The Alternating Knot 10₃₆

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_5,10,6,11 X₃₉₄₈ X_9,3,10,2 X_7,16,8,17 X_11,20,12,1 X_13,18,14,19 X_17,14,18,15 X_19,12,20,13 X_15,6,16,7

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

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

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 + 13t^-1 - 19 + 13t - 3t²

Conway Polynomial: 1 + z² - 3z⁴

Other knots with the same Alexander/Conway Polynomial: {K11a230, K11n29, ...}

Determinant and Signature: {51, -2}

Jones Polynomial: q^-9 - 3q^-8 + 4q^-7 - 6q^-6 + 8q^-5 - 8q^-4 + 8q^-3 - 6q^-2 + 4q^-1 - 2 + q

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

A2 (sl(3)) Invariant: q^-28 - q^-26 - q^-24 + q^-22 - 2q^-20 + q^-16 + 2q^-12 + q^-8 - 2q^-4 + 2q^-2 + q⁴

HOMFLY-PT Polynomial: 1 + z² - a² - a²z² - a²z⁴ + 2a⁴ + a⁴z² - a⁴z⁴ - a⁶ - a⁶z² - a⁶z⁴ + a⁸z²

Kauffman Polynomial: 1 - 2z² + z⁴ + az - 3az³ + 2az⁵ + a² - 3a²z² + 2a²z⁶ + a³z - 2a³z³ + 2a³z⁷ + 2a⁴ - 6a⁴z² + 6a⁴z⁴ - 3a⁴z⁶ + 2a⁴z⁸ - 3a⁵z + 8a⁵z³ - 6a⁵z⁵ + a⁵z⁷ + a⁵z⁹ + a⁶ - 8a⁶z² + 18a⁶z⁴ - 16a⁶z⁶ + 5a⁶z⁸ - 4a⁷z + 16a⁷z³ - 15a⁷z⁵ + 2a⁷z⁷ + a⁷z⁹ - 2a⁸z² + 8a⁸z⁴ - 10a⁸z⁶ + 3a⁸z⁸ - a⁹z + 9a⁹z³ - 11a⁹z⁵ + 3a⁹z⁷ + a¹⁰z² - 3a¹⁰z⁴ + a¹⁰z⁶

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

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=-2 is the signature of 10₃₆. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2

j = 3 1

j = 1 1

j = -1 3 1

j = -3 4 2

j = -5 4 2

j = -7 4 4

j = -9 4 4

j = -11 2 4

j = -13 2 4

j = -15 1 2

j = -17 2

j = -19 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-26 - 3q^-25 + 9q^-23 - 10q^-22 - 6q^-21 + 23q^-20 - 13q^-19 - 20q^-18 + 36q^-17 - 9q^-16 - 36q^-15 + 44q^-14 - q^-13 - 47q^-12 + 44q^-11 + 7q^-10 - 47q^-9 + 35q^-8 + 9q^-7 - 34q^-6 + 22q^-5 + 5q^-4 - 18q^-3 + 12q^-2 + q^-1 - 7 + 5q - 2q³ + q⁴

3 q^-51 - 3q^-50 + 5q^-48 + 4q^-47 - 10q^-46 - 12q^-45 + 16q^-44 + 22q^-43 - 16q^-42 - 38q^-41 + 12q^-40 + 52q^-39 + 2q^-38 - 67q^-37 - 15q^-36 + 67q^-35 + 40q^-34 - 68q^-33 - 56q^-32 + 55q^-31 + 75q^-30 - 40q^-29 - 88q^-28 + 22q^-27 + 95q^-26 + q^-25 - 105q^-24 - 16q^-23 + 101q^-22 + 39q^-21 - 103q^-20 - 48q^-19 + 88q^-18 + 63q^-17 - 79q^-16 - 58q^-15 + 54q^-14 + 57q^-13 - 39q^-12 - 42q^-11 + 21q^-10 + 28q^-9 - 12q^-8 - 10q^-7 + 4q^-6 + 3q^-5 - 6q^-4 + 7q^-3 + 3q^-2 - 6q^-1 - 6 + 8q + 3q² - 3q³ - 5q⁴ + 4q⁵ + q⁶ - 2q⁸ + q⁹

4 q^-84 - 3q^-83 + 5q^-81 + 4q^-79 - 17q^-78 - 5q^-77 + 17q^-76 + 8q^-75 + 28q^-74 - 47q^-73 - 36q^-72 + 20q^-71 + 22q^-70 + 95q^-69 - 60q^-68 - 83q^-67 - 19q^-66 - 6q^-65 + 188q^-64 - 18q^-63 - 86q^-62 - 70q^-61 - 106q^-60 + 222q^-59 + 34q^-58 - 47q^-56 - 227q^-55 + 154q^-54 + 5q^-53 + 121q^-52 + 89q^-51 - 275q^-50 + 35q^-49 - 135q^-48 + 193q^-47 + 284q^-46 - 226q^-45 - 63q^-44 - 329q^-43 + 192q^-42 + 466q^-41 - 122q^-40 - 122q^-39 - 509q^-38 + 153q^-37 + 599q^-36 - 7q^-35 - 150q^-34 - 644q^-33 + 88q^-32 + 671q^-31 + 114q^-30 - 135q^-29 - 717q^-28 - 14q^-27 + 644q^-26 + 220q^-25 - 42q^-24 - 681q^-23 - 141q^-22 + 489q^-21 + 260q^-20 + 97q^-19 - 516q^-18 - 213q^-17 + 264q^-16 + 194q^-15 + 190q^-14 - 290q^-13 - 187q^-12 + 83q^-11 + 80q^-10 + 185q^-9 - 115q^-8 - 107q^-7 + 2q^-6 - q^-5 + 123q^-4 - 30q^-3 - 41q^-2 - 12q^-1 - 26 + 61q - 4q² - 8q³ - 6q⁴ - 21q⁵ + 23q⁶ + q⁸ - q⁹ - 9q¹⁰ + 6q¹¹ + q¹³ - 2q¹⁵ + q¹⁶

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, 36]]

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

In[3]:=
GaussCode[Knot[10, 36]]

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

In[4]:=
DTCode[Knot[10, 36]]

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

In[5]:=
br = BR[Knot[10, 36]]

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

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

Out[6]=
{5, 12}

In[7]:=
BraidIndex[Knot[10, 36]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[10, 36]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 36]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 36]][t]

Out[10]=
3 13 2 -19 - -- + -- + 13 t - 3 t 2 t t

In[11]:=
Conway[Knot[10, 36]][z]

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

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

Out[12]=
{Knot[10, 36], Knot[11, Alternating, 230], Knot[11, NonAlternating, 29]}

In[13]:=
{KnotDet[Knot[10, 36]], KnotSignature[Knot[10, 36]]}

Out[13]=
{51, -2}

In[14]:=
Jones[Knot[10, 36]][q]

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

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

Out[15]=
{Knot[10, 36]}

In[16]:=
A2Invariant[Knot[10, 36]][q]

Out[16]=
-28 -26 -24 -22 2 -16 2 -8 2 2 4 q - q - q + q - --- + q + --- + q - -- + -- + q 20 12 4 2 q q q q

In[17]:=
HOMFLYPT[Knot[10, 36]][a, z]

Out[17]=
2 4 6 2 2 2 4 2 6 2 8 2 2 4 4 4 6 4 1 - a + 2 a - a + z - a z + a z - a z + a z - a z - a z - a z

In[18]:=
Kauffman[Knot[10, 36]][a, z]

Out[18]=
2 4 6 3 5 7 9 2 2 2 1 + a + 2 a + a + a z + a z - 3 a z - 4 a z - a z - 2 z - 3 a z - 4 2 6 2 8 2 10 2 3 3 3 5 3 > 6 a z - 8 a z - 2 a z + a z - 3 a z - 2 a z + 8 a z + 7 3 9 3 4 4 4 6 4 8 4 10 4 > 16 a z + 9 a z + z + 6 a z + 18 a z + 8 a z - 3 a z + 5 5 5 7 5 9 5 2 6 4 6 6 6 > 2 a z - 6 a z - 15 a z - 11 a z + 2 a z - 3 a z - 16 a z - 8 6 10 6 3 7 5 7 7 7 9 7 4 8 > 10 a z + a z + 2 a z + a z + 2 a z + 3 a z + 2 a z + 6 8 8 8 5 9 7 9 > 5 a z + 3 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 36]], Vassiliev[3][Knot[10, 36]]}

Out[19]=
{1, -2}

In[20]:=
Kh[Knot[10, 36]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 36], 2][q]

Out[21]=
-26 3 9 10 6 23 13 20 36 9 36 44 -7 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 25 23 22 21 20 19 18 17 16 15 14 q q q q q q q q q q q -13 47 44 7 47 35 9 34 22 5 18 12 1 > q - --- + --- + --- - -- + -- + -- - -- + -- + -- - -- + -- + - + 5 q - 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 3 4 > 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₃₆

10₃₅
10₃₇

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-26 - 3q^-25 + 9q^-23 - 10q^-22 - 6q^-21 + 23q^-20 - 13q^-19 - 20q^-18 + 36q^-17 - 9q^-16 - 36q^-15 + 44q^-14 - q^-13 - 47q^-12 + 44q^-11 + 7q^-10 - 47q^-9 + 35q^-8 + 9q^-7 - 34q^-6 + 22q^-5 + 5q^-4 - 18q^-3 + 12q^-2 + q^-1 - 7 + 5q - 2q³ + q⁴
3	q^-51 - 3q^-50 + 5q^-48 + 4q^-47 - 10q^-46 - 12q^-45 + 16q^-44 + 22q^-43 - 16q^-42 - 38q^-41 + 12q^-40 + 52q^-39 + 2q^-38 - 67q^-37 - 15q^-36 + 67q^-35 + 40q^-34 - 68q^-33 - 56q^-32 + 55q^-31 + 75q^-30 - 40q^-29 - 88q^-28 + 22q^-27 + 95q^-26 + q^-25 - 105q^-24 - 16q^-23 + 101q^-22 + 39q^-21 - 103q^-20 - 48q^-19 + 88q^-18 + 63q^-17 - 79q^-16 - 58q^-15 + 54q^-14 + 57q^-13 - 39q^-12 - 42q^-11 + 21q^-10 + 28q^-9 - 12q^-8 - 10q^-7 + 4q^-6 + 3q^-5 - 6q^-4 + 7q^-3 + 3q^-2 - 6q^-1 - 6 + 8q + 3q² - 3q³ - 5q⁴ + 4q⁵ + q⁶ - 2q⁸ + q⁹
4	q^-84 - 3q^-83 + 5q^-81 + 4q^-79 - 17q^-78 - 5q^-77 + 17q^-76 + 8q^-75 + 28q^-74 - 47q^-73 - 36q^-72 + 20q^-71 + 22q^-70 + 95q^-69 - 60q^-68 - 83q^-67 - 19q^-66 - 6q^-65 + 188q^-64 - 18q^-63 - 86q^-62 - 70q^-61 - 106q^-60 + 222q^-59 + 34q^-58 - 47q^-56 - 227q^-55 + 154q^-54 + 5q^-53 + 121q^-52 + 89q^-51 - 275q^-50 + 35q^-49 - 135q^-48 + 193q^-47 + 284q^-46 - 226q^-45 - 63q^-44 - 329q^-43 + 192q^-42 + 466q^-41 - 122q^-40 - 122q^-39 - 509q^-38 + 153q^-37 + 599q^-36 - 7q^-35 - 150q^-34 - 644q^-33 + 88q^-32 + 671q^-31 + 114q^-30 - 135q^-29 - 717q^-28 - 14q^-27 + 644q^-26 + 220q^-25 - 42q^-24 - 681q^-23 - 141q^-22 + 489q^-21 + 260q^-20 + 97q^-19 - 516q^-18 - 213q^-17 + 264q^-16 + 194q^-15 + 190q^-14 - 290q^-13 - 187q^-12 + 83q^-11 + 80q^-10 + 185q^-9 - 115q^-8 - 107q^-7 + 2q^-6 - q^-5 + 123q^-4 - 30q^-3 - 41q^-2 - 12q^-1 - 26 + 61q - 4q² - 8q³ - 6q⁴ - 21q⁵ + 23q⁶ + q⁸ - q⁹ - 9q¹⁰ + 6q¹¹ + q¹³ - 2q¹⁵ + q¹⁶

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 36]]
Out[2]=	PD[X[1, 4, 2, 5], X[5, 10, 6, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], > X[7, 16, 8, 17], X[11, 20, 12, 1], X[13, 18, 14, 19], X[17, 14, 18, 15], > X[19, 12, 20, 13], X[15, 6, 16, 7]]
In[3]:=	GaussCode[Knot[10, 36]]
Out[3]=	GaussCode[-1, 4, -3, 1, -2, 10, -5, 3, -4, 2, -6, 9, -7, 8, -10, 5, -8, 7, -9, > 6]
In[4]:=	DTCode[Knot[10, 36]]
Out[4]=	DTCode[4, 8, 10, 16, 2, 20, 18, 6, 14, 12]
In[5]:=	br = BR[Knot[10, 36]]
Out[5]=	BR[5, {-1, -1, -1, -2, 1, -2, -3, 2, -3, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 36]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 36]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 36]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 36]][t]
Out[10]=	3 13 2 -19 - -- + -- + 13 t - 3 t 2 t t
In[11]:=	Conway[Knot[10, 36]][z]
Out[11]=	2 4 1 + z - 3 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 36], Knot[11, Alternating, 230], Knot[11, NonAlternating, 29]}
In[13]:=	{KnotDet[Knot[10, 36]], KnotSignature[Knot[10, 36]]}
Out[13]=	{51, -2}
In[14]:=	Jones[Knot[10, 36]][q]
Out[14]=	-9 3 4 6 8 8 8 6 4 -2 + q - -- + -- - -- + -- - -- + -- - -- + - + q 8 7 6 5 4 3 2 q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 36]}
In[16]:=	A2Invariant[Knot[10, 36]][q]
Out[16]=	-28 -26 -24 -22 2 -16 2 -8 2 2 4 q - q - q + q - --- + q + --- + q - -- + -- + q 20 12 4 2 q q q q
In[17]:=	HOMFLYPT[Knot[10, 36]][a, z]
Out[17]=	2 4 6 2 2 2 4 2 6 2 8 2 2 4 4 4 6 4 1 - a + 2 a - a + z - a z + a z - a z + a z - a z - a z - a z
In[18]:=	Kauffman[Knot[10, 36]][a, z]
Out[18]=	2 4 6 3 5 7 9 2 2 2 1 + a + 2 a + a + a z + a z - 3 a z - 4 a z - a z - 2 z - 3 a z - 4 2 6 2 8 2 10 2 3 3 3 5 3 > 6 a z - 8 a z - 2 a z + a z - 3 a z - 2 a z + 8 a z + 7 3 9 3 4 4 4 6 4 8 4 10 4 > 16 a z + 9 a z + z + 6 a z + 18 a z + 8 a z - 3 a z + 5 5 5 7 5 9 5 2 6 4 6 6 6 > 2 a z - 6 a z - 15 a z - 11 a z + 2 a z - 3 a z - 16 a z - 8 6 10 6 3 7 5 7 7 7 9 7 4 8 > 10 a z + a z + 2 a z + a z + 2 a z + 3 a z + 2 a z + 6 8 8 8 5 9 7 9 > 5 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 36]], Vassiliev[3][Knot[10, 36]]}
Out[19]=	{1, -2}
In[20]:=	Kh[Knot[10, 36]][q, t]
Out[20]=	2 3 1 2 1 2 2 4 2 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 4 4 4 4 4 4 2 4 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + - + q t + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 3 2 > q t
In[21]:=	ColouredJones[Knot[10, 36], 2][q]
Out[21]=	-26 3 9 10 6 23 13 20 36 9 36 44 -7 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - 25 23 22 21 20 19 18 17 16 15 14 q q q q q q q q q q q -13 47 44 7 47 35 9 34 22 5 18 12 1 > q - --- + --- + --- - -- + -- + -- - -- + -- + -- - -- + -- + - + 5 q - 12 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 3 4 > 2 q + q

The Alternating Knot 1036

The Alternating Knot 10₃₆