The Alternating Knot 10₅₃

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X₃₈₄₉ X_5,14,6,15 X_15,20,16,1 X_9,16,10,17 X_19,10,20,11 X_11,18,12,19 X_17,12,18,13 X_13,6,14,7 X₇₂₈₃

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

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

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--3 2 3 / NotAvailable 1

Alexander Polynomial: 6t^-2 - 18t^-1 + 25 - 18t + 6t²

Conway Polynomial: 1 + 6z² + 6z⁴

Other knots with the same Alexander/Conway Polynomial: {K11a95, ...}

Determinant and Signature: {73, -4}

Jones Polynomial: q^-12 - 3q^-11 + 5q^-10 - 9q^-9 + 11q^-8 - 12q^-7 + 12q^-6 - 9q^-5 + 7q^-4 - 3q^-3 + q^-2

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

A2 (sl(3)) Invariant: q^-38 + q^-36 - 2q^-34 - q^-30 - 4q^-28 + q^-26 - q^-24 + q^-22 + 2q^-20 + 4q^-16 - q^-14 + q^-12 + 2q^-10 - 2q^-8 + q^-6

HOMFLY-PT Polynomial: a⁴z² + a⁴z⁴ + 3a⁶ + 6a⁶z² + 3a⁶z⁴ + 2a⁸z² + 2a⁸z⁴ - 3a¹⁰ - 3a¹⁰z² + a¹²

Kauffman Polynomial: - a⁴z² + a⁴z⁴ - 2a⁵z³ + 3a⁵z⁵ - 3a⁶ + 8a⁶z² - 9a⁶z⁴ + 6a⁶z⁶ + a⁷z + a⁷z³ - 6a⁷z⁵ + 6a⁷z⁷ + 4a⁸z² - 7a⁸z⁴ + 4a⁸z⁸ - 7a⁹z + 21a⁹z³ - 26a⁹z⁵ + 10a⁹z⁷ + a⁹z⁹ + 3a¹⁰ - 5a¹⁰z² + 6a¹⁰z⁴ - 13a¹⁰z⁶ + 7a¹⁰z⁸ - 11a¹¹z + 28a¹¹z³ - 27a¹¹z⁵ + 7a¹¹z⁷ + a¹¹z⁹ + a¹² + 2a¹²z² - 6a¹²z⁶ + 3a¹²z⁸ - 3a¹³z + 10a¹³z³ - 10a¹³z⁵ + 3a¹³z⁷ + 2a¹⁴z² - 3a¹⁴z⁴ + a¹⁴z⁶

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {6, -13}

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

t^rq^j r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0

j = -3 1

j = -5 3 1

j = -7 4

j = -9 5 3

j = -11 7 4

j = -13 5 5

j = -15 6 7

j = -17 3 5

j = -19 2 6

j = -21 1 3

j = -23 2

j = -25 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-34 - 3q^-33 + 10q^-31 - 13q^-30 - 7q^-29 + 35q^-28 - 25q^-27 - 32q^-26 + 73q^-25 - 28q^-24 - 71q^-23 + 105q^-22 - 16q^-21 - 106q^-20 + 115q^-19 + 4q^-18 - 117q^-17 + 98q^-16 + 18q^-15 - 95q^-14 + 62q^-13 + 20q^-12 - 54q^-11 + 27q^-10 + 11q^-9 - 19q^-8 + 7q^-7 + 3q^-6 - 3q^-5 + q^-4

3 q^-66 - 3q^-65 + 5q^-63 + 5q^-62 - 13q^-61 - 13q^-60 + 22q^-59 + 31q^-58 - 30q^-57 - 63q^-56 + 32q^-55 + 107q^-54 - 16q^-53 - 167q^-52 - 14q^-51 + 218q^-50 + 82q^-49 - 278q^-48 - 151q^-47 + 299q^-46 + 257q^-45 - 320q^-44 - 346q^-43 + 301q^-42 + 447q^-41 - 275q^-40 - 525q^-39 + 225q^-38 + 592q^-37 - 170q^-36 - 633q^-35 + 112q^-34 + 637q^-33 - 40q^-32 - 627q^-31 - 7q^-30 + 560q^-29 + 68q^-28 - 495q^-27 - 85q^-26 + 383q^-25 + 113q^-24 - 298q^-23 - 94q^-22 + 194q^-21 + 89q^-20 - 134q^-19 - 53q^-18 + 71q^-17 + 41q^-16 - 44q^-15 - 20q^-14 + 22q^-13 + 11q^-12 - 11q^-11 - 3q^-10 + 3q^-9 + 3q^-8 - 3q^-7 + q^-6

4 q^-108 - 3q^-107 + 5q^-105 + 5q^-103 - 20q^-102 - 6q^-101 + 23q^-100 + 11q^-99 + 34q^-98 - 71q^-97 - 56q^-96 + 41q^-95 + 58q^-94 + 160q^-93 - 135q^-92 - 208q^-91 - 44q^-90 + 98q^-89 + 492q^-88 - 60q^-87 - 415q^-86 - 382q^-85 - 86q^-84 + 975q^-83 + 349q^-82 - 387q^-81 - 908q^-80 - 746q^-79 + 1265q^-78 + 1019q^-77 + 172q^-76 - 1244q^-75 - 1796q^-74 + 1024q^-73 + 1567q^-72 + 1193q^-71 - 1077q^-70 - 2834q^-69 + 297q^-68 + 1702q^-67 + 2303q^-66 - 470q^-65 - 3538q^-64 - 613q^-63 + 1458q^-62 + 3208q^-61 + 304q^-60 - 3839q^-59 - 1452q^-58 + 993q^-57 + 3761q^-56 + 1064q^-55 - 3719q^-54 - 2086q^-53 + 372q^-52 + 3841q^-51 + 1701q^-50 - 3120q^-49 - 2342q^-48 - 337q^-47 + 3311q^-46 + 2030q^-45 - 2104q^-44 - 2058q^-43 - 901q^-42 + 2274q^-41 + 1854q^-40 - 1038q^-39 - 1322q^-38 - 1034q^-37 + 1163q^-36 + 1252q^-35 - 339q^-34 - 554q^-33 - 749q^-32 + 428q^-31 + 603q^-30 - 88q^-29 - 111q^-28 - 367q^-27 + 127q^-26 + 212q^-25 - 45q^-24 + 11q^-23 - 131q^-22 + 44q^-21 + 61q^-20 - 29q^-19 + 15q^-18 - 37q^-17 + 14q^-16 + 16q^-15 - 11q^-14 + 5q^-13 - 7q^-12 + 3q^-11 + 3q^-10 - 3q^-9 + q^-8

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, 53]]

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

In[3]:=
GaussCode[Knot[10, 53]]

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

In[4]:=
DTCode[Knot[10, 53]]

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

In[5]:=
br = BR[Knot[10, 53]]

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

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

Out[6]=
{5, 12}

In[7]:=
BraidIndex[Knot[10, 53]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[10, 53]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 53]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 53]][t]

Out[10]=
6 18 2 25 + -- - -- - 18 t + 6 t 2 t t

In[11]:=
Conway[Knot[10, 53]][z]

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

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

Out[12]=
{Knot[10, 53], Knot[11, Alternating, 95]}

In[13]:=
{KnotDet[Knot[10, 53]], KnotSignature[Knot[10, 53]]}

Out[13]=
{73, -4}

In[14]:=
Jones[Knot[10, 53]][q]

Out[14]=
-12 3 5 9 11 12 12 9 7 3 -2 q - --- + --- - -- + -- - -- + -- - -- + -- - -- + q 11 10 9 8 7 6 5 4 3 q 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, 53]}

In[16]:=
A2Invariant[Knot[10, 53]][q]

Out[16]=
-38 -36 2 -30 4 -26 -24 -22 2 4 -14 -12 q + q - --- - q - --- + q - q + q + --- + --- - q + q + 34 28 20 16 q q q q 2 2 -6 > --- - -- + q 10 8 q q

In[17]:=
HOMFLYPT[Knot[10, 53]][a, z]

Out[17]=
6 10 12 4 2 6 2 8 2 10 2 4 4 6 4 3 a - 3 a + a + a z + 6 a z + 2 a z - 3 a z + a z + 3 a z + 8 4 > 2 a z

In[18]:=
Kauffman[Knot[10, 53]][a, z]

Out[18]=
6 10 12 7 9 11 13 4 2 6 2 -3 a + 3 a + a + a z - 7 a z - 11 a z - 3 a z - a z + 8 a z + 8 2 10 2 12 2 14 2 5 3 7 3 9 3 > 4 a z - 5 a z + 2 a z + 2 a z - 2 a z + a z + 21 a z + 11 3 13 3 4 4 6 4 8 4 10 4 14 4 > 28 a z + 10 a z + a z - 9 a z - 7 a z + 6 a z - 3 a z + 5 5 7 5 9 5 11 5 13 5 6 6 > 3 a z - 6 a z - 26 a z - 27 a z - 10 a z + 6 a z - 10 6 12 6 14 6 7 7 9 7 11 7 13 7 > 13 a z - 6 a z + a z + 6 a z + 10 a z + 7 a z + 3 a z + 8 8 10 8 12 8 9 9 11 9 > 4 a z + 7 a z + 3 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 53]], Vassiliev[3][Knot[10, 53]]}

Out[19]=
{6, -13}

In[20]:=
Kh[Knot[10, 53]][q, t]

Out[20]=
-5 -3 1 2 1 3 2 6 3 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 25 10 23 9 21 9 21 8 19 8 19 7 17 7 q t q t q t q t q t q t q t 5 6 7 5 5 7 4 5 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 17 6 15 6 15 5 13 5 13 4 11 4 11 3 9 3 q t q t q t q t q t q t q t q t 3 4 3 > ----- + ----- + ---- 9 2 7 2 5 q t q t q t

In[21]:=
ColouredJones[Knot[10, 53], 2][q]

Out[21]=
-34 3 10 13 7 35 25 32 73 28 71 105 16 q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - --- - 33 31 30 29 28 27 26 25 24 23 22 21 q q q q q q q q q q q q 106 115 4 117 98 18 95 62 20 54 27 11 19 > --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + -- - -- + 20 19 18 17 16 15 14 13 12 11 10 9 8 q q q q q q q q q q q q q 7 3 3 -4 > -- + -- - -- + q 7 6 5 q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₅₃

10₅₂
10₅₄

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-34 - 3q^-33 + 10q^-31 - 13q^-30 - 7q^-29 + 35q^-28 - 25q^-27 - 32q^-26 + 73q^-25 - 28q^-24 - 71q^-23 + 105q^-22 - 16q^-21 - 106q^-20 + 115q^-19 + 4q^-18 - 117q^-17 + 98q^-16 + 18q^-15 - 95q^-14 + 62q^-13 + 20q^-12 - 54q^-11 + 27q^-10 + 11q^-9 - 19q^-8 + 7q^-7 + 3q^-6 - 3q^-5 + q^-4
3	q^-66 - 3q^-65 + 5q^-63 + 5q^-62 - 13q^-61 - 13q^-60 + 22q^-59 + 31q^-58 - 30q^-57 - 63q^-56 + 32q^-55 + 107q^-54 - 16q^-53 - 167q^-52 - 14q^-51 + 218q^-50 + 82q^-49 - 278q^-48 - 151q^-47 + 299q^-46 + 257q^-45 - 320q^-44 - 346q^-43 + 301q^-42 + 447q^-41 - 275q^-40 - 525q^-39 + 225q^-38 + 592q^-37 - 170q^-36 - 633q^-35 + 112q^-34 + 637q^-33 - 40q^-32 - 627q^-31 - 7q^-30 + 560q^-29 + 68q^-28 - 495q^-27 - 85q^-26 + 383q^-25 + 113q^-24 - 298q^-23 - 94q^-22 + 194q^-21 + 89q^-20 - 134q^-19 - 53q^-18 + 71q^-17 + 41q^-16 - 44q^-15 - 20q^-14 + 22q^-13 + 11q^-12 - 11q^-11 - 3q^-10 + 3q^-9 + 3q^-8 - 3q^-7 + q^-6
4	q^-108 - 3q^-107 + 5q^-105 + 5q^-103 - 20q^-102 - 6q^-101 + 23q^-100 + 11q^-99 + 34q^-98 - 71q^-97 - 56q^-96 + 41q^-95 + 58q^-94 + 160q^-93 - 135q^-92 - 208q^-91 - 44q^-90 + 98q^-89 + 492q^-88 - 60q^-87 - 415q^-86 - 382q^-85 - 86q^-84 + 975q^-83 + 349q^-82 - 387q^-81 - 908q^-80 - 746q^-79 + 1265q^-78 + 1019q^-77 + 172q^-76 - 1244q^-75 - 1796q^-74 + 1024q^-73 + 1567q^-72 + 1193q^-71 - 1077q^-70 - 2834q^-69 + 297q^-68 + 1702q^-67 + 2303q^-66 - 470q^-65 - 3538q^-64 - 613q^-63 + 1458q^-62 + 3208q^-61 + 304q^-60 - 3839q^-59 - 1452q^-58 + 993q^-57 + 3761q^-56 + 1064q^-55 - 3719q^-54 - 2086q^-53 + 372q^-52 + 3841q^-51 + 1701q^-50 - 3120q^-49 - 2342q^-48 - 337q^-47 + 3311q^-46 + 2030q^-45 - 2104q^-44 - 2058q^-43 - 901q^-42 + 2274q^-41 + 1854q^-40 - 1038q^-39 - 1322q^-38 - 1034q^-37 + 1163q^-36 + 1252q^-35 - 339q^-34 - 554q^-33 - 749q^-32 + 428q^-31 + 603q^-30 - 88q^-29 - 111q^-28 - 367q^-27 + 127q^-26 + 212q^-25 - 45q^-24 + 11q^-23 - 131q^-22 + 44q^-21 + 61q^-20 - 29q^-19 + 15q^-18 - 37q^-17 + 14q^-16 + 16q^-15 - 11q^-14 + 5q^-13 - 7q^-12 + 3q^-11 + 3q^-10 - 3q^-9 + q^-8

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 53]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 14, 6, 15], X[15, 20, 16, 1], > X[9, 16, 10, 17], X[19, 10, 20, 11], X[11, 18, 12, 19], X[17, 12, 18, 13], > X[13, 6, 14, 7], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 53]]
Out[3]=	GaussCode[-1, 10, -2, 1, -3, 9, -10, 2, -5, 6, -7, 8, -9, 3, -4, 5, -8, 7, -6, > 4]
In[4]:=	DTCode[Knot[10, 53]]
Out[4]=	DTCode[4, 8, 14, 2, 16, 18, 6, 20, 12, 10]
In[5]:=	br = BR[Knot[10, 53]]
Out[5]=	BR[5, {-1, -1, -2, 1, -2, 3, -2, -4, -3, -3, -3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 53]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 53]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 53]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, {2, 3}, 2, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 53]][t]
Out[10]=	6 18 2 25 + -- - -- - 18 t + 6 t 2 t t
In[11]:=	Conway[Knot[10, 53]][z]
Out[11]=	2 4 1 + 6 z + 6 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 53], Knot[11, Alternating, 95]}
In[13]:=	{KnotDet[Knot[10, 53]], KnotSignature[Knot[10, 53]]}
Out[13]=	{73, -4}
In[14]:=	Jones[Knot[10, 53]][q]
Out[14]=	-12 3 5 9 11 12 12 9 7 3 -2 q - --- + --- - -- + -- - -- + -- - -- + -- - -- + q 11 10 9 8 7 6 5 4 3 q 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, 53]}
In[16]:=	A2Invariant[Knot[10, 53]][q]
Out[16]=	-38 -36 2 -30 4 -26 -24 -22 2 4 -14 -12 q + q - --- - q - --- + q - q + q + --- + --- - q + q + 34 28 20 16 q q q q 2 2 -6 > --- - -- + q 10 8 q q
In[17]:=	HOMFLYPT[Knot[10, 53]][a, z]
Out[17]=	6 10 12 4 2 6 2 8 2 10 2 4 4 6 4 3 a - 3 a + a + a z + 6 a z + 2 a z - 3 a z + a z + 3 a z + 8 4 > 2 a z
In[18]:=	Kauffman[Knot[10, 53]][a, z]
Out[18]=	6 10 12 7 9 11 13 4 2 6 2 -3 a + 3 a + a + a z - 7 a z - 11 a z - 3 a z - a z + 8 a z + 8 2 10 2 12 2 14 2 5 3 7 3 9 3 > 4 a z - 5 a z + 2 a z + 2 a z - 2 a z + a z + 21 a z + 11 3 13 3 4 4 6 4 8 4 10 4 14 4 > 28 a z + 10 a z + a z - 9 a z - 7 a z + 6 a z - 3 a z + 5 5 7 5 9 5 11 5 13 5 6 6 > 3 a z - 6 a z - 26 a z - 27 a z - 10 a z + 6 a z - 10 6 12 6 14 6 7 7 9 7 11 7 13 7 > 13 a z - 6 a z + a z + 6 a z + 10 a z + 7 a z + 3 a z + 8 8 10 8 12 8 9 9 11 9 > 4 a z + 7 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 53]], Vassiliev[3][Knot[10, 53]]}
Out[19]=	{6, -13}
In[20]:=	Kh[Knot[10, 53]][q, t]
Out[20]=	-5 -3 1 2 1 3 2 6 3 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 25 10 23 9 21 9 21 8 19 8 19 7 17 7 q t q t q t q t q t q t q t 5 6 7 5 5 7 4 5 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 17 6 15 6 15 5 13 5 13 4 11 4 11 3 9 3 q t q t q t q t q t q t q t q t 3 4 3 > ----- + ----- + ---- 9 2 7 2 5 q t q t q t
In[21]:=	ColouredJones[Knot[10, 53], 2][q]
Out[21]=	-34 3 10 13 7 35 25 32 73 28 71 105 16 q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- + --- - --- - 33 31 30 29 28 27 26 25 24 23 22 21 q q q q q q q q q q q q 106 115 4 117 98 18 95 62 20 54 27 11 19 > --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + -- - -- + 20 19 18 17 16 15 14 13 12 11 10 9 8 q q q q q q q q q q q q q 7 3 3 -4 > -- + -- - -- + q 7 6 5 q q q

The Alternating Knot 1053

The Alternating Knot 10₅₃