The Alternating Knot 10₅₆

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Acknowledgement

KnotPlot

PD Presentation: X₄₂₅₁ X_10,4,11,3 X_12,6,13,5 X_18,14,19,13 X_16,7,17,8 X_6,17,7,18 X_20,16,1,15 X_14,20,15,19 X_8,12,9,11 X_2,10,3,9

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

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

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 + 8t^-2 - 14t^-1 + 17 - 14t + 8t² - 2t³

Conway Polynomial: 1 - 4z⁴ - 2z⁶

Other knots with the same Alexander/Conway Polynomial: {10₂₅, K11a140, ...}

Determinant and Signature: {65, 4}

Jones Polynomial: 1 - 2q + 5q² - 7q³ + 10q⁴ - 11q⁵ + 10q⁶ - 9q⁷ + 6q⁸ - 3q⁹ + q¹⁰

Other knots (up to mirrors) with the same Jones Polynomial: {10₂₅, ...}

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

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

Kauffman Polynomial: - a^-12z² + a^-12z⁴ - 3a^-11z³ + 3a^-11z⁵ + 2a^-10z² - 6a^-10z⁴ + 5a^-10z⁶ - 4a^-9z + 11a^-9z³ - 11a^-9z⁵ + 6a^-9z⁷ + a^-8 - 2a^-8z² + 4a^-8z⁴ - 5a^-8z⁶ + 4a^-8z⁸ - 8a^-7z + 21a^-7z³ - 21a^-7z⁵ + 7a^-7z⁷ + a^-7z⁹ + 2a^-6 - 7a^-6z² + 12a^-6z⁴ - 14a^-6z⁶ + 6a^-6z⁸ - 4a^-5z + 11a^-5z³ - 13a^-5z⁵ + 3a^-5z⁷ + a^-5z⁹ + 3a^-4z² - 3a^-4z⁴ - 3a^-4z⁶ + 2a^-4z⁸ + 4a^-3z³ - 6a^-3z⁵ + 2a^-3z⁷ - 2a^-2 + 5a^-2z² - 4a^-2z⁴ + a^-2z⁶

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {0, -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=4 is the signature of 10₅₆. 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 5 2

j = 13 5 4

j = 11 6 5

j = 9 4 5

j = 7 3 6

j = 5 2 4

j = 3 1 4

j = 1 1

j = -1 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-2 - 2q^-1 + 7q - 9q² - 4q³ + 25q⁴ - 21q⁵ - 20q⁶ + 55q⁷ - 26q⁸ - 50q⁹ + 83q¹⁰ - 19q¹¹ - 79q¹² + 94q¹³ - 4q¹⁴ - 90q¹⁵ + 83q¹⁶ + 9q¹⁷ - 76q¹⁸ + 54q¹⁹ + 13q²⁰ - 45q²¹ + 23q²² + 9q²³ - 16q²⁴ + 6q²⁵ + 2q²⁶ - 3q²⁷ + q²⁸

3 q^-6 - 2q^-5 + 2q^-3 + 4q^-2 - 8q^-1 - 5 + 11q + 17q² - 23q³ - 28q⁴ + 25q⁵ + 62q⁶ - 35q⁷ - 91q⁸ + 16q⁹ + 146q¹⁰ + 3q¹¹ - 183q¹² - 59q¹³ + 230q¹⁴ + 113q¹⁵ - 244q¹⁶ - 195q¹⁷ + 260q¹⁸ + 258q¹⁹ - 242q²⁰ - 331q²¹ + 222q²² + 385q²³ - 190q²⁴ - 422q²⁵ + 148q²⁶ + 445q²⁷ - 108q²⁸ - 438q²⁹ + 57q³⁰ + 419q³¹ - 20q³² - 365q³³ - 23q³⁴ + 305q³⁵ + 44q³⁶ - 227q³⁷ - 59q³⁸ + 159q³⁹ + 55q⁴⁰ - 99q⁴¹ - 42q⁴² + 53q⁴³ + 29q⁴⁴ - 28q⁴⁵ - 15q⁴⁶ + 14q⁴⁷ + 6q⁴⁸ - 7q⁴⁹ - q⁵⁰ + 2q⁵¹ + 2q⁵² - 3q⁵³ + q⁵⁴

4 q^-12 - 2q^-11 + 2q^-9 - q^-8 + 5q^-7 - 10q^-6 - q^-5 + 11q^-4 - 2q^-3 + 17q^-2 - 37q^-1 - 14 + 35q + 15q² + 60q³ - 100q⁴ - 80q⁵ + 45q⁶ + 69q⁷ + 216q⁸ - 153q⁹ - 241q¹⁰ - 74q¹¹ + 81q¹² + 550q¹³ - 31q¹⁴ - 386q¹⁵ - 410q¹⁶ - 170q¹⁷ + 921q¹⁸ + 375q¹⁹ - 244q²⁰ - 805q²¹ - 796q²² + 1019q²³ + 892q²⁴ + 312q²⁵ - 956q²⁶ - 1600q²⁷ + 713q²⁸ + 1224q²⁹ + 1089q³⁰ - 756q³¹ - 2275q³² + 163q³³ + 1276q³⁴ + 1807q³⁵ - 358q³⁶ - 2678q³⁷ - 404q³⁸ + 1132q³⁹ + 2309q⁴⁰ + 83q⁴¹ - 2784q⁴² - 888q⁴³ + 841q⁴⁴ + 2537q⁴⁵ + 520q⁴⁶ - 2554q⁴⁷ - 1216q⁴⁸ + 393q⁴⁹ + 2385q⁵⁰ + 897q⁵¹ - 1944q⁵² - 1264q⁵³ - 132q⁵⁴ + 1813q⁵⁵ + 1046q⁵⁶ - 1114q⁵⁷ - 954q⁵⁸ - 485q⁵⁹ + 1018q⁶⁰ + 852q⁶¹ - 416q⁶² - 461q⁶³ - 494q⁶⁴ + 377q⁶⁵ + 462q⁶⁶ - 83q⁶⁷ - 90q⁶⁸ - 287q⁶⁹ + 78q⁷⁰ + 160q⁷¹ - 17q⁷² + 33q⁷³ - 104q⁷⁴ + 12q⁷⁵ + 36q⁷⁶ - 19q⁷⁷ + 28q⁷⁸ - 26q⁷⁹ + 4q⁸⁰ + 8q⁸¹ - 10q⁸² + 8q⁸³ - 5q⁸⁴ + 2q⁸⁵ + 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, 56]]

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

In[3]:=
GaussCode[Knot[10, 56]]

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

In[4]:=
DTCode[Knot[10, 56]]

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

In[5]:=
br = BR[Knot[10, 56]]

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, 56]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 56]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 56]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 56]][t]

Out[10]=
2 8 14 2 3 17 - -- + -- - -- - 14 t + 8 t - 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 56]][z]

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

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

Out[12]=
{Knot[10, 25], Knot[10, 56], Knot[11, Alternating, 140]}

In[13]:=
{KnotDet[Knot[10, 56]], KnotSignature[Knot[10, 56]]}

Out[13]=
{65, 4}

In[14]:=
Jones[Knot[10, 56]][q]

Out[14]=
2 3 4 5 6 7 8 9 10 1 - 2 q + 5 q - 7 q + 10 q - 11 q + 10 q - 9 q + 6 q - 3 q + q

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

Out[15]=
{Knot[10, 25], Knot[10, 56]}

In[16]:=
A2Invariant[Knot[10, 56]][q]

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

In[17]:=
HOMFLYPT[Knot[10, 56]][a, z]

Out[17]=
2 2 2 2 4 4 4 4 6 6 -8 2 2 2 z 3 z 2 z 3 z z 3 z 3 z z z z a - -- + -- + ---- - ---- - ---- + ---- + -- - ---- - ---- + -- - -- - -- 6 2 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, 56]][a, z]

Out[18]=
2 2 2 2 2 2 -8 2 2 4 z 8 z 4 z z 2 z 2 z 7 z 3 z 5 z a + -- - -- - --- - --- - --- - --- + ---- - ---- - ---- + ---- + ---- - 6 2 9 7 5 12 10 8 6 4 2 a a a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 3 z 11 z 21 z 11 z 4 z z 6 z 4 z 12 z 3 z > ---- + ----- + ----- + ----- + ---- + --- - ---- + ---- + ----- - ---- - 11 9 7 5 3 12 10 8 6 4 a a a a a a a a a a 4 5 5 5 5 5 6 6 6 6 4 z 3 z 11 z 21 z 13 z 6 z 5 z 5 z 14 z 3 z > ---- + ---- - ----- - ----- - ----- - ---- + ---- - ---- - ----- - ---- + 2 11 9 7 5 3 10 8 6 4 a a a a a a a a a a 6 7 7 7 7 8 8 8 9 9 z 6 z 7 z 3 z 2 z 4 z 6 z 2 z z z > -- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + -- + -- 2 9 7 5 3 8 6 4 7 5 a a a a a a a a a a

In[19]:=
{Vassiliev[2][Knot[10, 56]], Vassiliev[3][Knot[10, 56]]}

Out[19]=
{0, -2}

In[20]:=
Kh[Knot[10, 56]][q, t]

Out[20]=
3 3 5 1 q q 5 7 7 2 9 2 9 3 4 q + 2 q + ---- + - + -- + 4 q t + 3 q t + 6 q t + 4 q t + 5 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 6 q t + 5 q t + 5 q t + 4 q t + 5 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, 56], 2][q]

Out[21]=
-2 2 2 3 4 5 6 7 8 9 q - - + 7 q - 9 q - 4 q + 25 q - 21 q - 20 q + 55 q - 26 q - 50 q + q 10 11 12 13 14 15 16 17 > 83 q - 19 q - 79 q + 94 q - 4 q - 90 q + 83 q + 9 q - 18 19 20 21 22 23 24 25 > 76 q + 54 q + 13 q - 45 q + 23 q + 9 q - 16 q + 6 q + 26 27 28 > 2 q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₅₆

10₅₅
10₅₇

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-2 - 2q^-1 + 7q - 9q² - 4q³ + 25q⁴ - 21q⁵ - 20q⁶ + 55q⁷ - 26q⁸ - 50q⁹ + 83q¹⁰ - 19q¹¹ - 79q¹² + 94q¹³ - 4q¹⁴ - 90q¹⁵ + 83q¹⁶ + 9q¹⁷ - 76q¹⁸ + 54q¹⁹ + 13q²⁰ - 45q²¹ + 23q²² + 9q²³ - 16q²⁴ + 6q²⁵ + 2q²⁶ - 3q²⁷ + q²⁸
3	q^-6 - 2q^-5 + 2q^-3 + 4q^-2 - 8q^-1 - 5 + 11q + 17q² - 23q³ - 28q⁴ + 25q⁵ + 62q⁶ - 35q⁷ - 91q⁸ + 16q⁹ + 146q¹⁰ + 3q¹¹ - 183q¹² - 59q¹³ + 230q¹⁴ + 113q¹⁵ - 244q¹⁶ - 195q¹⁷ + 260q¹⁸ + 258q¹⁹ - 242q²⁰ - 331q²¹ + 222q²² + 385q²³ - 190q²⁴ - 422q²⁵ + 148q²⁶ + 445q²⁷ - 108q²⁸ - 438q²⁹ + 57q³⁰ + 419q³¹ - 20q³² - 365q³³ - 23q³⁴ + 305q³⁵ + 44q³⁶ - 227q³⁷ - 59q³⁸ + 159q³⁹ + 55q⁴⁰ - 99q⁴¹ - 42q⁴² + 53q⁴³ + 29q⁴⁴ - 28q⁴⁵ - 15q⁴⁶ + 14q⁴⁷ + 6q⁴⁸ - 7q⁴⁹ - q⁵⁰ + 2q⁵¹ + 2q⁵² - 3q⁵³ + q⁵⁴
4	q^-12 - 2q^-11 + 2q^-9 - q^-8 + 5q^-7 - 10q^-6 - q^-5 + 11q^-4 - 2q^-3 + 17q^-2 - 37q^-1 - 14 + 35q + 15q² + 60q³ - 100q⁴ - 80q⁵ + 45q⁶ + 69q⁷ + 216q⁸ - 153q⁹ - 241q¹⁰ - 74q¹¹ + 81q¹² + 550q¹³ - 31q¹⁴ - 386q¹⁵ - 410q¹⁶ - 170q¹⁷ + 921q¹⁸ + 375q¹⁹ - 244q²⁰ - 805q²¹ - 796q²² + 1019q²³ + 892q²⁴ + 312q²⁵ - 956q²⁶ - 1600q²⁷ + 713q²⁸ + 1224q²⁹ + 1089q³⁰ - 756q³¹ - 2275q³² + 163q³³ + 1276q³⁴ + 1807q³⁵ - 358q³⁶ - 2678q³⁷ - 404q³⁸ + 1132q³⁹ + 2309q⁴⁰ + 83q⁴¹ - 2784q⁴² - 888q⁴³ + 841q⁴⁴ + 2537q⁴⁵ + 520q⁴⁶ - 2554q⁴⁷ - 1216q⁴⁸ + 393q⁴⁹ + 2385q⁵⁰ + 897q⁵¹ - 1944q⁵² - 1264q⁵³ - 132q⁵⁴ + 1813q⁵⁵ + 1046q⁵⁶ - 1114q⁵⁷ - 954q⁵⁸ - 485q⁵⁹ + 1018q⁶⁰ + 852q⁶¹ - 416q⁶² - 461q⁶³ - 494q⁶⁴ + 377q⁶⁵ + 462q⁶⁶ - 83q⁶⁷ - 90q⁶⁸ - 287q⁶⁹ + 78q⁷⁰ + 160q⁷¹ - 17q⁷² + 33q⁷³ - 104q⁷⁴ + 12q⁷⁵ + 36q⁷⁶ - 19q⁷⁷ + 28q⁷⁸ - 26q⁷⁹ + 4q⁸⁰ + 8q⁸¹ - 10q⁸² + 8q⁸³ - 5q⁸⁴ + 2q⁸⁵ + 2q⁸⁶ - 3q⁸⁷ + q⁸⁸

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 56]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[12, 6, 13, 5], X[18, 14, 19, 13], > X[16, 7, 17, 8], X[6, 17, 7, 18], X[20, 16, 1, 15], X[14, 20, 15, 19], > X[8, 12, 9, 11], X[2, 10, 3, 9]]
In[3]:=	GaussCode[Knot[10, 56]]
Out[3]=	GaussCode[1, -10, 2, -1, 3, -6, 5, -9, 10, -2, 9, -3, 4, -8, 7, -5, 6, -4, 8, > -7]
In[4]:=	DTCode[Knot[10, 56]]
Out[4]=	DTCode[4, 10, 12, 16, 2, 8, 18, 20, 6, 14]
In[5]:=	br = BR[Knot[10, 56]]
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, 56]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 56]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 56]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 56]][t]
Out[10]=	2 8 14 2 3 17 - -- + -- - -- - 14 t + 8 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 56]][z]
Out[11]=	4 6 1 - 4 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 25], Knot[10, 56], Knot[11, Alternating, 140]}
In[13]:=	{KnotDet[Knot[10, 56]], KnotSignature[Knot[10, 56]]}
Out[13]=	{65, 4}
In[14]:=	Jones[Knot[10, 56]][q]
Out[14]=	2 3 4 5 6 7 8 9 10 1 - 2 q + 5 q - 7 q + 10 q - 11 q + 10 q - 9 q + 6 q - 3 q + q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 25], Knot[10, 56]}
In[16]:=	A2Invariant[Knot[10, 56]][q]
Out[16]=	4 6 8 10 12 18 20 22 24 26 28 30 1 + q + 2 q - q + 3 q - q - 3 q + q - 2 q + q + q - q + q
In[17]:=	HOMFLYPT[Knot[10, 56]][a, z]
Out[17]=	2 2 2 2 4 4 4 4 6 6 -8 2 2 2 z 3 z 2 z 3 z z 3 z 3 z z z z a - -- + -- + ---- - ---- - ---- + ---- + -- - ---- - ---- + -- - -- - -- 6 2 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, 56]][a, z]
Out[18]=	2 2 2 2 2 2 -8 2 2 4 z 8 z 4 z z 2 z 2 z 7 z 3 z 5 z a + -- - -- - --- - --- - --- - --- + ---- - ---- - ---- + ---- + ---- - 6 2 9 7 5 12 10 8 6 4 2 a a a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 3 z 11 z 21 z 11 z 4 z z 6 z 4 z 12 z 3 z > ---- + ----- + ----- + ----- + ---- + --- - ---- + ---- + ----- - ---- - 11 9 7 5 3 12 10 8 6 4 a a a a a a a a a a 4 5 5 5 5 5 6 6 6 6 4 z 3 z 11 z 21 z 13 z 6 z 5 z 5 z 14 z 3 z > ---- + ---- - ----- - ----- - ----- - ---- + ---- - ---- - ----- - ---- + 2 11 9 7 5 3 10 8 6 4 a a a a a a a a a a 6 7 7 7 7 8 8 8 9 9 z 6 z 7 z 3 z 2 z 4 z 6 z 2 z z z > -- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + -- + -- 2 9 7 5 3 8 6 4 7 5 a a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 56]], Vassiliev[3][Knot[10, 56]]}
Out[19]=	{0, -2}
In[20]:=	Kh[Knot[10, 56]][q, t]
Out[20]=	3 3 5 1 q q 5 7 7 2 9 2 9 3 4 q + 2 q + ---- + - + -- + 4 q t + 3 q t + 6 q t + 4 q t + 5 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 6 q t + 5 q t + 5 q t + 4 q t + 5 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, 56], 2][q]
Out[21]=	-2 2 2 3 4 5 6 7 8 9 q - - + 7 q - 9 q - 4 q + 25 q - 21 q - 20 q + 55 q - 26 q - 50 q + q 10 11 12 13 14 15 16 17 > 83 q - 19 q - 79 q + 94 q - 4 q - 90 q + 83 q + 9 q - 18 19 20 21 22 23 24 25 > 76 q + 54 q + 13 q - 45 q + 23 q + 9 q - 16 q + 6 q + 26 27 28 > 2 q - 3 q + q

The Alternating Knot 1056

The Alternating Knot 10₅₆