - 14q^-22 + 16q^-21 + 25q^-20 - 14q^-19 - 46q^-18 + 11q^-17 + 64q^-16 + 5q^-15 - 85q^-14 - 29q^-13 + 102q^-12 + 53q^-11 - 107q^-10 - 86q^-9 + 114q^-8 + 108q^-7 - 106q^-6 - 135q^-5 + 107q^-4 + 142q^-3 - 87q^-2 - 158q^-1 + 82 + 149q - 55q² - 148q³ + 42q⁴ + 125q⁵ - 15q⁶ - 106q⁷ + 77q⁹ + 12q¹⁰ - 51q¹¹ - 16q¹² + 28q¹³ + 13q¹⁴ - 10q¹⁵ - 12q¹⁶ + 6q¹⁷ + 2q¹⁸ + 2q¹⁹ - 2q²⁰

4 q^-50 - 2q^-49 + 2q^-47 - q^-46 + 4q^-45 - 9q^-44 - q^-43 + 10q^-42 + 14q^-40 - 30q^-39 - 16q^-38 + 22q^-37 + 15q^-36 + 56q^-35 - 58q^-34 - 66q^-33 - 3q^-32 + 28q^-31 + 168q^-30 - 35q^-29 - 130q^-28 - 114q^-27 - 37q^-26 + 318q^-25 + 97q^-24 - 113q^-23 - 277q^-22 - 237q^-21 + 401q^-20 + 298q^-19 + 36q^-18 - 389q^-17 - 508q^-16 + 362q^-15 + 466q^-14 + 257q^-13 - 410q^-12 - 740q^-11 + 254q^-10 + 552q^-9 + 450q^-8 - 371q^-7 - 875q^-6 + 133q^-5 + 566q^-4 + 577q^-3 - 296q^-2 - 909q^-1 + 12 + 508q + 631q² - 173q³ - 827q⁴ - 115q⁵ + 361q⁶ + 598q⁷ - 11q⁸ - 622q⁹ - 200q¹⁰ + 152q¹¹ + 453q¹² + 119q¹³ - 343q¹⁴ - 188q¹⁵ - 19q¹⁶ + 241q¹⁷ + 140q¹⁸ - 112q¹⁹ - 97q²⁰ - 70q²¹ + 70q²² + 77q²³ - 8q²⁴ - 21q²⁵ - 37q²⁶ + 4q²⁷ + 18q²⁸ + 5q²⁹ + 2q³⁰ - 6q³¹ - 3q³² + q³³ + q³⁴

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, 135]]

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

In[3]:=
GaussCode[Knot[10, 135]]

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

In[4]:=
DTCode[Knot[10, 135]]

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

In[5]:=
br = BR[Knot[10, 135]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 135]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 135]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 135]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 2, 2, 3, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 135]][t]

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

In[11]:=
Conway[Knot[10, 135]][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, 135]], KnotSignature[Knot[10, 135]]}

Out[13]=
{37, 0}

In[14]:=
Jones[Knot[10, 135]][q]

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

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

Out[15]=
{Knot[10, 135]}

In[16]:=
A2Invariant[Knot[10, 135]][q]

Out[16]=
-16 2 -8 -4 3 2 4 10 1 - q - --- + q + q + -- + 3 q - q - 2 q 10 2 q q

In[17]:=
HOMFLYPT[Knot[10, 135]][a, z]

Out[17]=
2 2 4 2 2 z 2 2 4 2 4 2 4 4 - -- - a + 5 z - ---- + a z - a z + 2 z + a z 2 2 a a

In[18]:=
Kauffman[Knot[10, 135]][a, z]

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

In[19]:=
{Vassiliev[2][Knot[10, 135]], Vassiliev[3][Knot[10, 135]]}

Out[19]=
{3, -1}

In[20]:=
Kh[Knot[10, 135]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 135], 2][q]

Out[21]=
-15 2 6 8 4 19 14 16 35 13 30 44 7 42 + q - --- + --- - --- - --- + -- - -- - -- + -- - -- - -- + -- - -- - 14 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 38 2 3 4 5 6 7 8 9 10 > -- - q - 34 q + 28 q + 4 q - 21 q + 11 q + 4 q - 7 q + q + q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₃₅

10₁₃₄
10₁₃₆

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-15 - 2q^-14 + 6q^-12 - 8q^-11 - 4q^-10 + 19q^-9 - 14q^-8 - 16q^-7 + 35q^-6 - 13q^-5 - 30q^-4 + 44q^-3 - 7q^-2 - 38q^-1 + 42 - q - 34q² + 28q³ + 4q⁴ - 21q⁵ + 11q⁶ + 4q⁷ - 7q⁸ + q⁹ + q¹⁰
3	- q^-30 + 2q^-29 - 2q^-27 - 3q^-26 + 7q^-25 + 5q^-24 - 10q^-23 - 14q^-22 + 16q^-21 + 25q^-20 - 14q^-19 - 46q^-18 + 11q^-17 + 64q^-16 + 5q^-15 - 85q^-14 - 29q^-13 + 102q^-12 + 53q^-11 - 107q^-10 - 86q^-9 + 114q^-8 + 108q^-7 - 106q^-6 - 135q^-5 + 107q^-4 + 142q^-3 - 87q^-2 - 158q^-1 + 82 + 149q - 55q² - 148q³ + 42q⁴ + 125q⁵ - 15q⁶ - 106q⁷ + 77q⁹ + 12q¹⁰ - 51q¹¹ - 16q¹² + 28q¹³ + 13q¹⁴ - 10q¹⁵ - 12q¹⁶ + 6q¹⁷ + 2q¹⁸ + 2q¹⁹ - 2q²⁰
4	q^-50 - 2q^-49 + 2q^-47 - q^-46 + 4q^-45 - 9q^-44 - q^-43 + 10q^-42 + 14q^-40 - 30q^-39 - 16q^-38 + 22q^-37 + 15q^-36 + 56q^-35 - 58q^-34 - 66q^-33 - 3q^-32 + 28q^-31 + 168q^-30 - 35q^-29 - 130q^-28 - 114q^-27 - 37q^-26 + 318q^-25 + 97q^-24 - 113q^-23 - 277q^-22 - 237q^-21 + 401q^-20 + 298q^-19 + 36q^-18 - 389q^-17 - 508q^-16 + 362q^-15 + 466q^-14 + 257q^-13 - 410q^-12 - 740q^-11 + 254q^-10 + 552q^-9 + 450q^-8 - 371q^-7 - 875q^-6 + 133q^-5 + 566q^-4 + 577q^-3 - 296q^-2 - 909q^-1 + 12 + 508q + 631q² - 173q³ - 827q⁴ - 115q⁵ + 361q⁶ + 598q⁷ - 11q⁸ - 622q⁹ - 200q¹⁰ + 152q¹¹ + 453q¹² + 119q¹³ - 343q¹⁴ - 188q¹⁵ - 19q¹⁶ + 241q¹⁷ + 140q¹⁸ - 112q¹⁹ - 97q²⁰ - 70q²¹ + 70q²² + 77q²³ - 8q²⁴ - 21q²⁵ - 37q²⁶ + 4q²⁷ + 18q²⁸ + 5q²⁹ + 2q³⁰ - 6q³¹ - 3q³² + q³³ + q³⁴

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 135]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[9, 15, 10, 14], X[12, 5, 13, 6], > X[6, 13, 7, 14], X[11, 19, 12, 18], X[15, 1, 16, 20], X[19, 17, 20, 16], > X[17, 11, 18, 10], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 135]]
Out[3]=	GaussCode[-1, 10, -2, 1, 4, -5, -10, 2, -3, 9, -6, -4, 5, 3, -7, 8, -9, 6, -8, > 7]
In[4]:=	DTCode[Knot[10, 135]]
Out[4]=	DTCode[4, 8, -12, 2, 14, 18, -6, 20, 10, 16]
In[5]:=	br = BR[Knot[10, 135]]
Out[5]=	BR[4, {1, 1, 1, 2, -1, 2, -3, -2, -2, -2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 135]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 135]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 135]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 135]][t]
Out[10]=	3 9 2 13 + -- - - - 9 t + 3 t 2 t t
In[11]:=	Conway[Knot[10, 135]][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, 135]], KnotSignature[Knot[10, 135]]}
Out[13]=	{37, 0}
In[14]:=	Jones[Knot[10, 135]][q]
Out[14]=	-5 2 4 6 6 2 3 7 - q + -- - -- + -- - - - 5 q + 4 q - 2 q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 135]}
In[16]:=	A2Invariant[Knot[10, 135]][q]
Out[16]=	-16 2 -8 -4 3 2 4 10 1 - q - --- + q + q + -- + 3 q - q - 2 q 10 2 q q
In[17]:=	HOMFLYPT[Knot[10, 135]][a, z]
Out[17]=	2 2 4 2 2 z 2 2 4 2 4 2 4 4 - -- - a + 5 z - ---- + a z - a z + 2 z + a z 2 2 a a
In[18]:=	Kauffman[Knot[10, 135]][a, z]
Out[18]=	2 2 4 3 z 6 z 3 5 2 4 z 2 2 4 + -- - a - --- - --- - 4 a z + a z + 2 a z - 6 z - ---- + a z + 2 3 a 2 a a a 3 3 4 4 2 3 z 9 z 3 3 3 5 3 4 2 z 2 4 > 3 a z + ---- + ---- + 8 a z - a z - 3 a z + 3 z + ---- - 4 a z - 3 a 2 a a 5 6 7 4 4 4 z 5 3 5 5 5 z 2 6 4 6 2 z > 5 a z - ---- - 8 a z - 3 a z + a z + -- + a z + 2 a z + ---- + a 2 a a 7 3 7 8 2 8 > 4 a z + 2 a z + z + a z
In[19]:=	{Vassiliev[2][Knot[10, 135]], Vassiliev[3][Knot[10, 135]]}
Out[19]=	{3, -1}
In[20]:=	Kh[Knot[10, 135]][q, t]
Out[20]=	4 1 1 1 3 1 3 3 3 3 - + 4 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 7 3 > 2 q t + 3 q t + 2 q t + 2 q t + 2 q t
In[21]:=	ColouredJones[Knot[10, 135], 2][q]
Out[21]=	-15 2 6 8 4 19 14 16 35 13 30 44 7 42 + q - --- + --- - --- - --- + -- - -- - -- + -- - -- - -- + -- - -- - 14 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 38 2 3 4 5 6 7 8 9 10 > -- - q - 34 q + 28 q + 4 q - 21 q + 11 q + 4 q - 7 q + q + q q

The Non Alternating Knot 10135

The Non Alternating Knot 10₁₃₅