The Non Alternating Knot 10₁₆₀

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Acknowledgement

KnotPlot

PD Presentation: X₄₂₅₁ X_12,4,13,3 X_7,14,8,15 X_9,19,10,18 X_19,7,20,6 X_5,17,6,16 X_17,11,18,10 X_13,8,14,9 X_15,1,16,20 X_2,12,3,11

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

DT (Dowker-Thistlethwaite) Code: 4 12 -16 -14 -18 2 -8 -20 -10 -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 3 3 / NotAvailable 2

Alexander Polynomial: - t^-3 + 4t^-2 - 4t^-1 + 3 - 4t + 4t² - t³

Conway Polynomial: 1 + 3z² - 2z⁴ - z⁶

Other knots with the same Alexander/Conway Polynomial: {K11n118, ...}

Determinant and Signature: {21, 4}

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

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

A2 (sl(3)) Invariant: 1 + 2q¹⁰ + 2q¹⁴ - q²² - q²⁶

HOMFLY-PT Polynomial: - a^-8 + a^-6 + 3a^-6z² + a^-6z⁴ - 3a^-4z² - 4a^-4z⁴ - a^-4z⁶ + a^-2 + 3a^-2z² + a^-2z⁴

Kauffman Polynomial: 2a^-9z - a^-8 + a^-8z² + a^-8z⁴ + 3a^-7z³ - 3a^-7z⁵ + a^-7z⁷ - a^-6 + 2a^-6z⁴ - 3a^-6z⁶ + a^-6z⁸ - 3a^-5z + 10a^-5z³ - 11a^-5z⁵ + 3a^-5z⁷ + 3a^-4z² - 3a^-4z⁴ - 2a^-4z⁶ + a^-4z⁸ - a^-3z + 7a^-3z³ - 8a^-3z⁵ + 2a^-3z⁷ - a^-2 + 4a^-2z² - 4a^-2z⁴ + a^-2z⁶

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {3, 6}

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

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

j = 15 2

j = 13 1

j = 11 2 2

j = 9 2 1

j = 7 1 2

j = 5 2 2

j = 3 1 2

j = 1 1

j = -1 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-2 - 2q^-1 - 1 + 6q - 3q² - 6q³ + 9q⁴ - 9q⁶ + 7q⁷ + 4q⁸ - 9q⁹ + 3q¹⁰ + 7q¹¹ - 7q¹² - 2q¹³ + 8q¹⁴ - 4q¹⁵ - 4q¹⁶ + 4q¹⁷ - 2q¹⁹ + q²¹

3 q^-6 - 2q^-5 - q^-4 + 2q^-3 + 5q^-2 - 2q^-1 - 9 - q + 11q² + 6q³ - 9q⁴ - 11q⁵ + 6q⁶ + 11q⁷ + 3q⁸ - 10q⁹ - 7q¹⁰ + 2q¹¹ + 13q¹² + 4q¹³ - 13q¹⁴ - 14q¹⁵ + 16q¹⁶ + 19q¹⁷ - 12q¹⁸ - 29q¹⁹ + 15q²⁰ + 32q²¹ - 12q²² - 40q²³ + 14q²⁴ + 40q²⁵ - 9q²⁶ - 44q²⁷ + 5q²⁸ + 40q²⁹ + 2q³⁰ - 34q³¹ - 7q³² + 25q³³ + 8q³⁴ - 11q³⁵ - 11q³⁶ + 7q³⁷ + 4q³⁸ - 2q⁴⁰

4 q^-12 - 2q^-11 - q^-10 + 2q^-9 + q^-8 + 6q^-7 - 6q^-6 - 7q^-5 - 2q^-4 - q^-3 + 22q^-2 + q^-1 - 7 - 12q - 21q² + 21q³ + 9q⁴ + 15q⁵ + 5q⁶ - 33q⁷ - 3q⁸ - 16q⁹ + 16q¹⁰ + 41q¹¹ + q¹² - 2q¹³ - 55q¹⁴ - 31q¹⁵ + 45q¹⁶ + 47q¹⁷ + 43q¹⁸ - 57q¹⁹ - 90q²⁰ + 4q²¹ + 63q²² + 99q²³ - 21q²⁴ - 125q²⁵ - 47q²⁶ + 52q²⁷ + 136q²⁸ + 22q²⁹ - 139q³⁰ - 86q³¹ + 40q³² + 157q³³ + 53q³⁴ - 150q³⁵ - 114q³⁶ + 33q³⁷ + 173q³⁸ + 79q³⁹ - 154q⁴⁰ - 138q⁴¹ + 17q⁴² + 173q⁴³ + 107q⁴⁴ - 124q⁴⁵ - 142q⁴⁶ - 24q⁴⁷ + 131q⁴⁸ + 120q⁴⁹ - 58q⁵⁰ - 101q⁵¹ - 53q⁵² + 54q⁵³ + 82q⁵⁴ - 2q⁵⁵ - 35q⁵⁶ - 38q⁵⁷ + 2q⁵⁸ + 27q⁵⁹ + 7q⁶⁰ - 8q⁶² - 4q⁶³ + 2q⁶⁴ + q⁶⁶

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, 160]]

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

In[3]:=
GaussCode[Knot[10, 160]]

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

In[4]:=
DTCode[Knot[10, 160]]

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

In[5]:=
br = BR[Knot[10, 160]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 160]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 160]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 160]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 160]][t]

Out[10]=
-3 4 4 2 3 3 - t + -- - - - 4 t + 4 t - t 2 t t

In[11]:=
Conway[Knot[10, 160]][z]

Out[11]=
2 4 6 1 + 3 z - 2 z - z

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

Out[12]=
{Knot[10, 160], Knot[11, NonAlternating, 118]}

In[13]:=
{KnotDet[Knot[10, 160]], KnotSignature[Knot[10, 160]]}

Out[13]=
{21, 4}

In[14]:=
Jones[Knot[10, 160]][q]

Out[14]=
2 3 4 5 6 7 1 - 2 q + 3 q - 3 q + 4 q - 3 q + 3 q - 2 q

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

Out[15]=
{Knot[10, 160]}

In[16]:=
A2Invariant[Knot[10, 160]][q]

Out[16]=
10 14 22 26 1 + 2 q + 2 q - q - q

In[17]:=
HOMFLYPT[Knot[10, 160]][a, z]

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

In[18]:=
Kauffman[Knot[10, 160]][a, z]

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

In[19]:=
{Vassiliev[2][Knot[10, 160]], Vassiliev[3][Knot[10, 160]]}

Out[19]=
{3, 6}

In[20]:=
Kh[Knot[10, 160]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 160], 2][q]

Out[21]=
-2 2 2 3 4 6 7 8 9 10 -1 + q - - + 6 q - 3 q - 6 q + 9 q - 9 q + 7 q + 4 q - 9 q + 3 q + q 11 12 13 14 15 16 17 19 21 > 7 q - 7 q - 2 q + 8 q - 4 q - 4 q + 4 q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₆₀

10₁₅₉
10₁₆₁

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-2 - 2q^-1 - 1 + 6q - 3q² - 6q³ + 9q⁴ - 9q⁶ + 7q⁷ + 4q⁸ - 9q⁹ + 3q¹⁰ + 7q¹¹ - 7q¹² - 2q¹³ + 8q¹⁴ - 4q¹⁵ - 4q¹⁶ + 4q¹⁷ - 2q¹⁹ + q²¹
3	q^-6 - 2q^-5 - q^-4 + 2q^-3 + 5q^-2 - 2q^-1 - 9 - q + 11q² + 6q³ - 9q⁴ - 11q⁵ + 6q⁶ + 11q⁷ + 3q⁸ - 10q⁹ - 7q¹⁰ + 2q¹¹ + 13q¹² + 4q¹³ - 13q¹⁴ - 14q¹⁵ + 16q¹⁶ + 19q¹⁷ - 12q¹⁸ - 29q¹⁹ + 15q²⁰ + 32q²¹ - 12q²² - 40q²³ + 14q²⁴ + 40q²⁵ - 9q²⁶ - 44q²⁷ + 5q²⁸ + 40q²⁹ + 2q³⁰ - 34q³¹ - 7q³² + 25q³³ + 8q³⁴ - 11q³⁵ - 11q³⁶ + 7q³⁷ + 4q³⁸ - 2q⁴⁰
4	q^-12 - 2q^-11 - q^-10 + 2q^-9 + q^-8 + 6q^-7 - 6q^-6 - 7q^-5 - 2q^-4 - q^-3 + 22q^-2 + q^-1 - 7 - 12q - 21q² + 21q³ + 9q⁴ + 15q⁵ + 5q⁶ - 33q⁷ - 3q⁸ - 16q⁹ + 16q¹⁰ + 41q¹¹ + q¹² - 2q¹³ - 55q¹⁴ - 31q¹⁵ + 45q¹⁶ + 47q¹⁷ + 43q¹⁸ - 57q¹⁹ - 90q²⁰ + 4q²¹ + 63q²² + 99q²³ - 21q²⁴ - 125q²⁵ - 47q²⁶ + 52q²⁷ + 136q²⁸ + 22q²⁹ - 139q³⁰ - 86q³¹ + 40q³² + 157q³³ + 53q³⁴ - 150q³⁵ - 114q³⁶ + 33q³⁷ + 173q³⁸ + 79q³⁹ - 154q⁴⁰ - 138q⁴¹ + 17q⁴² + 173q⁴³ + 107q⁴⁴ - 124q⁴⁵ - 142q⁴⁶ - 24q⁴⁷ + 131q⁴⁸ + 120q⁴⁹ - 58q⁵⁰ - 101q⁵¹ - 53q⁵² + 54q⁵³ + 82q⁵⁴ - 2q⁵⁵ - 35q⁵⁶ - 38q⁵⁷ + 2q⁵⁸ + 27q⁵⁹ + 7q⁶⁰ - 8q⁶² - 4q⁶³ + 2q⁶⁴ + q⁶⁶

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 160]]
Out[2]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[7, 14, 8, 15], X[9, 19, 10, 18], > X[19, 7, 20, 6], X[5, 17, 6, 16], X[17, 11, 18, 10], X[13, 8, 14, 9], > X[15, 1, 16, 20], X[2, 12, 3, 11]]
In[3]:=	GaussCode[Knot[10, 160]]
Out[3]=	GaussCode[1, -10, 2, -1, -6, 5, -3, 8, -4, 7, 10, -2, -8, 3, -9, 6, -7, 4, -5, > 9]
In[4]:=	DTCode[Knot[10, 160]]
Out[4]=	DTCode[4, 12, -16, -14, -18, 2, -8, -20, -10, -6]
In[5]:=	br = BR[Knot[10, 160]]
Out[5]=	BR[4, {1, 1, 1, 2, 1, 1, -3, 2, -1, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 160]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 160]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 160]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 2}
In[10]:=	alex = Alexander[Knot[10, 160]][t]
Out[10]=	-3 4 4 2 3 3 - t + -- - - - 4 t + 4 t - t 2 t t
In[11]:=	Conway[Knot[10, 160]][z]
Out[11]=	2 4 6 1 + 3 z - 2 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 160], Knot[11, NonAlternating, 118]}
In[13]:=	{KnotDet[Knot[10, 160]], KnotSignature[Knot[10, 160]]}
Out[13]=	{21, 4}
In[14]:=	Jones[Knot[10, 160]][q]
Out[14]=	2 3 4 5 6 7 1 - 2 q + 3 q - 3 q + 4 q - 3 q + 3 q - 2 q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 160]}
In[16]:=	A2Invariant[Knot[10, 160]][q]
Out[16]=	10 14 22 26 1 + 2 q + 2 q - q - q
In[17]:=	HOMFLYPT[Knot[10, 160]][a, z]
Out[17]=	2 2 2 4 4 4 6 -8 -6 -2 3 z 3 z 3 z z 4 z z z -a + a + a + ---- - ---- + ---- + -- - ---- + -- - -- 6 4 2 6 4 2 4 a a a a a a a
In[18]:=	Kauffman[Knot[10, 160]][a, z]
Out[18]=	2 2 2 3 3 3 -8 -6 -2 2 z 3 z z z 3 z 4 z 3 z 10 z 7 z -a - a - a + --- - --- - -- + -- + ---- + ---- + ---- + ----- + ---- + 9 5 3 8 4 2 7 5 3 a a a a a a a a a 4 4 4 4 5 5 5 6 6 6 7 z 2 z 3 z 4 z 3 z 11 z 8 z 3 z 2 z z z > -- + ---- - ---- - ---- - ---- - ----- - ---- - ---- - ---- + -- + -- + 8 6 4 2 7 5 3 6 4 2 7 a a a a a a a a a a a 7 7 8 8 3 z 2 z z z > ---- + ---- + -- + -- 5 3 6 4 a a a a
In[19]:=	{Vassiliev[2][Knot[10, 160]], Vassiliev[3][Knot[10, 160]]}
Out[19]=	{3, 6}
In[20]:=	Kh[Knot[10, 160]][q, t]
Out[20]=	3 3 5 1 q q 5 7 7 2 9 2 9 3 2 q + 2 q + ---- + - + -- + 2 q t + q t + 2 q t + 2 q t + q t + 2 t t q t 11 3 11 4 13 4 15 5 > 2 q t + 2 q t + q t + 2 q t
In[21]:=	ColouredJones[Knot[10, 160], 2][q]
Out[21]=	-2 2 2 3 4 6 7 8 9 10 -1 + q - - + 6 q - 3 q - 6 q + 9 q - 9 q + 7 q + 4 q - 9 q + 3 q + q 11 12 13 14 15 16 17 19 21 > 7 q - 7 q - 2 q + 8 q - 4 q - 4 q + 4 q - 2 q + q

The Non Alternating Knot 10160

The Non Alternating Knot 10₁₆₀