The Alternating Knot 10₁₁₁

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Acknowledgement

KnotPlot

PD Presentation: X₆₂₇₁ X_10,4,11,3 X_14,8,15,7 X_8,14,9,13 X_2,10,3,9 X_18,12,19,11 X_16,5,17,6 X_4,17,5,18 X_20,16,1,15 X_12,20,13,19

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

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

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 1

Alexander Polynomial: - 2t^-3 + 9t^-2 - 17t^-1 + 21 - 17t + 9t² - 2t³

Conway Polynomial: 1 + z² - 3z⁴ - 2z⁶

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

Determinant and Signature: {77, 4}

Jones Polynomial: 1 - 3q + 7q² - 9q³ + 12q⁴ - 13q⁵ + 12q⁶ - 10q⁷ + 6q⁸ - 3q⁹ + q¹⁰

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

A2 (sl(3)) Invariant: 1 - q² + q⁴ + 2q⁶ - q⁸ + 4q¹⁰ - q¹² + q¹⁴ - 3q¹⁸ + q²⁰ - 3q²² + q²⁴ + q²⁶ - q²⁸ + q³⁰

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

Kauffman Polynomial: - a^-12z² + a^-12z⁴ - 3a^-11z³ + 3a^-11z⁵ + a^-10z² - 5a^-10z⁴ + 5a^-10z⁶ - 4a^-9z + 13a^-9z³ - 13a^-9z⁵ + 7a^-9z⁷ + a^-8 - 3a^-8z² + 10a^-8z⁴ - 11a^-8z⁶ + 6a^-8z⁸ - 10a^-7z + 30a^-7z³ - 28a^-7z⁵ + 7a^-7z⁷ + 2a^-7z⁹ + 3a^-6 - 10a^-6z² + 22a^-6z⁴ - 26a^-6z⁶ + 10a^-6z⁸ - 7a^-5z + 19a^-5z³ - 20a^-5z⁵ + 3a^-5z⁷ + 2a^-5z⁹ + 2a^-4 - 2a^-4z² + 3a^-4z⁴ - 9a^-4z⁶ + 4a^-4z⁸ - a^-3z + 5a^-3z³ - 8a^-3z⁵ + 3a^-3z⁷ - a^-2 + 3a^-2z² - 3a^-2z⁴ + a^-2z⁶

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {1, 0}

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 10₁₁₁. 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 r = 6 r = 7 r = 8

j = 21 1

j = 19 2

j = 17 4 1

j = 15 6 2

j = 13 6 4

j = 11 7 6

j = 9 5 6

j = 7 4 7

j = 5 3 5

j = 3 1 5

j = 1 2

j = -1 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-2 - 3q^-1 + 1 + 11q - 17q² - 7q³ + 44q⁴ - 32q⁵ - 39q⁶ + 86q⁷ - 27q⁸ - 86q⁹ + 113q¹⁰ - 4q¹¹ - 122q¹² + 114q¹³ + 22q¹⁴ - 129q¹⁵ + 88q¹⁶ + 37q¹⁷ - 98q¹⁸ + 47q¹⁹ + 31q²⁰ - 49q²¹ + 16q²² + 13q²³ - 15q²⁴ + 5q²⁵ + 2q²⁶ - 3q²⁷ + q²⁸

3 q^-6 - 3q^-5 + q^-4 + 5q^-3 + 3q^-2 - 17q^-1 - 10 + 34q + 34q² - 53q³ - 78q⁴ + 53q⁵ + 155q⁶ - 38q⁷ - 222q⁸ - 38q⁹ + 301q¹⁰ + 131q¹¹ - 328q¹² - 270q¹³ + 338q¹⁴ + 389q¹⁵ - 280q¹⁶ - 526q¹⁷ + 220q¹⁸ + 615q¹⁹ - 119q²⁰ - 696q²¹ + 24q²² + 737q²³ + 78q²⁴ - 752q²⁵ - 175q²⁶ + 733q²⁷ + 256q²⁸ - 668q²⁹ - 324q³⁰ + 571q³¹ + 356q³² - 437q³³ - 358q³⁴ + 307q³⁵ + 306q³⁶ - 173q³⁷ - 244q³⁸ + 86q³⁹ + 161q⁴⁰ - 30q⁴¹ - 92q⁴² + 7q⁴³ + 44q⁴⁴ - 2q⁴⁵ - 18q⁴⁶ + 3q⁴⁷ + 6q⁴⁸ - 4q⁴⁹ + q⁵¹ + 2q⁵² - 3q⁵³ + q⁵⁴

4 q^-12 - 3q^-11 + q^-10 + 5q^-9 - 3q^-8 + 3q^-7 - 20q^-6 + 2q^-5 + 36q^-4 + 7q^-3 + 14q^-2 - 107q^-1 - 55 + 105q + 120q² + 172q³ - 261q⁴ - 343q⁵ - 30q⁶ + 266q⁷ + 753q⁸ - 81q⁹ - 711q¹⁰ - 708q¹¹ - 152q¹² + 1466q¹³ + 834q¹⁴ - 361q¹⁵ - 1508q¹⁶ - 1546q¹⁷ + 1354q¹⁸ + 1924q¹⁹ + 1115q²⁰ - 1407q²¹ - 3224q²² + 30q²³ + 2152q²⁴ + 3007q²⁵ - 124q²⁶ - 4177q²⁷ - 1798q²⁸ + 1312q²⁹ + 4390q³⁰ + 1649q³¹ - 4202q³² - 3336q³³ + 11q³⁴ + 5058q³⁵ + 3230q³⁶ - 3697q³⁷ - 4377q³⁸ - 1267q³⁹ + 5178q⁴⁰ + 4445q⁴¹ - 2834q⁴² - 4930q⁴³ - 2484q⁴⁴ + 4652q⁴⁵ + 5204q⁴⁶ - 1478q⁴⁷ - 4688q⁴⁸ - 3517q⁴⁹ + 3213q⁵⁰ + 5079q⁵¹ + 164q⁵² - 3345q⁵³ - 3775q⁵⁴ + 1236q⁵⁵ + 3742q⁵⁶ + 1226q⁵⁷ - 1404q⁵⁸ - 2854q⁵⁹ - 203q⁶⁰ + 1830q⁶¹ + 1157q⁶² - 34q⁶³ - 1405q⁶⁴ - 514q⁶⁵ + 495q⁶⁶ + 509q⁶⁷ + 308q⁶⁸ - 426q⁶⁹ - 249q⁷⁰ + 50q⁷¹ + 87q⁷² + 171q⁷³ - 89q⁷⁴ - 53q⁷⁵ + 2q⁷⁶ - 14q⁷⁷ + 53q⁷⁸ - 19q⁷⁹ - 5q⁸⁰ + 4q⁸¹ - 11q⁸² + 11q⁸³ - 4q⁸⁴ + q⁸⁵ + 2q⁸⁶ - 3q⁸⁷ + q⁸⁸

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, 111]]

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

In[3]:=
GaussCode[Knot[10, 111]]

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

In[4]:=
DTCode[Knot[10, 111]]

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

In[5]:=
br = BR[Knot[10, 111]]

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

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

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 111]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 111]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 111]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 111]][t]

Out[10]=
2 9 17 2 3 21 - -- + -- - -- - 17 t + 9 t - 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 111]][z]

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

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

Out[12]=
{Knot[10, 111]}

In[13]:=
{KnotDet[Knot[10, 111]], KnotSignature[Knot[10, 111]]}

Out[13]=
{77, 4}

In[14]:=
Jones[Knot[10, 111]][q]

Out[14]=
2 3 4 5 6 7 8 9 10 1 - 3 q + 7 q - 9 q + 12 q - 13 q + 12 q - 10 q + 6 q - 3 q + q

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

Out[15]=
{Knot[10, 111]}

In[16]:=
A2Invariant[Knot[10, 111]][q]

Out[16]=
2 4 6 8 10 12 14 18 20 22 24 26 1 - q + q + 2 q - q + 4 q - q + q - 3 q + q - 3 q + q + q - 28 30 > q + q

In[17]:=
HOMFLYPT[Knot[10, 111]][a, z]

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

In[18]:=
Kauffman[Knot[10, 111]][a, z]

Out[18]=
2 2 2 2 2 -8 3 2 -2 4 z 10 z 7 z z z z 3 z 10 z 2 z a + -- + -- - a - --- - ---- - --- - -- - --- + --- - ---- - ----- - ---- + 6 4 9 7 5 3 12 10 8 6 4 a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 3 z 3 z 13 z 30 z 19 z 5 z z 5 z 10 z 22 z > ---- - ---- + ----- + ----- + ----- + ---- + --- - ---- + ----- + ----- + 2 11 9 7 5 3 12 10 8 6 a a a a a a a a a a 4 4 5 5 5 5 5 6 6 6 3 z 3 z 3 z 13 z 28 z 20 z 8 z 5 z 11 z 26 z > ---- - ---- + ---- - ----- - ----- - ----- - ---- + ---- - ----- - ----- - 4 2 11 9 7 5 3 10 8 6 a a a a a a a a a a 6 6 7 7 7 7 8 8 8 9 9 9 z z 7 z 7 z 3 z 3 z 6 z 10 z 4 z 2 z 2 z > ---- + -- + ---- + ---- + ---- + ---- + ---- + ----- + ---- + ---- + ---- 4 2 9 7 5 3 8 6 4 7 5 a a a a a a a a a a a

In[19]:=
{Vassiliev[2][Knot[10, 111]], Vassiliev[3][Knot[10, 111]]}

Out[19]=
{1, 0}

In[20]:=
Kh[Knot[10, 111]][q, t]

Out[20]=
3 3 5 1 2 q q 5 7 7 2 9 2 9 3 5 q + 3 q + ---- + --- + -- + 5 q t + 4 q t + 7 q t + 5 q t + 6 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 7 q t + 6 q t + 6 q t + 4 q t + 6 q t + 2 q t + 17 6 17 7 19 7 21 8 > 4 q t + q t + 2 q t + q t

In[21]:=
ColouredJones[Knot[10, 111], 2][q]

Out[21]=
-2 3 2 3 4 5 6 7 8 1 + q - - + 11 q - 17 q - 7 q + 44 q - 32 q - 39 q + 86 q - 27 q - q 9 10 11 12 13 14 15 16 > 86 q + 113 q - 4 q - 122 q + 114 q + 22 q - 129 q + 88 q + 17 18 19 20 21 22 23 24 > 37 q - 98 q + 47 q + 31 q - 49 q + 16 q + 13 q - 15 q + 25 26 27 28 > 5 q + 2 q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₁₁₁

10₁₁₀
10₁₁₂

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-2 - 3q^-1 + 1 + 11q - 17q² - 7q³ + 44q⁴ - 32q⁵ - 39q⁶ + 86q⁷ - 27q⁸ - 86q⁹ + 113q¹⁰ - 4q¹¹ - 122q¹² + 114q¹³ + 22q¹⁴ - 129q¹⁵ + 88q¹⁶ + 37q¹⁷ - 98q¹⁸ + 47q¹⁹ + 31q²⁰ - 49q²¹ + 16q²² + 13q²³ - 15q²⁴ + 5q²⁵ + 2q²⁶ - 3q²⁷ + q²⁸
3	q^-6 - 3q^-5 + q^-4 + 5q^-3 + 3q^-2 - 17q^-1 - 10 + 34q + 34q² - 53q³ - 78q⁴ + 53q⁵ + 155q⁶ - 38q⁷ - 222q⁸ - 38q⁹ + 301q¹⁰ + 131q¹¹ - 328q¹² - 270q¹³ + 338q¹⁴ + 389q¹⁵ - 280q¹⁶ - 526q¹⁷ + 220q¹⁸ + 615q¹⁹ - 119q²⁰ - 696q²¹ + 24q²² + 737q²³ + 78q²⁴ - 752q²⁵ - 175q²⁶ + 733q²⁷ + 256q²⁸ - 668q²⁹ - 324q³⁰ + 571q³¹ + 356q³² - 437q³³ - 358q³⁴ + 307q³⁵ + 306q³⁶ - 173q³⁷ - 244q³⁸ + 86q³⁹ + 161q⁴⁰ - 30q⁴¹ - 92q⁴² + 7q⁴³ + 44q⁴⁴ - 2q⁴⁵ - 18q⁴⁶ + 3q⁴⁷ + 6q⁴⁸ - 4q⁴⁹ + q⁵¹ + 2q⁵² - 3q⁵³ + q⁵⁴
4	q^-12 - 3q^-11 + q^-10 + 5q^-9 - 3q^-8 + 3q^-7 - 20q^-6 + 2q^-5 + 36q^-4 + 7q^-3 + 14q^-2 - 107q^-1 - 55 + 105q + 120q² + 172q³ - 261q⁴ - 343q⁵ - 30q⁶ + 266q⁷ + 753q⁸ - 81q⁹ - 711q¹⁰ - 708q¹¹ - 152q¹² + 1466q¹³ + 834q¹⁴ - 361q¹⁵ - 1508q¹⁶ - 1546q¹⁷ + 1354q¹⁸ + 1924q¹⁹ + 1115q²⁰ - 1407q²¹ - 3224q²² + 30q²³ + 2152q²⁴ + 3007q²⁵ - 124q²⁶ - 4177q²⁷ - 1798q²⁸ + 1312q²⁹ + 4390q³⁰ + 1649q³¹ - 4202q³² - 3336q³³ + 11q³⁴ + 5058q³⁵ + 3230q³⁶ - 3697q³⁷ - 4377q³⁸ - 1267q³⁹ + 5178q⁴⁰ + 4445q⁴¹ - 2834q⁴² - 4930q⁴³ - 2484q⁴⁴ + 4652q⁴⁵ + 5204q⁴⁶ - 1478q⁴⁷ - 4688q⁴⁸ - 3517q⁴⁹ + 3213q⁵⁰ + 5079q⁵¹ + 164q⁵² - 3345q⁵³ - 3775q⁵⁴ + 1236q⁵⁵ + 3742q⁵⁶ + 1226q⁵⁷ - 1404q⁵⁸ - 2854q⁵⁹ - 203q⁶⁰ + 1830q⁶¹ + 1157q⁶² - 34q⁶³ - 1405q⁶⁴ - 514q⁶⁵ + 495q⁶⁶ + 509q⁶⁷ + 308q⁶⁸ - 426q⁶⁹ - 249q⁷⁰ + 50q⁷¹ + 87q⁷² + 171q⁷³ - 89q⁷⁴ - 53q⁷⁵ + 2q⁷⁶ - 14q⁷⁷ + 53q⁷⁸ - 19q⁷⁹ - 5q⁸⁰ + 4q⁸¹ - 11q⁸² + 11q⁸³ - 4q⁸⁴ + q⁸⁵ + 2q⁸⁶ - 3q⁸⁷ + q⁸⁸

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 111]]
Out[2]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[14, 8, 15, 7], X[8, 14, 9, 13], > X[2, 10, 3, 9], X[18, 12, 19, 11], X[16, 5, 17, 6], X[4, 17, 5, 18], > X[20, 16, 1, 15], X[12, 20, 13, 19]]
In[3]:=	GaussCode[Knot[10, 111]]
Out[3]=	GaussCode[1, -5, 2, -8, 7, -1, 3, -4, 5, -2, 6, -10, 4, -3, 9, -7, 8, -6, 10, > -9]
In[4]:=	DTCode[Knot[10, 111]]
Out[4]=	DTCode[6, 10, 16, 14, 2, 18, 8, 20, 4, 12]
In[5]:=	br = BR[Knot[10, 111]]
Out[5]=	BR[4, {1, 1, 2, 2, -3, 2, 2, -1, 2, -3, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 111]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 111]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 111]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 111]][t]
Out[10]=	2 9 17 2 3 21 - -- + -- - -- - 17 t + 9 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 111]][z]
Out[11]=	2 4 6 1 + z - 3 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 111]}
In[13]:=	{KnotDet[Knot[10, 111]], KnotSignature[Knot[10, 111]]}
Out[13]=	{77, 4}
In[14]:=	Jones[Knot[10, 111]][q]
Out[14]=	2 3 4 5 6 7 8 9 10 1 - 3 q + 7 q - 9 q + 12 q - 13 q + 12 q - 10 q + 6 q - 3 q + q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 111]}
In[16]:=	A2Invariant[Knot[10, 111]][q]
Out[16]=	2 4 6 8 10 12 14 18 20 22 24 26 1 - q + q + 2 q - q + 4 q - q + q - 3 q + q - 3 q + q + q - 28 30 > q + q
In[17]:=	HOMFLYPT[Knot[10, 111]][a, z]
Out[17]=	2 2 2 2 4 4 4 4 6 6 -8 3 2 -2 2 z 4 z z 2 z z 3 z 2 z z z z a - -- + -- + a + ---- - ---- + -- + ---- + -- - ---- - ---- + -- - -- - -- 6 4 8 6 4 2 8 6 4 2 6 4 a a a a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 111]][a, z]
Out[18]=	2 2 2 2 2 -8 3 2 -2 4 z 10 z 7 z z z z 3 z 10 z 2 z a + -- + -- - a - --- - ---- - --- - -- - --- + --- - ---- - ----- - ---- + 6 4 9 7 5 3 12 10 8 6 4 a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 3 z 3 z 13 z 30 z 19 z 5 z z 5 z 10 z 22 z > ---- - ---- + ----- + ----- + ----- + ---- + --- - ---- + ----- + ----- + 2 11 9 7 5 3 12 10 8 6 a a a a a a a a a a 4 4 5 5 5 5 5 6 6 6 3 z 3 z 3 z 13 z 28 z 20 z 8 z 5 z 11 z 26 z > ---- - ---- + ---- - ----- - ----- - ----- - ---- + ---- - ----- - ----- - 4 2 11 9 7 5 3 10 8 6 a a a a a a a a a a 6 6 7 7 7 7 8 8 8 9 9 9 z z 7 z 7 z 3 z 3 z 6 z 10 z 4 z 2 z 2 z > ---- + -- + ---- + ---- + ---- + ---- + ---- + ----- + ---- + ---- + ---- 4 2 9 7 5 3 8 6 4 7 5 a a a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 111]], Vassiliev[3][Knot[10, 111]]}
Out[19]=	{1, 0}
In[20]:=	Kh[Knot[10, 111]][q, t]
Out[20]=	3 3 5 1 2 q q 5 7 7 2 9 2 9 3 5 q + 3 q + ---- + --- + -- + 5 q t + 4 q t + 7 q t + 5 q t + 6 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 7 q t + 6 q t + 6 q t + 4 q t + 6 q t + 2 q t + 17 6 17 7 19 7 21 8 > 4 q t + q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 111], 2][q]
Out[21]=	-2 3 2 3 4 5 6 7 8 1 + q - - + 11 q - 17 q - 7 q + 44 q - 32 q - 39 q + 86 q - 27 q - q 9 10 11 12 13 14 15 16 > 86 q + 113 q - 4 q - 122 q + 114 q + 22 q - 129 q + 88 q + 17 18 19 20 21 22 23 24 > 37 q - 98 q + 47 q + 31 q - 49 q + 16 q + 13 q - 15 q + 25 26 27 28 > 5 q + 2 q - 3 q + q

The Alternating Knot 10111

The Alternating Knot 10₁₁₁