The Alternating Knot 10₆₃

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Acknowledgement

KnotPlot

PD Presentation: X₁₄₂₅ X_3,10,4,11 X_5,16,6,17 X_17,20,18,1 X_11,18,12,19 X_19,12,20,13 X_7,14,8,15 X_13,8,14,9 X_15,6,16,7 X_9,2,10,3

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

DT (Dowker-Thistlethwaite) Code: 4 10 16 14 2 18 8 6 20 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 3 / NotAvailable 2

Alexander Polynomial: 5t^-2 - 14t^-1 + 19 - 14t + 5t²

Conway Polynomial: 1 + 6z² + 5z⁴

Other knots with the same Alexander/Conway Polynomial: {9₃₈, ...}

Determinant and Signature: {57, -4}

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

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

A2 (sl(3)) Invariant: q^-38 + q^-36 - 2q^-34 - q^-32 - 2q^-30 - 3q^-28 + 2q^-26 + q^-24 + 2q^-22 + q^-20 - q^-18 + 2q^-16 - q^-14 + q^-12 + 2q^-10 - q^-8 + q^-6

HOMFLY-PT Polynomial: a⁴ + 2a⁴z² + a⁴z⁴ + 3a⁶z² + 2a⁶z⁴ + 3a⁸ + 4a⁸z² + 2a⁸z⁴ - 4a¹⁰ - 3a¹⁰z² + a¹²

Kauffman Polynomial: a⁴ - 2a⁴z² + a⁴z⁴ - 2a⁵z³ + 2a⁵z⁵ + a⁶z² - 3a⁶z⁴ + 3a⁶z⁶ - 2a⁷z⁵ + 3a⁷z⁷ + 3a⁸ - 10a⁸z² + 11a⁸z⁴ - 6a⁸z⁶ + 3a⁸z⁸ - 8a⁹z + 20a⁹z³ - 16a⁹z⁵ + 4a⁹z⁷ + a⁹z⁹ + 4a¹⁰ - 16a¹⁰z² + 24a¹⁰z⁴ - 19a¹⁰z⁶ + 6a¹⁰z⁸ - 10a¹¹z + 28a¹¹z³ - 23a¹¹z⁵ + 4a¹¹z⁷ + a¹¹z⁹ + a¹² - 2a¹²z² + 6a¹²z⁴ - 9a¹²z⁶ + 3a¹²z⁸ - 2a¹³z + 10a¹³z³ - 11a¹³z⁵ + 3a¹³z⁷ + a¹⁴z² - 3a¹⁴z⁴ + a¹⁴z⁶

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {6, -14}

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 10₆₃. 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 2 1

j = -7 3

j = -9 4 2

j = -11 5 3

j = -13 4 4

j = -15 5 5

j = -17 2 4

j = -19 2 5

j = -21 1 2

j = -23 2

j = -25 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-34 - 3q^-33 + 9q^-31 - 11q^-30 - 5q^-29 + 26q^-28 - 17q^-27 - 22q^-26 + 45q^-25 - 15q^-24 - 44q^-23 + 59q^-22 - 5q^-21 - 61q^-20 + 61q^-19 + 7q^-18 - 64q^-17 + 49q^-16 + 12q^-15 - 49q^-14 + 30q^-13 + 10q^-12 - 26q^-11 + 14q^-10 + 4q^-9 - 9q^-8 + 5q^-7 + q^-6 - 2q^-5 + q^-4

3 q^-66 - 3q^-65 + 5q^-63 + 4q^-62 - 11q^-61 - 11q^-60 + 19q^-59 + 22q^-58 - 23q^-57 - 43q^-56 + 24q^-55 + 64q^-54 - 10q^-53 - 94q^-52 - 5q^-51 + 106q^-50 + 45q^-49 - 122q^-48 - 73q^-47 + 111q^-46 + 116q^-45 - 101q^-44 - 146q^-43 + 76q^-42 + 173q^-41 - 47q^-40 - 196q^-39 + 20q^-38 + 207q^-37 + 12q^-36 - 215q^-35 - 34q^-34 + 203q^-33 + 62q^-32 - 191q^-31 - 71q^-30 + 155q^-29 + 82q^-28 - 125q^-27 - 71q^-26 + 81q^-25 + 64q^-24 - 56q^-23 - 38q^-22 + 25q^-21 + 28q^-20 - 21q^-19 - 4q^-18 + 6q^-17 + 4q^-16 - 10q^-15 + 4q^-14 + 5q^-13 - 6q^-11 + 3q^-10 + q^-9 + q^-8 - 2q^-7 + q^-6

4 q^-108 - 3q^-107 + 5q^-105 + 4q^-103 - 18q^-102 - 4q^-101 + 20q^-100 + 8q^-99 + 25q^-98 - 57q^-97 - 37q^-96 + 36q^-95 + 37q^-94 + 100q^-93 - 98q^-92 - 119q^-91 - 2q^-90 + 48q^-89 + 258q^-88 - 61q^-87 - 194q^-86 - 131q^-85 - 61q^-84 + 411q^-83 + 89q^-82 - 118q^-81 - 243q^-80 - 325q^-79 + 393q^-78 + 224q^-77 + 148q^-76 - 161q^-75 - 603q^-74 + 162q^-73 + 175q^-72 + 460q^-71 + 135q^-70 - 731q^-69 - 145q^-68 - 61q^-67 + 667q^-66 + 505q^-65 - 695q^-64 - 401q^-63 - 353q^-62 + 759q^-61 + 813q^-60 - 579q^-59 - 572q^-58 - 604q^-57 + 761q^-56 + 1025q^-55 - 407q^-54 - 656q^-53 - 805q^-52 + 649q^-51 + 1114q^-50 - 167q^-49 - 586q^-48 - 918q^-47 + 383q^-46 + 1005q^-45 + 93q^-44 - 330q^-43 - 854q^-42 + 58q^-41 + 683q^-40 + 225q^-39 - 13q^-38 - 590q^-37 - 140q^-36 + 302q^-35 + 169q^-34 + 164q^-33 - 272q^-32 - 140q^-31 + 58q^-30 + 41q^-29 + 156q^-28 - 74q^-27 - 58q^-26 - 10q^-25 - 24q^-24 + 76q^-23 - 11q^-22 - 5q^-21 - 5q^-20 - 26q^-19 + 24q^-18 - 2q^-17 + 4q^-16 + q^-15 - 10q^-14 + 6q^-13 - q^-12 + q^-11 + q^-10 - 2q^-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, 63]]

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

In[3]:=
GaussCode[Knot[10, 63]]

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

In[4]:=
DTCode[Knot[10, 63]]

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

In[5]:=
br = BR[Knot[10, 63]]

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

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

Out[6]=
{5, 12}

In[7]:=
BraidIndex[Knot[10, 63]]

Out[7]=
5

In[8]:=
Show[DrawMorseLink[Knot[10, 63]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 63]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

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

In[10]:=
alex = Alexander[Knot[10, 63]][t]

Out[10]=
5 14 2 19 + -- - -- - 14 t + 5 t 2 t t

In[11]:=
Conway[Knot[10, 63]][z]

Out[11]=
2 4 1 + 6 z + 5 z

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

Out[12]=
{Knot[9, 38], Knot[10, 63]}

In[13]:=
{KnotDet[Knot[10, 63]], KnotSignature[Knot[10, 63]]}

Out[13]=
{57, -4}

In[14]:=
Jones[Knot[10, 63]][q]

Out[14]=
-12 3 4 7 9 9 9 7 5 2 -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, 63]}

In[16]:=
A2Invariant[Knot[10, 63]][q]

Out[16]=
-38 -36 2 -32 2 3 2 -24 2 -20 -18 2 q + q - --- - q - --- - --- + --- + q + --- + q - q + --- - 34 30 28 26 22 16 q q q q q q -14 -12 2 -8 -6 > q + q + --- - q + q 10 q

In[17]:=
HOMFLYPT[Knot[10, 63]][a, z]

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

In[18]:=
Kauffman[Knot[10, 63]][a, z]

Out[18]=
4 8 10 12 9 11 13 4 2 6 2 a + 3 a + 4 a + a - 8 a z - 10 a z - 2 a z - 2 a z + a z - 8 2 10 2 12 2 14 2 5 3 9 3 11 3 > 10 a z - 16 a z - 2 a z + a z - 2 a z + 20 a z + 28 a z + 13 3 4 4 6 4 8 4 10 4 12 4 14 4 > 10 a z + a z - 3 a z + 11 a z + 24 a z + 6 a z - 3 a z + 5 5 7 5 9 5 11 5 13 5 6 6 8 6 > 2 a z - 2 a z - 16 a z - 23 a z - 11 a z + 3 a z - 6 a z - 10 6 12 6 14 6 7 7 9 7 11 7 13 7 > 19 a z - 9 a z + a z + 3 a z + 4 a z + 4 a z + 3 a z + 8 8 10 8 12 8 9 9 11 9 > 3 a z + 6 a z + 3 a z + a z + a z

In[19]:=
{Vassiliev[2][Knot[10, 63]], Vassiliev[3][Knot[10, 63]]}

Out[19]=
{6, -14}

In[20]:=
Kh[Knot[10, 63]][q, t]

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

In[21]:=
ColouredJones[Knot[10, 63], 2][q]

Out[21]=
-34 3 9 11 5 26 17 22 45 15 44 59 5 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 61 61 7 64 49 12 49 30 10 26 14 4 9 > --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + -- - -- + 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 5 -6 2 -4 > -- + q - -- + q 7 5 q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₆₃

10₆₂
10₆₄

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-34 - 3q^-33 + 9q^-31 - 11q^-30 - 5q^-29 + 26q^-28 - 17q^-27 - 22q^-26 + 45q^-25 - 15q^-24 - 44q^-23 + 59q^-22 - 5q^-21 - 61q^-20 + 61q^-19 + 7q^-18 - 64q^-17 + 49q^-16 + 12q^-15 - 49q^-14 + 30q^-13 + 10q^-12 - 26q^-11 + 14q^-10 + 4q^-9 - 9q^-8 + 5q^-7 + q^-6 - 2q^-5 + q^-4
3	q^-66 - 3q^-65 + 5q^-63 + 4q^-62 - 11q^-61 - 11q^-60 + 19q^-59 + 22q^-58 - 23q^-57 - 43q^-56 + 24q^-55 + 64q^-54 - 10q^-53 - 94q^-52 - 5q^-51 + 106q^-50 + 45q^-49 - 122q^-48 - 73q^-47 + 111q^-46 + 116q^-45 - 101q^-44 - 146q^-43 + 76q^-42 + 173q^-41 - 47q^-40 - 196q^-39 + 20q^-38 + 207q^-37 + 12q^-36 - 215q^-35 - 34q^-34 + 203q^-33 + 62q^-32 - 191q^-31 - 71q^-30 + 155q^-29 + 82q^-28 - 125q^-27 - 71q^-26 + 81q^-25 + 64q^-24 - 56q^-23 - 38q^-22 + 25q^-21 + 28q^-20 - 21q^-19 - 4q^-18 + 6q^-17 + 4q^-16 - 10q^-15 + 4q^-14 + 5q^-13 - 6q^-11 + 3q^-10 + q^-9 + q^-8 - 2q^-7 + q^-6
4	q^-108 - 3q^-107 + 5q^-105 + 4q^-103 - 18q^-102 - 4q^-101 + 20q^-100 + 8q^-99 + 25q^-98 - 57q^-97 - 37q^-96 + 36q^-95 + 37q^-94 + 100q^-93 - 98q^-92 - 119q^-91 - 2q^-90 + 48q^-89 + 258q^-88 - 61q^-87 - 194q^-86 - 131q^-85 - 61q^-84 + 411q^-83 + 89q^-82 - 118q^-81 - 243q^-80 - 325q^-79 + 393q^-78 + 224q^-77 + 148q^-76 - 161q^-75 - 603q^-74 + 162q^-73 + 175q^-72 + 460q^-71 + 135q^-70 - 731q^-69 - 145q^-68 - 61q^-67 + 667q^-66 + 505q^-65 - 695q^-64 - 401q^-63 - 353q^-62 + 759q^-61 + 813q^-60 - 579q^-59 - 572q^-58 - 604q^-57 + 761q^-56 + 1025q^-55 - 407q^-54 - 656q^-53 - 805q^-52 + 649q^-51 + 1114q^-50 - 167q^-49 - 586q^-48 - 918q^-47 + 383q^-46 + 1005q^-45 + 93q^-44 - 330q^-43 - 854q^-42 + 58q^-41 + 683q^-40 + 225q^-39 - 13q^-38 - 590q^-37 - 140q^-36 + 302q^-35 + 169q^-34 + 164q^-33 - 272q^-32 - 140q^-31 + 58q^-30 + 41q^-29 + 156q^-28 - 74q^-27 - 58q^-26 - 10q^-25 - 24q^-24 + 76q^-23 - 11q^-22 - 5q^-21 - 5q^-20 - 26q^-19 + 24q^-18 - 2q^-17 + 4q^-16 + q^-15 - 10q^-14 + 6q^-13 - q^-12 + q^-11 + q^-10 - 2q^-9 + q^-8

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 63]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 16, 6, 17], X[17, 20, 18, 1], > X[11, 18, 12, 19], X[19, 12, 20, 13], X[7, 14, 8, 15], X[13, 8, 14, 9], > X[15, 6, 16, 7], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[10, 63]]
Out[3]=	GaussCode[-1, 10, -2, 1, -3, 9, -7, 8, -10, 2, -5, 6, -8, 7, -9, 3, -4, 5, -6, > 4]
In[4]:=	DTCode[Knot[10, 63]]
Out[4]=	DTCode[4, 10, 16, 14, 2, 18, 8, 6, 20, 12]
In[5]:=	br = BR[Knot[10, 63]]
Out[5]=	BR[5, {-1, -1, 2, -1, -3, -2, -2, -2, -3, -4, 3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 63]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 63]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 63]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 3, NotAvailable, 2}
In[10]:=	alex = Alexander[Knot[10, 63]][t]
Out[10]=	5 14 2 19 + -- - -- - 14 t + 5 t 2 t t
In[11]:=	Conway[Knot[10, 63]][z]
Out[11]=	2 4 1 + 6 z + 5 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 38], Knot[10, 63]}
In[13]:=	{KnotDet[Knot[10, 63]], KnotSignature[Knot[10, 63]]}
Out[13]=	{57, -4}
In[14]:=	Jones[Knot[10, 63]][q]
Out[14]=	-12 3 4 7 9 9 9 7 5 2 -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, 63]}
In[16]:=	A2Invariant[Knot[10, 63]][q]
Out[16]=	-38 -36 2 -32 2 3 2 -24 2 -20 -18 2 q + q - --- - q - --- - --- + --- + q + --- + q - q + --- - 34 30 28 26 22 16 q q q q q q -14 -12 2 -8 -6 > q + q + --- - q + q 10 q
In[17]:=	HOMFLYPT[Knot[10, 63]][a, z]
Out[17]=	4 8 10 12 4 2 6 2 8 2 10 2 4 4 a + 3 a - 4 a + a + 2 a z + 3 a z + 4 a z - 3 a z + a z + 6 4 8 4 > 2 a z + 2 a z
In[18]:=	Kauffman[Knot[10, 63]][a, z]
Out[18]=	4 8 10 12 9 11 13 4 2 6 2 a + 3 a + 4 a + a - 8 a z - 10 a z - 2 a z - 2 a z + a z - 8 2 10 2 12 2 14 2 5 3 9 3 11 3 > 10 a z - 16 a z - 2 a z + a z - 2 a z + 20 a z + 28 a z + 13 3 4 4 6 4 8 4 10 4 12 4 14 4 > 10 a z + a z - 3 a z + 11 a z + 24 a z + 6 a z - 3 a z + 5 5 7 5 9 5 11 5 13 5 6 6 8 6 > 2 a z - 2 a z - 16 a z - 23 a z - 11 a z + 3 a z - 6 a z - 10 6 12 6 14 6 7 7 9 7 11 7 13 7 > 19 a z - 9 a z + a z + 3 a z + 4 a z + 4 a z + 3 a z + 8 8 10 8 12 8 9 9 11 9 > 3 a z + 6 a z + 3 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 63]], Vassiliev[3][Knot[10, 63]]}
Out[19]=	{6, -14}
In[20]:=	Kh[Knot[10, 63]][q, t]
Out[20]=	-5 -3 1 2 1 2 2 5 2 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 4 5 5 4 4 5 3 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 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 2 3 2 > ----- + ----- + ---- 9 2 7 2 5 q t q t q t
In[21]:=	ColouredJones[Knot[10, 63], 2][q]
Out[21]=	-34 3 9 11 5 26 17 22 45 15 44 59 5 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 61 61 7 64 49 12 49 30 10 26 14 4 9 > --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + -- - -- + 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 5 -6 2 -4 > -- + q - -- + q 7 5 q q

The Alternating Knot 1063

The Alternating Knot 10₆₃